A common zero at the end point of the support of measure for the quasi-natured spectrally transformed polynomials (2024)

VIKASH KUMARDepartment of Mathematics
Indian Institute of Technology, Roorkee-247667, Uttarakhand, India
vikaskr0006@gmail.com, vkumar4@mt.iitr.ac.in
andA. SwaminathanDepartment of Mathematics
Indian Institute of Technology, Roorkee-247667, Uttarakhand, India
mathswami@gmail.com, a.swaminathan@ma.iitr.ac.in

Abstract.

In this work, the explicit expressions of coefficients involved in quasi-type kernel polynomials of order one and quasi-Geronimus polynomials of order one are determined for Jacobi polynomials. These coefficients are responsible for establishing the orthogonality of quasi-spectral polynomials for Jacobi polynomials. Additionally, the orthogonality of quasi-type kernel Laguerre polynomials of order one is derived. In the process of achieving orthogonality, one zero in both cases is located on the boundary of the support of the measure. This allows us to derive the chain sequence and minimal parameter sequence at the point lying at the end point of the support of the measure. Also, this leads to the question of characterizing such spectrally transformed polynomials.Furthermore, the interlacing properties among the zeros of quasi-spectral orthogonal Jacobi polynomials and Jacobi polynomials are illustrated.

Key words and phrases:

Quasi-orthogonal Polynomials; Kernel Polynomials; Jacobi polynomials; Laguerre polynomials; Linear Spectral Transformations

2020 Mathematics Subject Classification:

Primary 42C05, 33C45, 26C10

1. Introduction

The polynomial of degree n𝑛nitalic_n is termed quasi-orthogonal of order k𝑘kitalic_k with respect to a linear functional \mathcal{L}caligraphic_L over the interval (a,b)𝑎𝑏(a,b)( italic_a , italic_b ) if it satisfies the condition:

(xrp(x))=0forr=0,1,,nk1.formulae-sequencesuperscript𝑥𝑟𝑝𝑥0for𝑟01𝑛𝑘1\displaystyle\mathcal{L}(x^{r}p(x))=0~{}~{}\text{for}~{}~{}r=0,1,...,n-k-1.caligraphic_L ( italic_x start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT italic_p ( italic_x ) ) = 0 for italic_r = 0 , 1 , … , italic_n - italic_k - 1 .(1.1)

A necessary and sufficient condition for a monic polynomial p(x)𝑝𝑥p(x)italic_p ( italic_x ) of degree n𝑛nitalic_n to be a quasi-orthogonal polynomial of order k𝑘kitalic_k is that the polynomial p(x)𝑝𝑥p(x)italic_p ( italic_x ) can be expressed as a linear combination of orthogonal polynomials {j(x)}j=nknsuperscriptsubscriptsubscript𝑗𝑥𝑗𝑛𝑘𝑛\{\mathbb{P}_{j}(x)\}_{j=n-k}^{n}{ blackboard_P start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x ) } start_POSTSUBSCRIPT italic_j = italic_n - italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT with constant coefficients, i.e.,

p(x)=n(x)+bn(n)n1(x)+bn1(n)n2(x)++bnk+1(n)nk(x),𝑝𝑥subscript𝑛𝑥subscriptsuperscript𝑏𝑛𝑛subscript𝑛1𝑥subscriptsuperscript𝑏𝑛𝑛1subscript𝑛2𝑥subscriptsuperscript𝑏𝑛𝑛𝑘1subscript𝑛𝑘𝑥\displaystyle p(x)=\mathbb{P}_{n}(x)+b^{(n)}_{n}\mathbb{P}_{n-1}(x)+b^{(n)}_{n%-1}\mathbb{P}_{n-2}(x)+...+b^{(n)}_{n-k+1}\mathbb{P}_{n-k}(x),italic_p ( italic_x ) = blackboard_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ) + italic_b start_POSTSUPERSCRIPT ( italic_n ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT blackboard_P start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT ( italic_x ) + italic_b start_POSTSUPERSCRIPT ( italic_n ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT blackboard_P start_POSTSUBSCRIPT italic_n - 2 end_POSTSUBSCRIPT ( italic_x ) + … + italic_b start_POSTSUPERSCRIPT ( italic_n ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n - italic_k + 1 end_POSTSUBSCRIPT blackboard_P start_POSTSUBSCRIPT italic_n - italic_k end_POSTSUBSCRIPT ( italic_x ) ,(1.2)

where coefficients bj(n)subscriptsuperscript𝑏𝑛𝑗b^{(n)}_{j}italic_b start_POSTSUPERSCRIPT ( italic_n ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT’s cannot all be zero simultaneously. The orthogonality of quasi-orthogonal polynomials of order k𝑘kitalic_k is discussed in [4]. Riesz [16] introduced quasi-orthogonal polynomials of order one in the proof of the Hamburger moment problem. Subsequently, Shohat extended this concept to finite order in [18] and applied it in the study of mechanical quadrature formulas, is known as the Riesz-Shohat theorem [15, 17]. Particularly, the case k=1𝑘1k=1italic_k = 1 is of special interest. The necessary and sufficient conditions on the coefficients of quasi-orthogonal polynomials of order are imposed in [11] to achieve orthogonality. Additionally, in [11], the zeros of quasi-orthogonal polynomials of order one are utilized to investigate the electrostatic equilibrium problem. It is shown in [18] that at most k𝑘kitalic_k zeros of the polynomial p(x)𝑝𝑥p(x)italic_p ( italic_x ) defined in (1.2) lie outside the support of the measure. In [2], interlacing properties among the zeros of polynomial p(x)𝑝𝑥p(x)italic_p ( italic_x ) of degree mn𝑚𝑛m\leq nitalic_m ≤ italic_n and polynomial n(x)subscript𝑛𝑥\mathbb{P}_{n}(x)blackboard_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ) of degree n𝑛nitalic_n are discussed. The properties of quasi-orthogonality of discrete orthogonal polynomials, the Meixner polynomial, including the zeros and the interlacing property of zeros, are discussed in [9].

It is well known that Jacobi polynomials are orthogonal with respect to the Jacobi weight w(x)=(1x)α(1+x)β𝑤𝑥superscript1𝑥𝛼superscript1𝑥𝛽w(x)=(1-x)^{\alpha}(1+x)^{\beta}italic_w ( italic_x ) = ( 1 - italic_x ) start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT ( 1 + italic_x ) start_POSTSUPERSCRIPT italic_β end_POSTSUPERSCRIPT for α>1𝛼1\alpha>-1italic_α > - 1 and β>1𝛽1\beta>-1italic_β > - 1 in the interval (1,1)11(-1,1)( - 1 , 1 ). However, when other values of α𝛼\alphaitalic_α and β𝛽\betaitalic_β are allowed, Jacobi polynomials no longer maintain orthogonality. Nonetheless, by discarding some initial terms of the Jacobi polynomial sequence, orthogonality can still be achieved. For example, to achieve the standard orthogonality of Jacobi polynomials for α=1𝛼1\alpha=-1italic_α = - 1 and β>1𝛽1\beta>-1italic_β > - 1, it is necessary to eliminate the first term of the Jacobi sequence. Thus, the sequence {𝒫n(1,β)(x)}n=1superscriptsubscriptsuperscriptsubscript𝒫𝑛1𝛽𝑥𝑛1\{\mathcal{P}_{n}^{(-1,\beta)}(x)\}_{n=1}^{\infty}{ caligraphic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( - 1 , italic_β ) end_POSTSUPERSCRIPT ( italic_x ) } start_POSTSUBSCRIPT italic_n = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT becomes orthogonal under the Jacobi measure [5]. More generally, for the parameters α=m𝛼𝑚\alpha=-mitalic_α = - italic_m where m𝑚m\in\mathbb{N}italic_m ∈ blackboard_N and β>1𝛽1\beta>-1italic_β > - 1, orthogonality of the Jacobi polynomials {𝒫n(m,β)(x)}n=msuperscriptsubscriptsuperscriptsubscript𝒫𝑛𝑚𝛽𝑥𝑛𝑚\{\mathcal{P}_{n}^{(-m,\beta)}(x)\}_{n=m}^{\infty}{ caligraphic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( - italic_m , italic_β ) end_POSTSUPERSCRIPT ( italic_x ) } start_POSTSUBSCRIPT italic_n = italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT can be achieved by eliminating the first m𝑚mitalic_m terms of the Jacobi sequence. When seeking orthogonality of Jacobi polynomials for negative integral values of α𝛼\alphaitalic_α, utilizing (2.24), we encounter a multiplicity m𝑚mitalic_m for the zero x=1𝑥1x=1italic_x = 1.

A natural question that arises during this process is the possibility of the perturbed Jacobi polynomials to obtain a polynomial with a simple zero at x=1𝑥1x=1italic_x = 1, where the existence of the common zero x=1𝑥1x=1italic_x = 1 is independent of the parameters α𝛼\alphaitalic_α and β𝛽\betaitalic_β. This manuscript addresses this question by obtaining the orthogonality of quasi-Christoffel and quasi-Geronimus Jacobi polynomial of order one and providing an illustration by computing the zero. The same is obtained for the case of Laguerre polynomials. The emergence of a single, simple zero at the finite point of the support of the measure facilitates future investigations into these types of polynomials.

This manuscript serves a dual purpose. Firstly, it focuses on deriving the explicit expressions of coefficients, which are essential for achieving the orthogonality of quasi-Christoffel and quasi-Geronimus Jacobi polynomials. Secondly, it addresses the problem of identifying orthogonal polynomial systems by perturbing Jacobi families in such a way that only one zero lies at one of the end point of the support of the measure for the Jacobi polynomials.

The manuscript is organized as follows: In Section 2, we discuss the orthogonality of quasi-type kernel Laguerre polynomials of order one and quasi-type kernel Jacobi polynomials of order one. We present a graphical interpretation illustrating the interlacing of zeros among the Laguerre polynomials, Christoffel Laguerre polynomials, and quasi-type kernel Laguerre polynomials of order one. During this process, we observe that one zero of the quasi-Christoffel Jacobi polynomial of order one and the quasi-Christoffel Laguerre polynomial of order one lies at a finite end point of the support of the measure, which is true for the Laguerre case as well. The chain sequence and minimal parameter sequence are also computed at the point lying on the boundary of the support of the measure for the Jacobi as well as Laguerre polynomials. In Section 3, we explore the orthogonality of quasi-Geronimus Jacobi polynomials of order one and provide a graphical representation of the interlacing property and the positioning of the zeros of the corresponding polynomial.

2. Quasi-type kernel polynomial of order one

Let Csuperscript𝐶\mathcal{L}^{C}caligraphic_L start_POSTSUPERSCRIPT italic_C end_POSTSUPERSCRIPT denote the canonical Christoffel transformation of a linear functional \mathcal{L}caligraphic_L at point a𝑎aitalic_a. The new linear functional at a𝑎aitalic_a is defined as:

C(p(x))=((xa)p(x)).superscript𝐶𝑝𝑥𝑥𝑎𝑝𝑥\displaystyle\mathcal{L}^{C}(p(x))=\mathcal{L}((x-a)p(x)).caligraphic_L start_POSTSUPERSCRIPT italic_C end_POSTSUPERSCRIPT ( italic_p ( italic_x ) ) = caligraphic_L ( ( italic_x - italic_a ) italic_p ( italic_x ) ) .

If a𝑎aitalic_a does not belong to the support of the measure for the polynomial n(x)subscript𝑛𝑥\mathbb{P}_{n}(x)blackboard_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ), then there exists a sequence of orthogonal polynomials corresponding to Csuperscript𝐶\mathcal{L}^{C}caligraphic_L start_POSTSUPERSCRIPT italic_C end_POSTSUPERSCRIPT (see [1]). The polynomial corresponding to Csuperscript𝐶\mathcal{L}^{C}caligraphic_L start_POSTSUPERSCRIPT italic_C end_POSTSUPERSCRIPT is known as the kernel polynomial or Christoffel polynomial.

Theorem 1.

[13]Let 𝒞n(x;a)subscript𝒞𝑛𝑥𝑎\mathcal{C}_{n}(x;a)caligraphic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; italic_a ) be the monic polynomial with respect to canonical Christoffel transformation, which exists for some point a𝑎aitalic_a. The monic polynomial 𝒞n+1Q(x;a)superscriptsubscript𝒞𝑛1𝑄𝑥𝑎\mathcal{C}_{n+1}^{Q}(x;a)caligraphic_C start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT ( italic_x ; italic_a ) of degree n+1𝑛1n+1italic_n + 1 is a non trivial quasi-type kernel polynomial of order one if and only if there exists a sequence of constants γn0not-equivalent-tosubscript𝛾𝑛0\gamma_{n}\not\equiv 0italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ≢ 0, such that

𝒞n+1Q(x;a)=𝒞n+1(x;a)+γn+1𝒞n(x;a).superscriptsubscript𝒞𝑛1𝑄𝑥𝑎subscript𝒞𝑛1𝑥𝑎subscript𝛾𝑛1subscript𝒞𝑛𝑥𝑎\displaystyle\mathcal{C}_{n+1}^{Q}(x;a)=\mathcal{C}_{n+1}(x;a)+\gamma_{n+1}%\mathcal{C}_{n}(x;a).caligraphic_C start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT ( italic_x ; italic_a ) = caligraphic_C start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT ( italic_x ; italic_a ) + italic_γ start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT caligraphic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; italic_a ) .

Proposition 1 discusses the orthogonality of quasi-type kernel polynomials of order one under certain assumptions.

Proposition 1.

[13]Let 𝒞nQ(x;a)superscriptsubscript𝒞𝑛𝑄𝑥𝑎\mathcal{C}_{n}^{Q}(x;a)caligraphic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT ( italic_x ; italic_a ) be a monic quasi-type kernel polynomial of order one with parameter γnsubscript𝛾𝑛\gamma_{n}italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT such that

γn(cn+1ccnc+γnγn+1)+γnγn1λncλn+1c=0,n2.formulae-sequencesubscript𝛾𝑛superscriptsubscript𝑐𝑛1𝑐superscriptsubscript𝑐𝑛𝑐subscript𝛾𝑛subscript𝛾𝑛1subscript𝛾𝑛subscript𝛾𝑛1superscriptsubscript𝜆𝑛𝑐superscriptsubscript𝜆𝑛1𝑐0𝑛2\displaystyle\gamma_{n}(c_{n+1}^{c}-c_{n}^{c}+\gamma_{n}-\gamma_{n+1})+\frac{%\gamma_{n}}{\gamma_{n-1}}\lambda_{n}^{c}-\lambda_{n+1}^{c}=0,~{}n\geq 2.italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_c start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT - italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT + italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT - italic_γ start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT ) + divide start_ARG italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_γ start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT end_ARG italic_λ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT - italic_λ start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT = 0 , italic_n ≥ 2 .(2.1)

Then the polynomials 𝒞nQ(x;a)superscriptsubscript𝒞𝑛𝑄𝑥𝑎\mathcal{C}_{n}^{Q}(x;a)caligraphic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT ( italic_x ; italic_a ) satisfy the three-term recurrence relation

𝒞n+1Q(x;a)(xcn+1qc)𝒞nQ(x;a)+λn+1qc𝒞n1Q(x;a)=0,n0,formulae-sequencesuperscriptsubscript𝒞𝑛1𝑄𝑥𝑎𝑥superscriptsubscript𝑐𝑛1𝑞𝑐superscriptsubscript𝒞𝑛𝑄𝑥𝑎superscriptsubscript𝜆𝑛1𝑞𝑐superscriptsubscript𝒞𝑛1𝑄𝑥𝑎0𝑛0\displaystyle\mathcal{C}_{n+1}^{Q}(x;a)-(x-c_{n+1}^{qc})\mathcal{C}_{n}^{Q}(x;%a)+\lambda_{n+1}^{qc}\mathcal{C}_{n-1}^{Q}(x;a)=0,~{}n\geq 0,caligraphic_C start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT ( italic_x ; italic_a ) - ( italic_x - italic_c start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_q italic_c end_POSTSUPERSCRIPT ) caligraphic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT ( italic_x ; italic_a ) + italic_λ start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_q italic_c end_POSTSUPERSCRIPT caligraphic_C start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT ( italic_x ; italic_a ) = 0 , italic_n ≥ 0 ,

where the recurrence parameters are given by

λn+1qc=γnγn1λnc,cn+1qc=cn+1c+γnγn+1.formulae-sequencesuperscriptsubscript𝜆𝑛1𝑞𝑐subscript𝛾𝑛subscript𝛾𝑛1superscriptsubscript𝜆𝑛𝑐superscriptsubscript𝑐𝑛1𝑞𝑐superscriptsubscript𝑐𝑛1𝑐subscript𝛾𝑛subscript𝛾𝑛1\displaystyle\lambda_{n+1}^{qc}=\frac{\gamma_{n}}{\gamma_{n-1}}\lambda_{n}^{c}%,~{}~{}~{}~{}~{}~{}~{}c_{n+1}^{qc}=c_{n+1}^{c}+\gamma_{n}-\gamma_{n+1}.italic_λ start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_q italic_c end_POSTSUPERSCRIPT = divide start_ARG italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_γ start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT end_ARG italic_λ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT , italic_c start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_q italic_c end_POSTSUPERSCRIPT = italic_c start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT + italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT - italic_γ start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT .

If λn+1qc0superscriptsubscript𝜆𝑛1𝑞𝑐0\lambda_{n+1}^{qc}\neq 0italic_λ start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_q italic_c end_POSTSUPERSCRIPT ≠ 0, then {𝒞nQ(x;a)}n=2superscriptsubscriptsubscriptsuperscript𝒞𝑄𝑛𝑥𝑎𝑛2\{\mathcal{C}^{Q}_{n}(x;a)\}_{n=2}^{\infty}{ caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; italic_a ) } start_POSTSUBSCRIPT italic_n = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT forms a monic orthogonal polynomial sequence.

2.1. Laguerre polynomials

The monic Laguerre polynomials are characterized by the following three-term recurrence relation [7, page 154]:

n+1(α)(x)=(xcn+1)n(α)(x)λn+1n1(α)(x),subscriptsuperscript𝛼𝑛1𝑥𝑥subscript𝑐𝑛1subscriptsuperscript𝛼𝑛𝑥subscript𝜆𝑛1subscriptsuperscript𝛼𝑛1𝑥\displaystyle\mathcal{L}^{(\alpha)}_{n+1}(x)=(x-c_{n+1})\mathcal{L}^{(\alpha)}%_{n}(x)-\lambda_{n+1}\mathcal{L}^{(\alpha)}_{n-1}(x),caligraphic_L start_POSTSUPERSCRIPT ( italic_α ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT ( italic_x ) = ( italic_x - italic_c start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT ) caligraphic_L start_POSTSUPERSCRIPT ( italic_α ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ) - italic_λ start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT caligraphic_L start_POSTSUPERSCRIPT ( italic_α ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT ( italic_x ) ,(2.2)

with initial data 1(α)(x)=0,0(α)(x)=1formulae-sequencesubscriptsuperscript𝛼1𝑥0subscriptsuperscript𝛼0𝑥1\mathcal{L}^{(\alpha)}_{-1}(x)=0,\mathcal{L}^{(\alpha)}_{0}(x)=1caligraphic_L start_POSTSUPERSCRIPT ( italic_α ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT ( italic_x ) = 0 , caligraphic_L start_POSTSUPERSCRIPT ( italic_α ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x ) = 1. The recurrence parameters, denoted by cn+1subscript𝑐𝑛1c_{n+1}italic_c start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT and λn+1subscript𝜆𝑛1\lambda_{n+1}italic_λ start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT, are expressed as cn+1=2n+α+1subscript𝑐𝑛12𝑛𝛼1c_{n+1}=2n+\alpha+1italic_c start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT = 2 italic_n + italic_α + 1 and λn+1=n(n+α)subscript𝜆𝑛1𝑛𝑛𝛼\lambda_{n+1}=n(n+\alpha)italic_λ start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT = italic_n ( italic_n + italic_α ). These Laguerre polynomials exhibit orthogonality within the interval (0,)0(0,\infty)( 0 , ∞ ) concerning the weight function w(x;α)=xαex𝑤𝑥𝛼superscript𝑥𝛼superscript𝑒𝑥w(x;\alpha)=x^{\alpha}e^{-x}italic_w ( italic_x ; italic_α ) = italic_x start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - italic_x end_POSTSUPERSCRIPT, where α>1𝛼1\alpha>-1italic_α > - 1. Upon applying the Christoffel transformation to the Laguerre weight with a=0𝑎0a=0italic_a = 0, the resulting transformed weight is w~(x;α)=xα+1ex~𝑤𝑥𝛼superscript𝑥𝛼1superscript𝑒𝑥\tilde{w}(x;\alpha)=x^{\alpha+1}e^{-x}over~ start_ARG italic_w end_ARG ( italic_x ; italic_α ) = italic_x start_POSTSUPERSCRIPT italic_α + 1 end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - italic_x end_POSTSUPERSCRIPT, α>1𝛼1\alpha>-1italic_α > - 1. Consequently, the Christoffel Laguerre polynomials at a=0𝑎0a=0italic_a = 0 assume the form of the Laguerre polynomial with parameter α+1𝛼1\alpha+1italic_α + 1. The monic Christoffel Laguerre polynomials at a=0𝑎0a=0italic_a = 0, denoted by 𝒞n(x;0):=n(α+1)(x)assignsubscript𝒞𝑛𝑥0subscriptsuperscript𝛼1𝑛𝑥\mathcal{C}_{n}(x;0):=\mathcal{L}^{(\alpha+1)}_{n}(x)caligraphic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; 0 ) := caligraphic_L start_POSTSUPERSCRIPT ( italic_α + 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ), are generated by the three-term recurrence relation

n+1(α+1)(x)=(xcn+1c)n(α+1)(x)λn+1cn1(α+1)(x),subscriptsuperscript𝛼1𝑛1𝑥𝑥subscriptsuperscript𝑐𝑐𝑛1subscriptsuperscript𝛼1𝑛𝑥subscriptsuperscript𝜆𝑐𝑛1subscriptsuperscript𝛼1𝑛1𝑥\displaystyle\mathcal{L}^{(\alpha+1)}_{n+1}(x)=(x-c^{c}_{n+1})\mathcal{L}^{(%\alpha+1)}_{n}(x)-\lambda^{c}_{n+1}\mathcal{L}^{(\alpha+1)}_{n-1}(x),caligraphic_L start_POSTSUPERSCRIPT ( italic_α + 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT ( italic_x ) = ( italic_x - italic_c start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT ) caligraphic_L start_POSTSUPERSCRIPT ( italic_α + 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ) - italic_λ start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT caligraphic_L start_POSTSUPERSCRIPT ( italic_α + 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT ( italic_x ) ,(2.3)

with initial conditions 1(α+1)(x)=0,0(α+1)(x)=1formulae-sequencesubscriptsuperscript𝛼11𝑥0subscriptsuperscript𝛼10𝑥1\mathcal{L}^{(\alpha+1)}_{-1}(x)=0,\mathcal{L}^{(\alpha+1)}_{0}(x)=1caligraphic_L start_POSTSUPERSCRIPT ( italic_α + 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT ( italic_x ) = 0 , caligraphic_L start_POSTSUPERSCRIPT ( italic_α + 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x ) = 1. The recurrence coefficients are cn+1c=2n+α+2subscriptsuperscript𝑐𝑐𝑛12𝑛𝛼2c^{c}_{n+1}=2n+\alpha+2italic_c start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT = 2 italic_n + italic_α + 2 and λn+1c=n(n+α+1)subscriptsuperscript𝜆𝑐𝑛1𝑛𝑛𝛼1\lambda^{c}_{n+1}=n(n+\alpha+1)italic_λ start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT = italic_n ( italic_n + italic_α + 1 ).
The monic quasi-type kernel Laguerre polynomial of order one is given by

𝒞nQ(x;0)=n(α+1)(x)+γnn1(α+1)(x).subscriptsuperscript𝒞𝑄𝑛𝑥0subscriptsuperscript𝛼1𝑛𝑥subscript𝛾𝑛subscriptsuperscript𝛼1𝑛1𝑥\displaystyle\mathcal{C}^{Q}_{n}(x;0)=\mathcal{L}^{(\alpha+1)}_{n}(x)+\gamma_{%n}\mathcal{L}^{(\alpha+1)}_{n-1}(x).caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; 0 ) = caligraphic_L start_POSTSUPERSCRIPT ( italic_α + 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ) + italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT caligraphic_L start_POSTSUPERSCRIPT ( italic_α + 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT ( italic_x ) .(2.4)

When the quasi-Christoffel polynomials are considered, the orthogonality is disrupted, resulting in at most one zero lying outside the support of the measure for the Laguerre polynomials. Table 1 presents the behavior of zeros of 𝒞nQ(x;0)subscriptsuperscript𝒞𝑄𝑛𝑥0\mathcal{C}^{Q}_{n}(x;0)caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; 0 ).

Zeros of 𝒞nQ(x;0)subscriptsuperscript𝒞𝑄𝑛𝑥0\mathcal{C}^{Q}_{n}(x;0)caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; 0 )
n=5𝑛5n=5italic_n = 5, α=0𝛼0\alpha=0italic_α = 0, γn=7subscript𝛾𝑛7\gamma_{n}=7italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = 7n=6𝑛6n=6italic_n = 6, α=1.5𝛼1.5\alpha=1.5italic_α = 1.5, γn=9subscript𝛾𝑛9\gamma_{n}=9italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = 9
-0.407194-0.219116
1.086911.67954
3.26373.90364
6.751217.07314
12.305411.5115
-18.0513

For α=0𝛼0\alpha=0italic_α = 0 and γn=7subscript𝛾𝑛7\gamma_{n}=7italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = 7, we observe that one zero (x0=0.404714)subscript𝑥00.404714(x_{0}=-0.404714)( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = - 0.404714 ) of 𝒞nQ(x;0)subscriptsuperscript𝒞𝑄𝑛𝑥0\mathcal{C}^{Q}_{n}(x;0)caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; 0 ) lies outside the support of the measure for the Laguerre polynomials, while all other zeros are positive. Similarly, for α=1.5𝛼1.5\alpha=1.5italic_α = 1.5 and γn=9subscript𝛾𝑛9\gamma_{n}=9italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = 9, Table 1 shows that at most one negative zero of the polynomial 𝒞nQ(x;0)subscriptsuperscript𝒞𝑄𝑛𝑥0\mathcal{C}^{Q}_{n}(x;0)caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; 0 ) exists. To ensure the orthogonality of a quasi-type kernel Laguerre polynomial of order one, the condition (2.1) must be satisfied, which gives

(2+γjγj+1)+1γj1(j1)(j+α)1γjj(j+α+1)=0.2subscript𝛾𝑗subscript𝛾𝑗11subscript𝛾𝑗1𝑗1𝑗𝛼1subscript𝛾𝑗𝑗𝑗𝛼10\displaystyle(2+\gamma_{j}-\gamma_{j+1})+\frac{1}{\gamma_{j-1}}(j-1)(j+\alpha)%-\frac{1}{\gamma_{j}}j(j+\alpha+1)=0.( 2 + italic_γ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - italic_γ start_POSTSUBSCRIPT italic_j + 1 end_POSTSUBSCRIPT ) + divide start_ARG 1 end_ARG start_ARG italic_γ start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT end_ARG ( italic_j - 1 ) ( italic_j + italic_α ) - divide start_ARG 1 end_ARG start_ARG italic_γ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG italic_j ( italic_j + italic_α + 1 ) = 0 .(2.5)

Taking sum over j=2𝑗2j=2italic_j = 2 to n+1𝑛1n+1italic_n + 1, the equation is expressed as follows:

(2n+γ2γn+2)+1γ1(α+2)1γn+1(n+1)(n+α+2)=0.2𝑛subscript𝛾2subscript𝛾𝑛21subscript𝛾1𝛼21subscript𝛾𝑛1𝑛1𝑛𝛼20\displaystyle(2n+\gamma_{2}-\gamma_{n+2})+\frac{1}{\gamma_{1}}(\alpha+2)-\frac%{1}{\gamma_{n+1}}{(n+1)}(n+\alpha+2)=0.( 2 italic_n + italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - italic_γ start_POSTSUBSCRIPT italic_n + 2 end_POSTSUBSCRIPT ) + divide start_ARG 1 end_ARG start_ARG italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ( italic_α + 2 ) - divide start_ARG 1 end_ARG start_ARG italic_γ start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT end_ARG ( italic_n + 1 ) ( italic_n + italic_α + 2 ) = 0 .(2.6)

The two possible solutions to (2.5) are as follows:

Solution 1.By choosing γ1=α+2subscript𝛾1𝛼2\gamma_{1}=\alpha+2italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_α + 2 and γ2=α+3subscript𝛾2𝛼3\gamma_{2}=\alpha+3italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_α + 3, we recursively determine γn=n+α+1subscript𝛾𝑛𝑛𝛼1\gamma_{n}=n+\alpha+1italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = italic_n + italic_α + 1. As a result, the quasi-type kernel Laguerre polynomial of order one is given by

𝒞nQ(x;0)=n(α+1)(x)+(n+α+1)n1(α+1)(x),subscriptsuperscript𝒞𝑄𝑛𝑥0subscriptsuperscript𝛼1𝑛𝑥𝑛𝛼1subscriptsuperscript𝛼1𝑛1𝑥\displaystyle\mathcal{C}^{Q}_{n}(x;0)=\mathcal{L}^{(\alpha+1)}_{n}(x)+(n+%\alpha+1)\mathcal{L}^{(\alpha+1)}_{n-1}(x),caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; 0 ) = caligraphic_L start_POSTSUPERSCRIPT ( italic_α + 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ) + ( italic_n + italic_α + 1 ) caligraphic_L start_POSTSUPERSCRIPT ( italic_α + 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT ( italic_x ) ,(2.7)

which satisfies the three-term recurrence relation

𝒞n+1Q(x;0)=(xcn+1qc)𝒞nQ(x;0)λn+1qc𝒞n1Q(x;0),subscriptsuperscript𝒞𝑄𝑛1𝑥0𝑥subscriptsuperscript𝑐𝑞𝑐𝑛1subscriptsuperscript𝒞𝑄𝑛𝑥0subscriptsuperscript𝜆𝑞𝑐𝑛1subscriptsuperscript𝒞𝑄𝑛1𝑥0\displaystyle\mathcal{C}^{Q}_{n+1}(x;0)=(x-c^{qc}_{n+1})\mathcal{C}^{Q}_{n}(x;%0)-\lambda^{qc}_{n+1}\mathcal{C}^{Q}_{n-1}(x;0),caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT ( italic_x ; 0 ) = ( italic_x - italic_c start_POSTSUPERSCRIPT italic_q italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT ) caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; 0 ) - italic_λ start_POSTSUPERSCRIPT italic_q italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT ( italic_x ; 0 ) ,

with recurrence coefficients given by

λn+1qc=γnγn1λnc=(n1)(n+α+1),cn+1qc=cn+1c+γnγn+1=2n+α+1.formulae-sequencesubscriptsuperscript𝜆𝑞𝑐𝑛1subscript𝛾𝑛subscript𝛾𝑛1superscriptsubscript𝜆𝑛𝑐𝑛1𝑛𝛼1subscriptsuperscript𝑐𝑞𝑐𝑛1superscriptsubscript𝑐𝑛1𝑐subscript𝛾𝑛subscript𝛾𝑛12𝑛𝛼1\displaystyle\lambda^{qc}_{n+1}=\frac{\gamma_{n}}{\gamma_{n-1}}\lambda_{n}^{c}%=(n-1)(n+\alpha+1),~{}c^{qc}_{n+1}=c_{n+1}^{c}+\gamma_{n}-\gamma_{n+1}=2n+%\alpha+1.italic_λ start_POSTSUPERSCRIPT italic_q italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT = divide start_ARG italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_γ start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT end_ARG italic_λ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT = ( italic_n - 1 ) ( italic_n + italic_α + 1 ) , italic_c start_POSTSUPERSCRIPT italic_q italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT = italic_c start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT + italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT - italic_γ start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT = 2 italic_n + italic_α + 1 .(2.8)

If we put α=1𝛼1\alpha=-1italic_α = - 1 into equation (2.7), then the polynomial 𝒞nQ(x;0)subscriptsuperscript𝒞𝑄𝑛𝑥0\mathcal{C}^{Q}_{n}(x;0)caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; 0 ) coincides with the Laguerre polynomial of degree n𝑛nitalic_n with parameter α=1𝛼1\alpha=-1italic_α = - 1, i.e.,

n(1)(x)=n(0)(x)+nn1(0)(x).subscriptsuperscript1𝑛𝑥subscriptsuperscript0𝑛𝑥𝑛subscriptsuperscript0𝑛1𝑥\displaystyle\mathcal{L}^{(-1)}_{n}(x)=\mathcal{L}^{(0)}_{n}(x)+n\mathcal{L}^{%(0)}_{n-1}(x).caligraphic_L start_POSTSUPERSCRIPT ( - 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ) = caligraphic_L start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ) + italic_n caligraphic_L start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT ( italic_x ) .(2.9)

By setting γn=n+α+1subscript𝛾𝑛𝑛𝛼1\gamma_{n}=n+\alpha+1italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = italic_n + italic_α + 1, we ensure the orthogonality of the monic quasi-type kernel Laguerre polynomials of order one. Consequently, the constant term in the polynomial 𝒞nQ(x;0)subscriptsuperscript𝒞𝑄𝑛𝑥0\mathcal{C}^{Q}_{n}(x;0)caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; 0 ) vanishes, implying that p(x)=x𝑝𝑥𝑥p(x)=xitalic_p ( italic_x ) = italic_x is a factor of the polynomial 𝒞nQ(x;0)subscriptsuperscript𝒞𝑄𝑛𝑥0\mathcal{C}^{Q}_{n}(x;0)caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; 0 ) for each degree n1𝑛1n\geq 1italic_n ≥ 1. Table 2 illustrates that one zero of 𝒞nQ(x;0)subscriptsuperscript𝒞𝑄𝑛𝑥0\mathcal{C}^{Q}_{n}(x;0)caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; 0 ) lies on the boundary of the support of the measure for the Laguerre polynomials, while the other zeros lie inside the interval (0,)0(0,\infty)( 0 , ∞ ).

Interlacing of 𝒞nQ(x;0)subscriptsuperscript𝒞𝑄𝑛𝑥0\mathcal{C}^{Q}_{n}(x;0)caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; 0 ) and 𝒞n+1Q(x;0)subscriptsuperscript𝒞𝑄𝑛1𝑥0\mathcal{C}^{Q}_{n+1}(x;0)caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT ( italic_x ; 0 )
α=0.5𝛼0.5\alpha=-0.5italic_α = - 0.5, n=5𝑛5n=5italic_n = 5, γn=n+α+1subscript𝛾𝑛𝑛𝛼1\gamma_{n}=n+\alpha+1italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = italic_n + italic_α + 1α=0.5𝛼0.5\alpha=-0.5italic_α = - 0.5, n=6𝑛6n=6italic_n = 6, γn=n+α+1subscript𝛾𝑛𝑛𝛼1\gamma_{n}=n+\alpha+1italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = italic_n + italic_α + 1
0.00.0
0.9785070.817632
2.990382.47233
6.31935.11601
11.71189.04415
-15.0499

Zeros of nα(x)subscriptsuperscript𝛼𝑛𝑥\mathcal{L}^{\alpha}_{n}(x)caligraphic_L start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x )Zeros of 𝒞n(x;0)subscript𝒞𝑛𝑥0\mathcal{C}_{n}(x;0)caligraphic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; 0 )Zeros of 𝒞nQ(x;0)superscriptsubscript𝒞𝑛𝑄𝑥0\mathcal{C}_{n}^{Q}(x;0)caligraphic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT ( italic_x ; 0 )
α=0.5𝛼0.5\alpha=-0.5italic_α = - 0.5, n=5𝑛5n=5italic_n = 5α=0.5𝛼0.5\alpha=-0.5italic_α = - 0.5, n=5𝑛5n=5italic_n = 5α=2𝛼2\alpha=2italic_α = 2, n=5𝑛5n=5italic_n = 5
0.1175810.4313990.0
1.074561.759752.31916
3.085944.104475.12867
6.414737.74679.20089
11.807213.457715.3513
---

A common zero at the end point of the support of measure for the quasi-natured spectrally transformed polynomials (1)
A common zero at the end point of the support of measure for the quasi-natured spectrally transformed polynomials (2)
A common zero at the end point of the support of measure for the quasi-natured spectrally transformed polynomials (3)

In Figure 1, for α=0.5𝛼0.5\alpha=-0.5italic_α = - 0.5, we observe the interlacing of zeros between 𝒞5Q(x;0)subscriptsuperscript𝒞𝑄5𝑥0\mathcal{C}^{Q}_{5}(x;0)caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT ( italic_x ; 0 ) and 𝒞6Q(x;0)subscriptsuperscript𝒞𝑄6𝑥0\mathcal{C}^{Q}_{6}(x;0)caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT ( italic_x ; 0 ). Additionally, Figure 2 illustrates the interlacing between the zeros of 𝒞5Q(x;0)subscriptsuperscript𝒞𝑄5𝑥0\mathcal{C}^{Q}_{5}(x;0)caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT ( italic_x ; 0 ) and 5(0.5)(x)subscriptsuperscript0.55𝑥\mathcal{L}^{(-0.5)}_{5}(x)caligraphic_L start_POSTSUPERSCRIPT ( - 0.5 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT ( italic_x ). Similarly, interlacing between the Christoffel polynomial and the quasi-type Christoffel polynomial of order one is also demonstrated in Figure 3.

It is worth noting that the sequence of Laguerre polynomials {n(α)(x)}n=0superscriptsubscriptsuperscriptsubscript𝑛𝛼𝑥𝑛0\{\mathcal{L}_{n}^{(\alpha)}(x)\}_{n=0}^{\infty}{ caligraphic_L start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_α ) end_POSTSUPERSCRIPT ( italic_x ) } start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT becomes classically orthogonal when α>1𝛼1\alpha>-1italic_α > - 1. However, substituting α=1𝛼1\alpha=-1italic_α = - 1 breaks this classical orthogonality condition, necessitating orthogonality in the non-classical sense, such as Sobolev orthogonality. The tail-end sequence of Laguerre polynomials {n(1)(x)}n=1superscriptsubscriptsuperscriptsubscript𝑛1𝑥𝑛1\{\mathcal{L}_{n}^{(-1)}(x)\}_{n=1}^{\infty}{ caligraphic_L start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( - 1 ) end_POSTSUPERSCRIPT ( italic_x ) } start_POSTSUBSCRIPT italic_n = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT becomes orthogonal with respect to the usual inner product. The more general framework of the orthogonality of sequence of Laguerre polynomials {n(α)(x)}n=α,α=m,mformulae-sequencesuperscriptsubscriptsuperscriptsubscript𝑛𝛼𝑥𝑛𝛼𝛼𝑚𝑚\{\mathcal{L}_{n}^{(\alpha)}(x)\}_{n=\alpha}^{\infty},\alpha=-m,m\in\mathbb{N}{ caligraphic_L start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_α ) end_POSTSUPERSCRIPT ( italic_x ) } start_POSTSUBSCRIPT italic_n = italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT , italic_α = - italic_m , italic_m ∈ blackboard_N is discussed in [8]. Further exploration of Laguerre polynomial orthogonality in the non-classical sense can be found in [8, 10].

To extend the applicability of Laguerre polynomials to negative integral values of α𝛼\alphaitalic_α, i.e., α=m,mformulae-sequence𝛼𝑚𝑚\alpha=-m,m\in\mathbb{N}italic_α = - italic_m , italic_m ∈ blackboard_N, we can utilize the following formula (see [19, equation (5.2.1)]):

n(m)(x)=(1)mxmΓ(nm+1)Γ(n+1)nm(m)(x),formn.formulae-sequencesuperscriptsubscript𝑛𝑚𝑥superscript1𝑚superscript𝑥𝑚Γ𝑛𝑚1Γ𝑛1superscriptsubscript𝑛𝑚𝑚𝑥for𝑚𝑛\displaystyle\mathcal{L}_{n}^{(-m)}(x)=(-1)^{m}x^{m}\frac{\Gamma(n-m+1)}{%\Gamma(n+1)}\mathcal{L}_{n-m}^{(m)}(x),~{}~{}~{}\text{for}~{}~{}m\leq n.caligraphic_L start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( - italic_m ) end_POSTSUPERSCRIPT ( italic_x ) = ( - 1 ) start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT divide start_ARG roman_Γ ( italic_n - italic_m + 1 ) end_ARG start_ARG roman_Γ ( italic_n + 1 ) end_ARG caligraphic_L start_POSTSUBSCRIPT italic_n - italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_m ) end_POSTSUPERSCRIPT ( italic_x ) , for italic_m ≤ italic_n .(2.10)

When m=1𝑚1m=1italic_m = 1 in (2.10), we observe that x=0𝑥0x=0italic_x = 0 becomes a common zero of the polynomial n(1)(x)superscriptsubscript𝑛1𝑥\mathcal{L}_{n}^{(-1)}(x)caligraphic_L start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( - 1 ) end_POSTSUPERSCRIPT ( italic_x ) for n1𝑛1n\geq 1italic_n ≥ 1. The polynomial described in equation (2.7), which is obtained by achieving orthogonality of the quasi-type kernel Laguerre polynomial of order one, has a common zero at x=0𝑥0x=0italic_x = 0. This zero remains consistent regardless of the values of the parameter α𝛼\alphaitalic_α, as also shown in Table 2 and Table 3.

Remark 2.1.

The derivative of a Laguerre polynomial yields another Laguerre polynomial [7, page 149]. Specifically,

dnα(x)dx=nn1α+1(x).𝑑superscriptsubscript𝑛𝛼𝑥𝑑𝑥𝑛superscriptsubscript𝑛1𝛼1𝑥\displaystyle\frac{d\mathcal{L}_{n}^{\alpha}(x)}{dx}=n\mathcal{L}_{n-1}^{%\alpha+1}(x).divide start_ARG italic_d caligraphic_L start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT ( italic_x ) end_ARG start_ARG italic_d italic_x end_ARG = italic_n caligraphic_L start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_α + 1 end_POSTSUPERSCRIPT ( italic_x ) .(2.11)

Thus, it follows that a linear combination of the Christoffel Laguerre polynomial and the derivative of a Laguerre polynomial constitutes an orthogonal polynomial. This can be achieved by substituting (2.11) into (2.7).

The connection between the Chain sequence and orthogonal polynomials is well known. Specifically, the chain sequence enables us to establish a relationship between the support of measure and the recurrence coefficients. A sequence {sn}n=1superscriptsubscriptsubscript𝑠𝑛𝑛1\{s_{n}\}_{n=1}^{\infty}{ italic_s start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_n = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT is defined as a chain sequence if there exists a parameter sequence {dn}n=0superscriptsubscriptsubscript𝑑𝑛𝑛0\{d_{n}\}_{n=0}^{\infty}{ italic_d start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT such that

sn=(1dn1)dn,subscript𝑠𝑛1subscript𝑑𝑛1subscript𝑑𝑛\displaystyle s_{n}=(1-d_{n-1})d_{n},italic_s start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = ( 1 - italic_d start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT ) italic_d start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ,(2.12)

where d0[0,1)subscript𝑑001d_{0}\in[0,1)italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ [ 0 , 1 ) and dn(0,1)subscript𝑑𝑛01d_{n}\in(0,1)italic_d start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∈ ( 0 , 1 ) for n1𝑛1n\geq 1italic_n ≥ 1. If [a,b]𝑎𝑏[a,b][ italic_a , italic_b ] represents the support of the measure for the orthogonal polynomial n(x)subscript𝑛𝑥\mathbb{P}_{n}(x)blackboard_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ) and ta𝑡𝑎t\leq aitalic_t ≤ italic_a, then {sn(t)}subscript𝑠𝑛𝑡\{s_{n}(t)\}{ italic_s start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) }, defined by

sn(t)=λn+1(cnt)(cn+1t),subscript𝑠𝑛𝑡subscript𝜆𝑛1subscript𝑐𝑛𝑡subscript𝑐𝑛1𝑡\displaystyle s_{n}(t)=\frac{\lambda_{n+1}}{(c_{n}-t)(c_{n+1}-t)},italic_s start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) = divide start_ARG italic_λ start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT end_ARG start_ARG ( italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT - italic_t ) ( italic_c start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT - italic_t ) end_ARG ,(2.13)

forms a chain sequence. The expression of the minimal parameter sequence in terms of orthogonal polynomials is also well known. If t(a,b)𝑡𝑎𝑏t\not\in(a,b)italic_t ∉ ( italic_a , italic_b ), then the chain sequence can be expressed as

sn(t)=(1mn1(t))mn(t),subscript𝑠𝑛𝑡1subscript𝑚𝑛1𝑡subscript𝑚𝑛𝑡\displaystyle s_{n}(t)=(1-m_{n-1}(t))m_{n}(t),italic_s start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) = ( 1 - italic_m start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT ( italic_t ) ) italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) ,(2.14)

where the parameter chain sequence {mn(t)}subscript𝑚𝑛𝑡\{m_{n}(t)\}{ italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) } is given by

mn(t)=1n+1(t)(tcn+1)n(t),forn{0}.formulae-sequencesubscript𝑚𝑛𝑡1subscript𝑛1𝑡𝑡subscript𝑐𝑛1subscript𝑛𝑡for𝑛0\displaystyle m_{n}(t)=1-\frac{\mathbb{P}_{n+1}(t)}{(t-c_{n+1})\mathbb{P}_{n}(%t)},~{}~{}\text{for}~{}~{}n\in\mathbb{N}\cup\{0\}.italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) = 1 - divide start_ARG blackboard_P start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT ( italic_t ) end_ARG start_ARG ( italic_t - italic_c start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT ) blackboard_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) end_ARG , for italic_n ∈ blackboard_N ∪ { 0 } .(2.15)

For more detailed information about the chain sequence, we refer to [7].

We observe that exactly one zero of 𝒞nq(x;0)superscriptsubscript𝒞𝑛𝑞𝑥0\mathcal{C}_{n}^{q}(x;0)caligraphic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT ( italic_x ; 0 ) defined in (2.7) lies at the finite end point of the support of the measure for the Laguerre polynomial. Nevertheless, we can still derive the chain sequence and minimal parameter sequence. This is made possible by the cancellation of the common zero in (2.15). Utilizing the recurrence parameters for the quasi-type kernel Laguerre polynomial of order one, as defined in (2.8), we derive the chain sequence {sn(x)}n=2superscriptsubscriptsubscript𝑠𝑛𝑥𝑛2\{s_{n}(x)\}_{n=2}^{\infty}{ italic_s start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ) } start_POSTSUBSCRIPT italic_n = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT at x=0𝑥0x=0italic_x = 0. The corresponding chain sequence is given by

sn(0)=(n1)(n+α+1)(2n+α+1)(2n+α1)=(1mn1(0))mn(0),subscript𝑠𝑛0𝑛1𝑛𝛼12𝑛𝛼12𝑛𝛼11subscript𝑚𝑛10subscript𝑚𝑛0\displaystyle s_{n}(0)=\frac{(n-1)(n+\alpha+1)}{(2n+\alpha+1)(2n+\alpha-1)}=(1%-m_{n-1}(0))m_{n}(0),italic_s start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( 0 ) = divide start_ARG ( italic_n - 1 ) ( italic_n + italic_α + 1 ) end_ARG start_ARG ( 2 italic_n + italic_α + 1 ) ( 2 italic_n + italic_α - 1 ) end_ARG = ( 1 - italic_m start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT ( 0 ) ) italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( 0 ) ,(2.16)

where the parameter sequence {mn(0)}n=1superscriptsubscriptsubscript𝑚𝑛0𝑛1\{m_{n}(0)\}_{n=1}^{\infty}{ italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( 0 ) } start_POSTSUBSCRIPT italic_n = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT is given by

mn(0)=n12n+α+1.subscript𝑚𝑛0𝑛12𝑛𝛼1\displaystyle m_{n}(0)=\frac{n-1}{2n+\alpha+1}.italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( 0 ) = divide start_ARG italic_n - 1 end_ARG start_ARG 2 italic_n + italic_α + 1 end_ARG .(2.17)

In fact, the parameter sequence mn(0)subscript𝑚𝑛0m_{n}(0)italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( 0 ) is a minimal parameter sequence, as m1(0)=0subscript𝑚100m_{1}(0)=0italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( 0 ) = 0. It is evident that the minimal parameter sequence mn(0)>0subscript𝑚𝑛00m_{n}(0)>0italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( 0 ) > 0 for n2𝑛2n\geq 2italic_n ≥ 2 and α>1𝛼1\alpha>-1italic_α > - 1. Additionally, the strict upper bound for mn(0)subscript𝑚𝑛0m_{n}(0)italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( 0 ) is 1212\frac{1}{2}divide start_ARG 1 end_ARG start_ARG 2 end_ARG for n2𝑛2n\geq 2italic_n ≥ 2 and α>1𝛼1\alpha>-1italic_α > - 1. This can be shown as follows:

mn(0)=n12n+α+1n12n=12(11n)<12.subscript𝑚𝑛0𝑛12𝑛𝛼1𝑛12𝑛1211𝑛12\displaystyle m_{n}(0)=\frac{n-1}{2n+\alpha+1}\leq\frac{n-1}{2n}=\frac{1}{2}%\left(1-\frac{1}{n}\right)<\frac{1}{2}.italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( 0 ) = divide start_ARG italic_n - 1 end_ARG start_ARG 2 italic_n + italic_α + 1 end_ARG ≤ divide start_ARG italic_n - 1 end_ARG start_ARG 2 italic_n end_ARG = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( 1 - divide start_ARG 1 end_ARG start_ARG italic_n end_ARG ) < divide start_ARG 1 end_ARG start_ARG 2 end_ARG .

Thus, according to [3, Lemma 2.5], the complementary chain sequence of the chain sequence sn(0)subscript𝑠𝑛0s_{n}(0)italic_s start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( 0 ) is single parameter positive chain sequence (SPPCS).

Solution 2. For any value of α𝛼\alphaitalic_α, another solution to equation (2.5) is γn=nsubscript𝛾𝑛𝑛\gamma_{n}=nitalic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = italic_n. Thus, the sequence 𝒞nQ(x;0)subscriptsuperscript𝒞𝑄𝑛𝑥0{\mathcal{C}^{Q}_{n}(x;0)}caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; 0 ) forms an orthogonal polynomial sequence. With γn=nsubscript𝛾𝑛𝑛\gamma_{n}=nitalic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = italic_n, we recover the Laguerre polynomial with parameter α𝛼\alphaitalic_α. Specifically,

𝒞nQ(x;0):=n(α)(x)=n(α+1)(x)+nn1(α+1)(x),assignsubscriptsuperscript𝒞𝑄𝑛𝑥0subscriptsuperscript𝛼𝑛𝑥subscriptsuperscript𝛼1𝑛𝑥𝑛subscriptsuperscript𝛼1𝑛1𝑥\displaystyle\mathcal{C}^{Q}_{n}(x;0):=\mathcal{L}^{(\alpha)}_{n}(x)=\mathcal{%L}^{(\alpha+1)}_{n}(x)+n\mathcal{L}^{(\alpha+1)}_{n-1}(x),caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; 0 ) := caligraphic_L start_POSTSUPERSCRIPT ( italic_α ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ) = caligraphic_L start_POSTSUPERSCRIPT ( italic_α + 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ) + italic_n caligraphic_L start_POSTSUPERSCRIPT ( italic_α + 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT ( italic_x ) ,(2.18)

which becomes an orthogonal polynomial with recurrence parameters given by

λn+1qc=n(n+α),cn+1qc=2n+α+1.formulae-sequencesubscriptsuperscript𝜆𝑞𝑐𝑛1𝑛𝑛𝛼subscriptsuperscript𝑐𝑞𝑐𝑛12𝑛𝛼1\displaystyle\lambda^{qc}_{n+1}=n(n+\alpha),~{}c^{qc}_{n+1}=2n+\alpha+1.italic_λ start_POSTSUPERSCRIPT italic_q italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT = italic_n ( italic_n + italic_α ) , italic_c start_POSTSUPERSCRIPT italic_q italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT = 2 italic_n + italic_α + 1 .
Remark 2.2.

In equation (2.18), it is observed that the quasi-Christoffel Laguerre polynomials transition into Laguerre polynomials with parameter α𝛼\alphaitalic_α. This similarity corresponds to the decomposition of Laguerre polynomials as depicted in [19, page 102].

2.2. Quasi-Christoffel Jacobi polynomial

Let 𝒫n(α,β)(x)subscriptsuperscript𝒫𝛼𝛽𝑛𝑥\mathcal{P}^{(\alpha,\beta)}_{n}(x)caligraphic_P start_POSTSUPERSCRIPT ( italic_α , italic_β ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ) denote the monic Jacobi polynomials defined by the three-term recurrence relation:

𝒫n+1(α,β)(x)=(xcn+1)𝒫n(α,β)(x)λn+1𝒫n1(α,β)(x),subscriptsuperscript𝒫𝛼𝛽𝑛1𝑥𝑥subscript𝑐𝑛1subscriptsuperscript𝒫𝛼𝛽𝑛𝑥subscript𝜆𝑛1subscriptsuperscript𝒫𝛼𝛽𝑛1𝑥\displaystyle\mathcal{P}^{(\alpha,\beta)}_{n+1}(x)=(x-c_{n+1})\mathcal{P}^{(%\alpha,\beta)}_{n}(x)-\lambda_{n+1}\mathcal{P}^{(\alpha,\beta)}_{n-1}(x),caligraphic_P start_POSTSUPERSCRIPT ( italic_α , italic_β ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT ( italic_x ) = ( italic_x - italic_c start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT ) caligraphic_P start_POSTSUPERSCRIPT ( italic_α , italic_β ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ) - italic_λ start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT caligraphic_P start_POSTSUPERSCRIPT ( italic_α , italic_β ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT ( italic_x ) ,

with recurrence coefficients given by

λn+1subscript𝜆𝑛1\displaystyle\lambda_{n+1}italic_λ start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT=4n(n+α)(n+β)(n+α+β)(2n+α+β)2(2n+α+β+1)(2n+α+β1),absent4𝑛𝑛𝛼𝑛𝛽𝑛𝛼𝛽superscript2𝑛𝛼𝛽22𝑛𝛼𝛽12𝑛𝛼𝛽1\displaystyle=\frac{4n(n+\alpha)(n+\beta)(n+\alpha+\beta)}{(2n+\alpha+\beta)^{%2}(2n+\alpha+\beta+1)(2n+\alpha+\beta-1)},= divide start_ARG 4 italic_n ( italic_n + italic_α ) ( italic_n + italic_β ) ( italic_n + italic_α + italic_β ) end_ARG start_ARG ( 2 italic_n + italic_α + italic_β ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 2 italic_n + italic_α + italic_β + 1 ) ( 2 italic_n + italic_α + italic_β - 1 ) end_ARG ,
cn+1subscript𝑐𝑛1\displaystyle c_{n+1}italic_c start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT=β2α2(2n+α+β)(2n+α+β+2).absentsuperscript𝛽2superscript𝛼22𝑛𝛼𝛽2𝑛𝛼𝛽2\displaystyle=\frac{\beta^{2}-\alpha^{2}}{(2n+\alpha+\beta)(2n+\alpha+\beta+2)}.= divide start_ARG italic_β start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ( 2 italic_n + italic_α + italic_β ) ( 2 italic_n + italic_α + italic_β + 2 ) end_ARG .

The Jacobi polynomials form an orthogonal sequence on the interval (1,1)11(-1,1)( - 1 , 1 ) with respect to the weight function w(x)=(1x)α(1+x)β𝑤𝑥superscript1𝑥𝛼superscript1𝑥𝛽w(x)=(1-x)^{\alpha}(1+x)^{\beta}italic_w ( italic_x ) = ( 1 - italic_x ) start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT ( 1 + italic_x ) start_POSTSUPERSCRIPT italic_β end_POSTSUPERSCRIPT, where α>1𝛼1\alpha>-1italic_α > - 1 and β>1𝛽1\beta>-1italic_β > - 1. The significance of Jacobi polynomials, including their special instances like ultraspherical polynomials, Legendre polynomials and Chebyshev polynomials extends across various mathematical domains. One notable application lies in their connection to the spectral analysis of Laplacian and sub-Laplacian operators. Pertaining to this, [6] delves into the discussion on pointwise estimations for ultraspherical polynomials. The exploration of Pell’s equation as it relates to Chebyshev polynomials is detailed in [14]. Moreover, these orthogonal polynomials are intricately connected to convex optimization and real algebraic geometry, as elucidated in the same source [14].

Upon applying the Christoffel transformation at a=1𝑎1a=-1italic_a = - 1 to this weight function, we obtain w~(x)=(1x)α(1+x)β+1~𝑤𝑥superscript1𝑥𝛼superscript1𝑥𝛽1\tilde{w}(x)=(1-x)^{\alpha}(1+x)^{\beta+1}over~ start_ARG italic_w end_ARG ( italic_x ) = ( 1 - italic_x ) start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT ( 1 + italic_x ) start_POSTSUPERSCRIPT italic_β + 1 end_POSTSUPERSCRIPT, with α>1𝛼1\alpha>-1italic_α > - 1 and β>1𝛽1\beta>-1italic_β > - 1. The Christoffel transformed polynomial of the Jacobi polynomial is another Jacobi polynomial with parameter (α,β+1)𝛼𝛽1(\alpha,\beta+1)( italic_α , italic_β + 1 ). This Christoffel Jacobi polynomial with parameter (α,β+1)𝛼𝛽1(\alpha,\beta+1)( italic_α , italic_β + 1 ) is denoted by 𝒞n(x;1):=𝒫n(α,β+1)(x)assignsubscript𝒞𝑛𝑥1subscriptsuperscript𝒫𝛼𝛽1𝑛𝑥\mathcal{C}_{n}(x;-1):=\mathcal{P}^{(\alpha,\beta+1)}_{n}(x)caligraphic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; - 1 ) := caligraphic_P start_POSTSUPERSCRIPT ( italic_α , italic_β + 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ), and it can be obtained using the recursion formula:

𝒫n+1(α,β+1)(x)=(xcn+1c)𝒫n(α,β+1)(x)λn+1c𝒫n1(α,β+1)(x),subscriptsuperscript𝒫𝛼𝛽1𝑛1𝑥𝑥subscriptsuperscript𝑐𝑐𝑛1subscriptsuperscript𝒫𝛼𝛽1𝑛𝑥subscriptsuperscript𝜆𝑐𝑛1subscriptsuperscript𝒫𝛼𝛽1𝑛1𝑥\displaystyle\mathcal{P}^{(\alpha,\beta+1)}_{n+1}(x)=(x-c^{c}_{n+1})\mathcal{P%}^{(\alpha,\beta+1)}_{n}(x)-\lambda^{c}_{n+1}\mathcal{P}^{(\alpha,\beta+1)}_{n%-1}(x),caligraphic_P start_POSTSUPERSCRIPT ( italic_α , italic_β + 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT ( italic_x ) = ( italic_x - italic_c start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT ) caligraphic_P start_POSTSUPERSCRIPT ( italic_α , italic_β + 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ) - italic_λ start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT caligraphic_P start_POSTSUPERSCRIPT ( italic_α , italic_β + 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT ( italic_x ) ,

where the transformed recurrence parameters are given by:

λn+1csubscriptsuperscript𝜆𝑐𝑛1\displaystyle\lambda^{c}_{n+1}italic_λ start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT=4n(n+α)(n+β+1)(n+α+β+1)(2n+α+β+1)2(2n+α+β+2)(2n+α+β),absent4𝑛𝑛𝛼𝑛𝛽1𝑛𝛼𝛽1superscript2𝑛𝛼𝛽122𝑛𝛼𝛽22𝑛𝛼𝛽\displaystyle=\frac{4n(n+\alpha)(n+\beta+1)(n+\alpha+\beta+1)}{(2n+\alpha+%\beta+1)^{2}(2n+\alpha+\beta+2)(2n+\alpha+\beta)},= divide start_ARG 4 italic_n ( italic_n + italic_α ) ( italic_n + italic_β + 1 ) ( italic_n + italic_α + italic_β + 1 ) end_ARG start_ARG ( 2 italic_n + italic_α + italic_β + 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 2 italic_n + italic_α + italic_β + 2 ) ( 2 italic_n + italic_α + italic_β ) end_ARG ,
cn+1csubscriptsuperscript𝑐𝑐𝑛1\displaystyle c^{c}_{n+1}italic_c start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT=(β+1)2α2(2n+α+β+1)(2n+α+β+3).absentsuperscript𝛽12superscript𝛼22𝑛𝛼𝛽12𝑛𝛼𝛽3\displaystyle=\frac{(\beta+1)^{2}-\alpha^{2}}{(2n+\alpha+\beta+1)(2n+\alpha+%\beta+3)}.= divide start_ARG ( italic_β + 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ( 2 italic_n + italic_α + italic_β + 1 ) ( 2 italic_n + italic_α + italic_β + 3 ) end_ARG .

By forming a linear combination of two consecutive degrees of Christoffel Jacobi polynomials, we define the quasi-type kernel Jacobi polynomial of order one as:

𝒞nQ(x;1)=𝒫n(α,β+1)(x)+γn𝒫n1(α,β+1)(x).subscriptsuperscript𝒞𝑄𝑛𝑥1subscriptsuperscript𝒫𝛼𝛽1𝑛𝑥subscript𝛾𝑛subscriptsuperscript𝒫𝛼𝛽1𝑛1𝑥\displaystyle\mathcal{C}^{Q}_{n}(x;-1)=\mathcal{P}^{(\alpha,\beta+1)}_{n}(x)+%\gamma_{n}\mathcal{P}^{(\alpha,\beta+1)}_{n-1}(x).caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; - 1 ) = caligraphic_P start_POSTSUPERSCRIPT ( italic_α , italic_β + 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ) + italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT caligraphic_P start_POSTSUPERSCRIPT ( italic_α , italic_β + 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT ( italic_x ) .(2.19)

Due to the partial orthogonality of the polynomial 𝒞nQ(x;1)subscriptsuperscript𝒞𝑄𝑛𝑥1\mathcal{C}^{Q}_{n}(x;-1)caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; - 1 ), it is possible for some zeros to lie outside the support of a Jacobi measure. Subsequently, we observe numerically that at most one zero lies outside the support of the measure for the Jacobi polynomials. Furthermore, we note that a zero can be situated on either side of the support of a measure, either to the left or right.

Zeros of 𝒞5Q(x;1)subscriptsuperscript𝒞𝑄5𝑥1\mathcal{C}^{Q}_{5}(x;-1)caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT ( italic_x ; - 1 )Zeros of 𝒞6Q(x;1)subscriptsuperscript𝒞𝑄6𝑥1\mathcal{C}^{Q}_{6}(x;-1)caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT ( italic_x ; - 1 )
α=0.5𝛼0.5\alpha=-0.5italic_α = - 0.5, β=0𝛽0\beta=0italic_β = 0, γn=3subscript𝛾𝑛3\gamma_{n}=3italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = 3α=0𝛼0\alpha=0italic_α = 0, β=0.5𝛽0.5\beta=0.5italic_β = 0.5, γn=2subscript𝛾𝑛2\gamma_{n}=2italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = 2α=1𝛼1\alpha=1italic_α = 1, β=0.5𝛽0.5\beta=-0.5italic_β = - 0.5, γn=1subscript𝛾𝑛1\gamma_{n}=-1italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = - 1α=0.5𝛼0.5\alpha=0.5italic_α = 0.5, β=1𝛽1\beta=1italic_β = 1, γn=2subscript𝛾𝑛2\gamma_{n}=-2italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = - 2
-1.23179-2.09864-0.88766-0.73675
-0.60752-0.62066-0.57465-0.34365
-0.00608-0.04931-0.127920.11967
0.595280.518350.356370.56019
0.952230.902440.770890.88114
--1.240752.14008

Table 4 demonstrates that for α=0.5𝛼0.5\alpha=-0.5italic_α = - 0.5 and β=0𝛽0\beta=0italic_β = 0 with γn=3subscript𝛾𝑛3\gamma_{n}=3italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = 3, one zero (x0=1.23179)subscript𝑥01.23179(x_{0}=-1.23179)( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = - 1.23179 ) of 𝒞5Q(x;1)subscriptsuperscript𝒞𝑄5𝑥1\mathcal{C}^{Q}_{5}(x;-1)caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT ( italic_x ; - 1 ) lies outside the left side of the interval (1,1)11(-1,1)( - 1 , 1 ). Similarly, for α=0.5𝛼0.5\alpha=0.5italic_α = 0.5 and β=1𝛽1\beta=1italic_β = 1 with γn=2subscript𝛾𝑛2\gamma_{n}=-2italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = - 2, one zero (x0=2.14008)subscript𝑥02.14008(x_{0}=2.14008)( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 2.14008 ) of 𝒞5Q(x;1)subscriptsuperscript𝒞𝑄5𝑥1\mathcal{C}^{Q}_{5}(x;-1)caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT ( italic_x ; - 1 ) lies outside the right side of the interval (1,1)11(-1,1)( - 1 , 1 ). These instances are illustrated in Table 4 for various values of α𝛼\alphaitalic_α, β𝛽\betaitalic_β, and γnsubscript𝛾𝑛\gamma_{n}italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT.

To achieve the orthogonality of the quasi-Christoffel Jacobi polynomial of order one, we must determine the value of γnsubscript𝛾𝑛\gamma_{n}italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT that satisfies (2.1). We have

(β+1)2α2(α+β+2n1)(α+β+2n+1)+(β+1)2α2(α+β+2n+1)(α+β+2n+3)+γnγn+1superscript𝛽12superscript𝛼2𝛼𝛽2𝑛1𝛼𝛽2𝑛1superscript𝛽12superscript𝛼2𝛼𝛽2𝑛1𝛼𝛽2𝑛3subscript𝛾𝑛subscript𝛾𝑛1\displaystyle-\frac{(\beta+1)^{2}-\alpha^{2}}{(\alpha+\beta+2n-1)(\alpha+\beta%+2n+1)}+\frac{(\beta+1)^{2}-\alpha^{2}}{(\alpha+\beta+2n+1)(\alpha+\beta+2n+3)%}+\gamma_{n}-\gamma_{n+1}- divide start_ARG ( italic_β + 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ( italic_α + italic_β + 2 italic_n - 1 ) ( italic_α + italic_β + 2 italic_n + 1 ) end_ARG + divide start_ARG ( italic_β + 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ( italic_α + italic_β + 2 italic_n + 1 ) ( italic_α + italic_β + 2 italic_n + 3 ) end_ARG + italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT - italic_γ start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT
1γn4n(α+n)(β+n+1)(α+β+n+1)(α+β+2n)(α+β+2n+1)2(α+β+2n+2)1subscript𝛾𝑛4𝑛𝛼𝑛𝛽𝑛1𝛼𝛽𝑛1𝛼𝛽2𝑛superscript𝛼𝛽2𝑛12𝛼𝛽2𝑛2\displaystyle\hskip 28.45274pt-\frac{1}{\gamma_{n}}\frac{4n(\alpha+n)(\beta+n+%1)(\alpha+\beta+n+1)}{(\alpha+\beta+2n)(\alpha+\beta+2n+1)^{2}(\alpha+\beta+2n%+2)}- divide start_ARG 1 end_ARG start_ARG italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG divide start_ARG 4 italic_n ( italic_α + italic_n ) ( italic_β + italic_n + 1 ) ( italic_α + italic_β + italic_n + 1 ) end_ARG start_ARG ( italic_α + italic_β + 2 italic_n ) ( italic_α + italic_β + 2 italic_n + 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_α + italic_β + 2 italic_n + 2 ) end_ARG
+1γn14(n1)(α+n1)(β+n)(α+β+n)(α+β+2n2)(α+β+2n1)2(α+β+2n)=0.1subscript𝛾𝑛14𝑛1𝛼𝑛1𝛽𝑛𝛼𝛽𝑛𝛼𝛽2𝑛2superscript𝛼𝛽2𝑛12𝛼𝛽2𝑛0\displaystyle\hskip 85.35826pt+\frac{1}{\gamma_{n-1}}\frac{4(n-1)(\alpha+n-1)(%\beta+n)(\alpha+\beta+n)}{(\alpha+\beta+2n-2)(\alpha+\beta+2n-1)^{2}(\alpha+%\beta+2n)}=0.+ divide start_ARG 1 end_ARG start_ARG italic_γ start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT end_ARG divide start_ARG 4 ( italic_n - 1 ) ( italic_α + italic_n - 1 ) ( italic_β + italic_n ) ( italic_α + italic_β + italic_n ) end_ARG start_ARG ( italic_α + italic_β + 2 italic_n - 2 ) ( italic_α + italic_β + 2 italic_n - 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_α + italic_β + 2 italic_n ) end_ARG = 0 .(2.20)

The two possible solutions to (2.2) are as follows:

Solution 1. Recursively, we obtain the explicit expression for γnsubscript𝛾𝑛\gamma_{n}italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT as follows:

γn=2(α+n)(n+α+β+1)(2n+α+β+1)(2n+α+β),subscript𝛾𝑛2𝛼𝑛𝑛𝛼𝛽12𝑛𝛼𝛽12𝑛𝛼𝛽\displaystyle\gamma_{n}=-\frac{2(\alpha+n)(n+\alpha+\beta+1)}{(2n+\alpha+\beta%+1)(2n+\alpha+\beta)},italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = - divide start_ARG 2 ( italic_α + italic_n ) ( italic_n + italic_α + italic_β + 1 ) end_ARG start_ARG ( 2 italic_n + italic_α + italic_β + 1 ) ( 2 italic_n + italic_α + italic_β ) end_ARG ,(2.21)

which satisfies the nonlinear difference equation (2.2). Thus, the polynomial 𝒞n+1Q(x;1)subscriptsuperscript𝒞𝑄𝑛1𝑥1\mathcal{C}^{Q}_{n+1}(x;-1)caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT ( italic_x ; - 1 ) is defined as:

𝒞nQ(x;1)=𝒫n(α,β+1)(x)2(α+n)(n+α+β+1)(2n+α+β+1)(2n+α+β)𝒫n1(α,β+1)(x),subscriptsuperscript𝒞𝑄𝑛𝑥1subscriptsuperscript𝒫𝛼𝛽1𝑛𝑥2𝛼𝑛𝑛𝛼𝛽12𝑛𝛼𝛽12𝑛𝛼𝛽subscriptsuperscript𝒫𝛼𝛽1𝑛1𝑥\displaystyle\mathcal{C}^{Q}_{n}(x;-1)=\mathcal{P}^{(\alpha,\beta+1)}_{n}(x)-%\frac{2(\alpha+n)(n+\alpha+\beta+1)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta)}%\mathcal{P}^{(\alpha,\beta+1)}_{n-1}(x),caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; - 1 ) = caligraphic_P start_POSTSUPERSCRIPT ( italic_α , italic_β + 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ) - divide start_ARG 2 ( italic_α + italic_n ) ( italic_n + italic_α + italic_β + 1 ) end_ARG start_ARG ( 2 italic_n + italic_α + italic_β + 1 ) ( 2 italic_n + italic_α + italic_β ) end_ARG caligraphic_P start_POSTSUPERSCRIPT ( italic_α , italic_β + 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT ( italic_x ) ,(2.22)

which becomes orthogonal and satisfies the three-term recurrence relation

𝒞n+1Q(x;1)=(xcn+1qc)𝒞nQ(x;1)λn+1qc𝒞n1Q(x;1),subscriptsuperscript𝒞𝑄𝑛1𝑥1𝑥subscriptsuperscript𝑐𝑞𝑐𝑛1subscriptsuperscript𝒞𝑄𝑛𝑥1subscriptsuperscript𝜆𝑞𝑐𝑛1subscriptsuperscript𝒞𝑄𝑛1𝑥1\displaystyle\mathcal{C}^{Q}_{n+1}(x;-1)=(x-c^{qc}_{n+1})\mathcal{C}^{Q}_{n}(x%;-1)-\lambda^{qc}_{n+1}\mathcal{C}^{Q}_{n-1}(x;-1),caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT ( italic_x ; - 1 ) = ( italic_x - italic_c start_POSTSUPERSCRIPT italic_q italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT ) caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; - 1 ) - italic_λ start_POSTSUPERSCRIPT italic_q italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT ( italic_x ; - 1 ) ,

with recurrence coefficients given by:

λn+1qcsubscriptsuperscript𝜆𝑞𝑐𝑛1\displaystyle\lambda^{qc}_{n+1}italic_λ start_POSTSUPERSCRIPT italic_q italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT=4(n1)(n+α)(n+β)(n+α+β+1)(2n+α+β)2(2n+α+β+1)(2n+α+β1),absent4𝑛1𝑛𝛼𝑛𝛽𝑛𝛼𝛽1superscript2𝑛𝛼𝛽22𝑛𝛼𝛽12𝑛𝛼𝛽1\displaystyle=\frac{4(n-1)(n+\alpha)(n+\beta)(n+\alpha+\beta+1)}{(2n+\alpha+%\beta)^{2}(2n+\alpha+\beta+1)(2n+\alpha+\beta-1)},= divide start_ARG 4 ( italic_n - 1 ) ( italic_n + italic_α ) ( italic_n + italic_β ) ( italic_n + italic_α + italic_β + 1 ) end_ARG start_ARG ( 2 italic_n + italic_α + italic_β ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 2 italic_n + italic_α + italic_β + 1 ) ( 2 italic_n + italic_α + italic_β - 1 ) end_ARG ,
cn+1qcsubscriptsuperscript𝑐𝑞𝑐𝑛1\displaystyle c^{qc}_{n+1}italic_c start_POSTSUPERSCRIPT italic_q italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT=(βα)(2+β+α)(2n+α+β)(2n+α+β+2).absent𝛽𝛼2𝛽𝛼2𝑛𝛼𝛽2𝑛𝛼𝛽2\displaystyle=\frac{(\beta-\alpha)(2+\beta+\alpha)}{(2n+\alpha+\beta)(2n+%\alpha+\beta+2)}.= divide start_ARG ( italic_β - italic_α ) ( 2 + italic_β + italic_α ) end_ARG start_ARG ( 2 italic_n + italic_α + italic_β ) ( 2 italic_n + italic_α + italic_β + 2 ) end_ARG .(2.23)

We observe that using the value of γnsubscript𝛾𝑛\gamma_{n}italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT defined in (2.21) yields the orthogonality of 𝒞nQ(x;1)superscriptsubscript𝒞𝑛𝑄𝑥1\mathcal{C}_{n}^{Q}(x;-1)caligraphic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT ( italic_x ; - 1 ). Subsequently, we illustrate the behavior of the zeros of 𝒞nQ(x;1)superscriptsubscript𝒞𝑛𝑄𝑥1\mathcal{C}_{n}^{Q}(x;-1)caligraphic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT ( italic_x ; - 1 ) in Table 5. For each n1𝑛1n\geq 1italic_n ≥ 1, we also note that p(x)=x1𝑝𝑥𝑥1p(x)=x-1italic_p ( italic_x ) = italic_x - 1 is a factor of the polynomial 𝒞nQ(x;1)superscriptsubscript𝒞𝑛𝑄𝑥1\mathcal{C}_{n}^{Q}(x;-1)caligraphic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT ( italic_x ; - 1 ). This implies that exactly one zero, x=1𝑥1x=1italic_x = 1, lies on the boundary of the true interval of orthogonality for Jacobi polynomials.

Zeros of 𝒞nQ(x;1)subscriptsuperscript𝒞𝑄𝑛𝑥1\mathcal{C}^{Q}_{n}(x;-1)caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; - 1 )Zeros of 𝒞nQ(x;1)subscriptsuperscript𝒞𝑄𝑛𝑥1\mathcal{C}^{Q}_{n}(x;-1)caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; - 1 )
n=5𝑛5n=5italic_n = 5, α=0.5𝛼0.5\alpha=-0.5italic_α = - 0.5, β=0𝛽0\beta=0italic_β = 0n=6𝑛6n=6italic_n = 6, α=1𝛼1\alpha=1italic_α = 1, β=0.5𝛽0.5\beta=-0.5italic_β = - 0.5n=5𝑛5n=5italic_n = 5, α=0.2𝛼0.2\alpha=-0.2italic_α = - 0.2, β=0.2𝛽0.2\beta=0.2italic_β = 0.2n=6𝑛6n=6italic_n = 6, α=0.2𝛼0.2\alpha=-0.2italic_α = - 0.2, β=0.2𝛽0.2\beta=0.2italic_β = 0.2
-0.844012-0.886199-0.733177-0.806761
-0.443904-0.586069-0.236142-0.430117
0.088564-0.1615510.3337840.044001
0.6067590.3009730.7955350.506945
10.70784710.852599
-1-1

A common zero at the end point of the support of measure for the quasi-natured spectrally transformed polynomials (4)

In Table 5 and Figure 4, it is observed that for a particular set of α𝛼\alphaitalic_α and β𝛽\betaitalic_β values, such as α=0.2𝛼0.2\alpha=-0.2italic_α = - 0.2 and β=0.2𝛽0.2\beta=0.2italic_β = 0.2, the zeros of 𝒞6Q(x;1)subscriptsuperscript𝒞𝑄6𝑥1\mathcal{C}^{Q}_{6}(x;-1)caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT ( italic_x ; - 1 ) and 𝒞5Q(x;1)subscriptsuperscript𝒞𝑄5𝑥1\mathcal{C}^{Q}_{5}(x;-1)caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT ( italic_x ; - 1 ) exhibit interlacing behavior. Additionally, Table 5 illustrates that at most one zero lies on the boundary of the interval (1,1)11(-1,1)( - 1 , 1 ) while all others lie within the interval.

Zeros of 𝒫n(α,β)(x)subscriptsuperscript𝒫𝛼𝛽𝑛𝑥\mathcal{P}^{(\alpha,\beta)}_{n}(x)caligraphic_P start_POSTSUPERSCRIPT ( italic_α , italic_β ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x )Zeros of 𝒞n(x;1)subscript𝒞𝑛𝑥1\mathcal{C}_{n}(x;-1)caligraphic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; - 1 )Zeros of 𝒞nQ(x;1)subscriptsuperscript𝒞𝑄𝑛𝑥1\mathcal{C}^{Q}_{n}(x;-1)caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; - 1 )
n=6𝑛6n=6italic_n = 6, α=1𝛼1\alpha=1italic_α = 1, β=2𝛽2\beta=2italic_β = 2n=6𝑛6n=6italic_n = 6, α=1𝛼1\alpha=1italic_α = 1, β=2𝛽2\beta=2italic_β = 2n=6𝑛6n=6italic_n = 6, α=1𝛼1\alpha=1italic_α = 1, β=2𝛽2\beta=2italic_β = 2
-0.799382-0.727904-0.690458
-0.491906-0.402917-0.32652
-0.111734-0.0288540.082337
0.283540.3448380.475179
0.6337930.6679460.792794
0.8856880.8968921

A common zero at the end point of the support of measure for the quasi-natured spectrally transformed polynomials (5)
A common zero at the end point of the support of measure for the quasi-natured spectrally transformed polynomials (6)

Interlacing between the zeros of 𝒫6(α,β)(x)superscriptsubscript𝒫6𝛼𝛽𝑥\mathcal{P}_{6}^{(\alpha,\beta)}(x)caligraphic_P start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_α , italic_β ) end_POSTSUPERSCRIPT ( italic_x ) and 𝒞6Q(x;1)subscriptsuperscript𝒞𝑄6𝑥1\mathcal{C}^{Q}_{6}(x;-1)caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT ( italic_x ; - 1 ) for α=1𝛼1\alpha=1italic_α = 1 and β=2𝛽2\beta=2italic_β = 2 is depicted in Table 6 and Figure 5. Furthermore, interlacing is also evident between the zeros of the Christoffel polynomial 𝒞6(x;1)subscript𝒞6𝑥1\mathcal{C}_{6}(x;-1)caligraphic_C start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT ( italic_x ; - 1 ) and the quasi-type kernel Jacobi orthogonal polynomial 𝒞6Q(x;1)subscriptsuperscript𝒞𝑄6𝑥1\mathcal{C}^{Q}_{6}(x;-1)caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT ( italic_x ; - 1 ), as shown in Table 6 and Figure 5.

For α>1𝛼1\alpha>-1italic_α > - 1 and β>1𝛽1\beta>-1italic_β > - 1, the zeros of the Jacobi polynomials 𝒫n(α,β)(x)superscriptsubscript𝒫𝑛𝛼𝛽𝑥\mathcal{P}_{n}^{(\alpha,\beta)}(x)caligraphic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_α , italic_β ) end_POSTSUPERSCRIPT ( italic_x ) are located inside the interval (1,1)11(-1,1)( - 1 , 1 ) and are simple. However, when we extend the values of the parameters (α,β)𝛼𝛽(\alpha,\beta)( italic_α , italic_β ), this result no longer holds. Specifically, for α=1𝛼1\alpha=-1italic_α = - 1 and β>1𝛽1\beta>-1italic_β > - 1, x=1𝑥1x=1italic_x = 1 becomes the common zero of all the polynomials 𝒫n(1,β)(x)superscriptsubscript𝒫𝑛1𝛽𝑥\mathcal{P}_{n}^{(-1,\beta)}(x)caligraphic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( - 1 , italic_β ) end_POSTSUPERSCRIPT ( italic_x ). A more general result for α=k𝛼𝑘\alpha=-kitalic_α = - italic_k, where kn𝑘𝑛k\leq nitalic_k ≤ italic_n and k𝑘k\in\mathbb{N}italic_k ∈ blackboard_N, can be derived from the following formula (see [19, equation 4.22.2]):

𝒫n(k,β)(x)=12kΓ(n+β+1)Γ(nk+1)Γ(n+β+1k)Γ(n+1)(x1)k𝒫nk(k,β)(x).superscriptsubscript𝒫𝑛𝑘𝛽𝑥1superscript2𝑘Γ𝑛𝛽1Γ𝑛𝑘1Γ𝑛𝛽1𝑘Γ𝑛1superscript𝑥1𝑘superscriptsubscript𝒫𝑛𝑘𝑘𝛽𝑥\displaystyle\mathcal{P}_{n}^{(-k,\beta)}(x)=\frac{1}{2^{k}}\frac{\Gamma(n+%\beta+1)\Gamma(n-k+1)}{\Gamma(n+\beta+1-k)\Gamma(n+1)}(x-1)^{k}\mathcal{P}_{n-%k}^{(k,\beta)}(x).caligraphic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( - italic_k , italic_β ) end_POSTSUPERSCRIPT ( italic_x ) = divide start_ARG 1 end_ARG start_ARG 2 start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_ARG divide start_ARG roman_Γ ( italic_n + italic_β + 1 ) roman_Γ ( italic_n - italic_k + 1 ) end_ARG start_ARG roman_Γ ( italic_n + italic_β + 1 - italic_k ) roman_Γ ( italic_n + 1 ) end_ARG ( italic_x - 1 ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT caligraphic_P start_POSTSUBSCRIPT italic_n - italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k , italic_β ) end_POSTSUPERSCRIPT ( italic_x ) .(2.24)

However, the polynomial obtained by restoring orthogonality as described in equation (2.22) has the common zero x=1𝑥1x=1italic_x = 1 with multiplicity one, which remains independent of the parameters α𝛼\alphaitalic_α and β𝛽\betaitalic_β. The independence from the parameters α𝛼\alphaitalic_α and β𝛽\betaitalic_β of the factor p(x)=x1𝑝𝑥𝑥1p(x)=x-1italic_p ( italic_x ) = italic_x - 1 is also evident in Table 5.

We can use the recurrence parameter λnqcsuperscriptsubscript𝜆𝑛𝑞𝑐\lambda_{n}^{qc}italic_λ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_q italic_c end_POSTSUPERSCRIPT and cnqcsuperscriptsubscript𝑐𝑛𝑞𝑐c_{n}^{qc}italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_q italic_c end_POSTSUPERSCRIPT of the polynomial 𝒞nQ(x;1)superscriptsubscript𝒞𝑛𝑄𝑥1\mathcal{C}_{n}^{Q}(x;-1)caligraphic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT ( italic_x ; - 1 ) to obtain the chain sequence. For x=1𝑥1x=-1italic_x = - 1, by substituting (2.2) in (2.13) we obtain the chain sequence sn(1)subscript𝑠𝑛1s_{n}(-1)italic_s start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( - 1 ) for n=2,3,𝑛23n=2,3,...italic_n = 2 , 3 , … is given by

sn(1)=4(n1)(n+α)(n+β)(n+α+β+1)(2n+α+β+2)(2n+α+β2)Mn(α,β)(2n+α+β+1)(2n+α+β1),subscript𝑠𝑛14𝑛1𝑛𝛼𝑛𝛽𝑛𝛼𝛽12𝑛𝛼𝛽22𝑛𝛼𝛽2subscript𝑀𝑛𝛼𝛽2𝑛𝛼𝛽12𝑛𝛼𝛽1\displaystyle s_{n}(-1)=\frac{4(n-1)(n+\alpha)(n+\beta)(n+\alpha+\beta+1)(2n+%\alpha+\beta+2)(2n+\alpha+\beta-2)}{M_{n}(\alpha,\beta)(2n+\alpha+\beta+1)(2n+%\alpha+\beta-1)},italic_s start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( - 1 ) = divide start_ARG 4 ( italic_n - 1 ) ( italic_n + italic_α ) ( italic_n + italic_β ) ( italic_n + italic_α + italic_β + 1 ) ( 2 italic_n + italic_α + italic_β + 2 ) ( 2 italic_n + italic_α + italic_β - 2 ) end_ARG start_ARG italic_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_α , italic_β ) ( 2 italic_n + italic_α + italic_β + 1 ) ( 2 italic_n + italic_α + italic_β - 1 ) end_ARG ,

where

Mn(α,β)=[(βα)(β+α+2)\displaystyle M_{n}(\alpha,\beta)=[(\beta-\alpha)(\beta+\alpha+2)italic_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_α , italic_β ) = [ ( italic_β - italic_α ) ( italic_β + italic_α + 2 )+(2n+α+β)(2n+α+β+2)]×\displaystyle+(2n+\alpha+\beta)(2n+\alpha+\beta+2)]\times+ ( 2 italic_n + italic_α + italic_β ) ( 2 italic_n + italic_α + italic_β + 2 ) ] ×
[(βα)(β+α+2)+(2n+α+β)(2n+α+β2)].delimited-[]𝛽𝛼𝛽𝛼22𝑛𝛼𝛽2𝑛𝛼𝛽2\displaystyle[(\beta-\alpha)(\beta+\alpha+2)+(2n+\alpha+\beta)(2n+\alpha+\beta%-2)].[ ( italic_β - italic_α ) ( italic_β + italic_α + 2 ) + ( 2 italic_n + italic_α + italic_β ) ( 2 italic_n + italic_α + italic_β - 2 ) ] .

The minimal parameter sequence can be derived at one of the end point, say x=1𝑥1x=-1italic_x = - 1, of the support of the measure for the Jacobi polynomials. Utilizing equations (2.15) and (2.22), we deduce the minimal parameter sequence as

m1(1)=0subscript𝑚110\displaystyle m_{1}(-1)=0italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( - 1 ) = 0
1mn(1)=2(2n+α+β)(n+β+1)(n+α+β+2)(2n+α+β+1)[(2n+α+β)(2n+α+β+2)+(βα)(α+β+2)].1subscript𝑚𝑛122𝑛𝛼𝛽𝑛𝛽1𝑛𝛼𝛽22𝑛𝛼𝛽1delimited-[]2𝑛𝛼𝛽2𝑛𝛼𝛽2𝛽𝛼𝛼𝛽2\displaystyle 1-m_{n}(-1)=\frac{2(2n+\alpha+\beta)(n+\beta+1)(n+\alpha+\beta+2%)}{(2n+\alpha+\beta+1)[(2n+\alpha+\beta)(2n+\alpha+\beta+2)+(\beta-\alpha)(%\alpha+\beta+2)]}.1 - italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( - 1 ) = divide start_ARG 2 ( 2 italic_n + italic_α + italic_β ) ( italic_n + italic_β + 1 ) ( italic_n + italic_α + italic_β + 2 ) end_ARG start_ARG ( 2 italic_n + italic_α + italic_β + 1 ) [ ( 2 italic_n + italic_α + italic_β ) ( 2 italic_n + italic_α + italic_β + 2 ) + ( italic_β - italic_α ) ( italic_α + italic_β + 2 ) ] end_ARG .

If we take α=β>1𝛼𝛽1\alpha=\beta>-1italic_α = italic_β > - 1 in the above equation, then for n2𝑛2n\geq 2italic_n ≥ 2 we have

mn(1)=1n+2α+22n+2α+1=n12n+2α+1<n12n1<12.subscript𝑚𝑛11𝑛2𝛼22𝑛2𝛼1𝑛12𝑛2𝛼1𝑛12𝑛112\displaystyle m_{n}(-1)=1-\frac{n+2\alpha+2}{2n+2\alpha+1}=\frac{n-1}{2n+2%\alpha+1}<\frac{n-1}{2n-1}<\frac{1}{2}.italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( - 1 ) = 1 - divide start_ARG italic_n + 2 italic_α + 2 end_ARG start_ARG 2 italic_n + 2 italic_α + 1 end_ARG = divide start_ARG italic_n - 1 end_ARG start_ARG 2 italic_n + 2 italic_α + 1 end_ARG < divide start_ARG italic_n - 1 end_ARG start_ARG 2 italic_n - 1 end_ARG < divide start_ARG 1 end_ARG start_ARG 2 end_ARG .

This shows that the complementary chain sequence of sn(1)subscript𝑠𝑛1s_{n}(-1)italic_s start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( - 1 ) is SPPCS for α=β𝛼𝛽\alpha=\betaitalic_α = italic_β. This result also holds true when α>1𝛼1\alpha>-1italic_α > - 1 and β>1𝛽1\beta>-1italic_β > - 1.

In equation (2.22), we observe that precisely one zero of 𝒞nQ(x;1)subscriptsuperscript𝒞𝑄𝑛𝑥1\mathcal{C}^{Q}_{n}(x;-1)caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; - 1 ) is situated at one of the extreme points of the support of the measure. Despite this, we are able to derive the minimal parameter sequence at the extreme point x=1𝑥1x=1italic_x = 1. This is achievable because the involvement of the ratio of 𝒞n+1qc(x;1)superscriptsubscript𝒞𝑛1𝑞𝑐𝑥1\mathcal{C}_{n+1}^{qc}(x;-1)caligraphic_C start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_q italic_c end_POSTSUPERSCRIPT ( italic_x ; - 1 ) and 𝒞nqc(x;1)superscriptsubscript𝒞𝑛𝑞𝑐𝑥1\mathcal{C}_{n}^{qc}(x;-1)caligraphic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_q italic_c end_POSTSUPERSCRIPT ( italic_x ; - 1 ) in (2.15) which results in the cancellation of the common zero of the quasi-Christoffel Jacobi orthogonal polynomials.

Solution 2. Alternatively,

γn=2n(α+n)(2n+α+β+1)(2n+α+β),subscript𝛾𝑛2𝑛𝛼𝑛2𝑛𝛼𝛽12𝑛𝛼𝛽\displaystyle\gamma_{n}=\frac{2n(\alpha+n)}{(2n+\alpha+\beta+1)(2n+\alpha+%\beta)},italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = divide start_ARG 2 italic_n ( italic_α + italic_n ) end_ARG start_ARG ( 2 italic_n + italic_α + italic_β + 1 ) ( 2 italic_n + italic_α + italic_β ) end_ARG ,(2.25)

presents another solution to equation (2.2). Utilizing this γnsubscript𝛾𝑛\gamma_{n}italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, the polynomial 𝒞nQ(x;1)subscriptsuperscript𝒞𝑄𝑛𝑥1\mathcal{C}^{Q}_{n}(x;-1)caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; - 1 ) becomes orthogonal and reverts to the original orthogonal polynomial:

𝒞nQ(x;1):=𝒫n(α,β)(x)=𝒫n(α,β+1)(x)+2n(α+n)(2n+α+β+1)(2n+α+β)𝒫n1(α,β+1)(x).assignsubscriptsuperscript𝒞𝑄𝑛𝑥1subscriptsuperscript𝒫𝛼𝛽𝑛𝑥subscriptsuperscript𝒫𝛼𝛽1𝑛𝑥2𝑛𝛼𝑛2𝑛𝛼𝛽12𝑛𝛼𝛽subscriptsuperscript𝒫𝛼𝛽1𝑛1𝑥\displaystyle\mathcal{C}^{Q}_{n}(x;-1):=\mathcal{P}^{(\alpha,\beta)}_{n}(x)=%\mathcal{P}^{(\alpha,\beta+1)}_{n}(x)+\frac{2n(\alpha+n)}{(2n+\alpha+\beta+1)(%2n+\alpha+\beta)}\mathcal{P}^{(\alpha,\beta+1)}_{n-1}(x).caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; - 1 ) := caligraphic_P start_POSTSUPERSCRIPT ( italic_α , italic_β ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ) = caligraphic_P start_POSTSUPERSCRIPT ( italic_α , italic_β + 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ) + divide start_ARG 2 italic_n ( italic_α + italic_n ) end_ARG start_ARG ( 2 italic_n + italic_α + italic_β + 1 ) ( 2 italic_n + italic_α + italic_β ) end_ARG caligraphic_P start_POSTSUPERSCRIPT ( italic_α , italic_β + 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT ( italic_x ) .(2.26)

The recurrence parameter are given by:

λn+1qc=λn+1,cn+1qc=cn+1formulae-sequencesubscriptsuperscript𝜆𝑞𝑐𝑛1subscript𝜆𝑛1subscriptsuperscript𝑐𝑞𝑐𝑛1subscript𝑐𝑛1\displaystyle\lambda^{qc}_{n+1}=\lambda_{n+1},c^{qc}_{n+1}=c_{n+1}italic_λ start_POSTSUPERSCRIPT italic_q italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT = italic_λ start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT , italic_c start_POSTSUPERSCRIPT italic_q italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT = italic_c start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT
Remark 2.3.

The representation (2.26) can also be seen as a decomposition of the Jacobi polynomial with parameter (α,β)𝛼𝛽(\alpha,\beta)( italic_α , italic_β ) in terms of the Jacobi polynomial with extended parameters (α,β+1)𝛼𝛽1(\alpha,\beta+1)( italic_α , italic_β + 1 ).

3. Quasi-Geronimus polynomial of order one

Let Gsuperscript𝐺\mathcal{L}^{G}caligraphic_L start_POSTSUPERSCRIPT italic_G end_POSTSUPERSCRIPT denote the Geronimus transformation of a linear functional \mathcal{L}caligraphic_L at point a𝑎aitalic_a. The linear functional Gsuperscript𝐺\mathcal{L}^{G}caligraphic_L start_POSTSUPERSCRIPT italic_G end_POSTSUPERSCRIPT is defined in terms of the initial one as:

G(p(x))=(p(x)p(a)xa)+Mp(a).superscript𝐺𝑝𝑥𝑝𝑥𝑝𝑎𝑥𝑎𝑀𝑝𝑎\displaystyle\mathcal{L}^{G}(p(x))=\mathcal{L}\left(\frac{p(x)-p(a)}{x-a}%\right)+Mp(a).caligraphic_L start_POSTSUPERSCRIPT italic_G end_POSTSUPERSCRIPT ( italic_p ( italic_x ) ) = caligraphic_L ( divide start_ARG italic_p ( italic_x ) - italic_p ( italic_a ) end_ARG start_ARG italic_x - italic_a end_ARG ) + italic_M italic_p ( italic_a ) .

If M0𝑀0M\neq 0italic_M ≠ 0 and a𝑎aitalic_a does not belong to the support of the measure for n(x)subscript𝑛𝑥\mathbb{P}_{n}(x)blackboard_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ), then there exists a sequence of orthogonal polynomials known as Geronimus polynomials with respect to the linear functional Gsuperscript𝐺\mathcal{L}^{G}caligraphic_L start_POSTSUPERSCRIPT italic_G end_POSTSUPERSCRIPT.

Theorem 2.

[12]Let 𝒢n(x;a)subscript𝒢𝑛𝑥𝑎\mathcal{G}_{n}(x;a)caligraphic_G start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; italic_a ) denote the monic polynomial associated with the canonical Geronimus transformation at a certain point a𝑎aitalic_a. The monic polynomial 𝒢n+1Q(x;a)superscriptsubscript𝒢𝑛1𝑄𝑥𝑎\mathcal{G}_{n+1}^{Q}(x;a)caligraphic_G start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT ( italic_x ; italic_a ) of degree n+1𝑛1n+1italic_n + 1 is a non-trivial quasi-Geronimus polynomial of order one if and only if there exists a sequence of non-zero constants χnsubscript𝜒𝑛\chi_{n}italic_χ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, such that:

𝒢n+1Q(x;a)=𝒢n+1(x;a)+χn+1𝒢n(x;a).superscriptsubscript𝒢𝑛1𝑄𝑥𝑎subscript𝒢𝑛1𝑥𝑎subscript𝜒𝑛1subscript𝒢𝑛𝑥𝑎\displaystyle\mathcal{G}_{n+1}^{Q}(x;a)=\mathcal{G}_{n+1}(x;a)+\chi_{n+1}%\mathcal{G}_{n}(x;a).caligraphic_G start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT ( italic_x ; italic_a ) = caligraphic_G start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT ( italic_x ; italic_a ) + italic_χ start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT caligraphic_G start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; italic_a ) .

The orthogonality of quasi-type kernel polynomial of order one under certain assumptions on χnsubscript𝜒𝑛\chi_{n}italic_χ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT is discussed in [12, Propostion 1].

3.1. Jacobi polynomials

When applying the canonical Geronimus transformation to the Jacobi weight, w(x)=(1x)α(1+x)β𝑤𝑥superscript1𝑥𝛼superscript1𝑥𝛽w(x)=(1-x)^{\alpha}(1+x)^{\beta}italic_w ( italic_x ) = ( 1 - italic_x ) start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT ( 1 + italic_x ) start_POSTSUPERSCRIPT italic_β end_POSTSUPERSCRIPT, with α>1𝛼1\alpha>-1italic_α > - 1 and β>1𝛽1\beta>-1italic_β > - 1, using a=1𝑎1a=-1italic_a = - 1 and M=2α+β𝔹(α+1,β)𝑀superscript2𝛼𝛽𝔹𝛼1𝛽M=2^{\alpha+\beta}\mathbb{B}(\alpha+1,\beta)italic_M = 2 start_POSTSUPERSCRIPT italic_α + italic_β end_POSTSUPERSCRIPT blackboard_B ( italic_α + 1 , italic_β ), where 𝔹(,)𝔹\mathbb{B}(\cdot,\cdot)blackboard_B ( ⋅ , ⋅ ) denotes the Beta function, the resulting transformed weight becomes w~(x)=(1x)α(1+x)β1~𝑤𝑥superscript1𝑥𝛼superscript1𝑥𝛽1\tilde{w}(x)=(1-x)^{\alpha}(1+x)^{\beta-1}over~ start_ARG italic_w end_ARG ( italic_x ) = ( 1 - italic_x ) start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT ( 1 + italic_x ) start_POSTSUPERSCRIPT italic_β - 1 end_POSTSUPERSCRIPT for α>1𝛼1\alpha>-1italic_α > - 1 and β>0𝛽0\beta>0italic_β > 0. Therefore, the orthogonal polynomial corresponding to the Geronimus Jacobi weight is again the Jacobi polynomial with parameter (α,β1)𝛼𝛽1(\alpha,\beta-1)( italic_α , italic_β - 1 ). Denoted by 𝒢n(x;1):=𝒫n(α,β1)(x)assignsubscript𝒢𝑛𝑥1superscriptsubscript𝒫𝑛𝛼𝛽1𝑥\mathcal{G}_{n}(x;-1):=\mathcal{P}_{n}^{(\alpha,\beta-1)}(x)caligraphic_G start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; - 1 ) := caligraphic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_α , italic_β - 1 ) end_POSTSUPERSCRIPT ( italic_x ), this polynomial can be obtained using a recurrence relation with recurrence parameters:

λn+1gsubscriptsuperscript𝜆𝑔𝑛1\displaystyle\lambda^{g}_{n+1}italic_λ start_POSTSUPERSCRIPT italic_g end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT=4n(n+α)(n+β1)(n+α+β1)(2n+α+β1)2(2n+α+β)(2n+α+β2),absent4𝑛𝑛𝛼𝑛𝛽1𝑛𝛼𝛽1superscript2𝑛𝛼𝛽122𝑛𝛼𝛽2𝑛𝛼𝛽2\displaystyle=\frac{4n(n+\alpha)(n+\beta-1)(n+\alpha+\beta-1)}{(2n+\alpha+%\beta-1)^{2}(2n+\alpha+\beta)(2n+\alpha+\beta-2)},= divide start_ARG 4 italic_n ( italic_n + italic_α ) ( italic_n + italic_β - 1 ) ( italic_n + italic_α + italic_β - 1 ) end_ARG start_ARG ( 2 italic_n + italic_α + italic_β - 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 2 italic_n + italic_α + italic_β ) ( 2 italic_n + italic_α + italic_β - 2 ) end_ARG ,
cn+1gsubscriptsuperscript𝑐𝑔𝑛1\displaystyle c^{g}_{n+1}italic_c start_POSTSUPERSCRIPT italic_g end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT=(β1)2α2(2n+α+β1)(2n+α+β+1).absentsuperscript𝛽12superscript𝛼22𝑛𝛼𝛽12𝑛𝛼𝛽1\displaystyle=\frac{(\beta-1)^{2}-\alpha^{2}}{(2n+\alpha+\beta-1)(2n+\alpha+%\beta+1)}.= divide start_ARG ( italic_β - 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ( 2 italic_n + italic_α + italic_β - 1 ) ( 2 italic_n + italic_α + italic_β + 1 ) end_ARG .

The quasi-Geronimus Jacobi polynomial of order one is defined as:

𝒞nQ(x;1)=𝒫n(α,β1)(x)+χn𝒫n1(α,β1)(x),subscriptsuperscript𝒞𝑄𝑛𝑥1subscriptsuperscript𝒫𝛼𝛽1𝑛𝑥subscript𝜒𝑛subscriptsuperscript𝒫𝛼𝛽1𝑛1𝑥\displaystyle\mathcal{C}^{Q}_{n}(x;-1)=\mathcal{P}^{(\alpha,\beta-1)}_{n}(x)+%\chi_{n}\mathcal{P}^{(\alpha,\beta-1)}_{n-1}(x),caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; - 1 ) = caligraphic_P start_POSTSUPERSCRIPT ( italic_α , italic_β - 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ) + italic_χ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT caligraphic_P start_POSTSUPERSCRIPT ( italic_α , italic_β - 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT ( italic_x ) ,(3.1)

which becomes orthogonal if χnsubscript𝜒𝑛\chi_{n}italic_χ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT satisfies the following non-linear difference equation:

(β1)2α2(α+β+2n3)(α+β+2n1)+(β1)2α2(α+β+2n1)(α+β+2n+1)+χnχn+1superscript𝛽12superscript𝛼2𝛼𝛽2𝑛3𝛼𝛽2𝑛1superscript𝛽12superscript𝛼2𝛼𝛽2𝑛1𝛼𝛽2𝑛1subscript𝜒𝑛subscript𝜒𝑛1\displaystyle-\frac{(\beta-1)^{2}-\alpha^{2}}{(\alpha+\beta+2n-3)(\alpha+\beta%+2n-1)}+\frac{(\beta-1)^{2}-\alpha^{2}}{(\alpha+\beta+2n-1)(\alpha+\beta+2n+1)%}+\chi_{n}-\chi_{n+1}- divide start_ARG ( italic_β - 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ( italic_α + italic_β + 2 italic_n - 3 ) ( italic_α + italic_β + 2 italic_n - 1 ) end_ARG + divide start_ARG ( italic_β - 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ( italic_α + italic_β + 2 italic_n - 1 ) ( italic_α + italic_β + 2 italic_n + 1 ) end_ARG + italic_χ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT - italic_χ start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT
1χn4n(α+n)(β+n1)(α+β+n1)(α+β+2n2)(α+β+2n1)2(α+β+2n)1subscript𝜒𝑛4𝑛𝛼𝑛𝛽𝑛1𝛼𝛽𝑛1𝛼𝛽2𝑛2superscript𝛼𝛽2𝑛12𝛼𝛽2𝑛\displaystyle\hskip 28.45274pt-\frac{1}{\chi_{n}}\frac{4n(\alpha+n)(\beta+n-1)%(\alpha+\beta+n-1)}{(\alpha+\beta+2n-2)(\alpha+\beta+2n-1)^{2}(\alpha+\beta+2n)}- divide start_ARG 1 end_ARG start_ARG italic_χ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG divide start_ARG 4 italic_n ( italic_α + italic_n ) ( italic_β + italic_n - 1 ) ( italic_α + italic_β + italic_n - 1 ) end_ARG start_ARG ( italic_α + italic_β + 2 italic_n - 2 ) ( italic_α + italic_β + 2 italic_n - 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_α + italic_β + 2 italic_n ) end_ARG
+1χn14(n1)(α+n1)(β+n2)(α+β+n2)(α+β+2n4)(α+β+2n3)2(α+β+2n2)=0.1subscript𝜒𝑛14𝑛1𝛼𝑛1𝛽𝑛2𝛼𝛽𝑛2𝛼𝛽2𝑛4superscript𝛼𝛽2𝑛32𝛼𝛽2𝑛20\displaystyle\hskip 85.35826pt+\frac{1}{\chi_{n-1}}\frac{4(n-1)(\alpha+n-1)(%\beta+n-2)(\alpha+\beta+n-2)}{(\alpha+\beta+2n-4)(\alpha+\beta+2n-3)^{2}(%\alpha+\beta+2n-2)}=0.+ divide start_ARG 1 end_ARG start_ARG italic_χ start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT end_ARG divide start_ARG 4 ( italic_n - 1 ) ( italic_α + italic_n - 1 ) ( italic_β + italic_n - 2 ) ( italic_α + italic_β + italic_n - 2 ) end_ARG start_ARG ( italic_α + italic_β + 2 italic_n - 4 ) ( italic_α + italic_β + 2 italic_n - 3 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_α + italic_β + 2 italic_n - 2 ) end_ARG = 0 .(3.2)

The two possible solutions to (3.1) are as follows:

Solution 1. It is easy to see that

χn=2(α+n)(n+α+β1)(2n+α+β1)(2n+α+β2),subscript𝜒𝑛2𝛼𝑛𝑛𝛼𝛽12𝑛𝛼𝛽12𝑛𝛼𝛽2\displaystyle\chi_{n}=-\frac{2(\alpha+n)(n+\alpha+\beta-1)}{(2n+\alpha+\beta-1%)(2n+\alpha+\beta-2)},italic_χ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = - divide start_ARG 2 ( italic_α + italic_n ) ( italic_n + italic_α + italic_β - 1 ) end_ARG start_ARG ( 2 italic_n + italic_α + italic_β - 1 ) ( 2 italic_n + italic_α + italic_β - 2 ) end_ARG ,(3.3)

solves the non-linear difference equation (3.1).Thus, the quasi-Geronimus Jacobi polynomial of order one becomes orthogonal with the χnsubscript𝜒𝑛\chi_{n}italic_χ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT defined in (3.3). Moreover, the recurrence parameters for obtaining 𝒞nQ(x;1)subscriptsuperscript𝒞𝑄𝑛𝑥1\mathcal{C}^{Q}_{n}(x;-1)caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; - 1 ) are given by:

λn+1qgsubscriptsuperscript𝜆𝑞𝑔𝑛1\displaystyle\lambda^{qg}_{n+1}italic_λ start_POSTSUPERSCRIPT italic_q italic_g end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT=4(n1)(n+α)(n+β2)(n+α+β1)(2n+α+β2)2(2n+α+β1)(2n+α+β3),absent4𝑛1𝑛𝛼𝑛𝛽2𝑛𝛼𝛽1superscript2𝑛𝛼𝛽222𝑛𝛼𝛽12𝑛𝛼𝛽3\displaystyle=\frac{4(n-1)(n+\alpha)(n+\beta-2)(n+\alpha+\beta-1)}{(2n+\alpha+%\beta-2)^{2}(2n+\alpha+\beta-1)(2n+\alpha+\beta-3)},= divide start_ARG 4 ( italic_n - 1 ) ( italic_n + italic_α ) ( italic_n + italic_β - 2 ) ( italic_n + italic_α + italic_β - 1 ) end_ARG start_ARG ( 2 italic_n + italic_α + italic_β - 2 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 2 italic_n + italic_α + italic_β - 1 ) ( 2 italic_n + italic_α + italic_β - 3 ) end_ARG ,
cn+1qgsubscriptsuperscript𝑐𝑞𝑔𝑛1\displaystyle c^{qg}_{n+1}italic_c start_POSTSUPERSCRIPT italic_q italic_g end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT=(βα2)(β+α)(2n+α+β2)(2n+α+β).absent𝛽𝛼2𝛽𝛼2𝑛𝛼𝛽22𝑛𝛼𝛽\displaystyle=\frac{(\beta-\alpha-2)(\beta+\alpha)}{(2n+\alpha+\beta-2)(2n+%\alpha+\beta)}.= divide start_ARG ( italic_β - italic_α - 2 ) ( italic_β + italic_α ) end_ARG start_ARG ( 2 italic_n + italic_α + italic_β - 2 ) ( 2 italic_n + italic_α + italic_β ) end_ARG .

Next, we demonstrate the interlacing property between the zeros of the quasi-type kernel Jacobi polynomial of order one and the quasi-Geronimus Jacobi polynomial of order one.

Zeros of 𝒞nQ(x;1)subscriptsuperscript𝒞𝑄𝑛𝑥1\mathcal{C}^{Q}_{n}(x;-1)caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; - 1 )Zeros of 𝒢nQ(x;1)subscriptsuperscript𝒢𝑄𝑛𝑥1\mathcal{G}^{Q}_{n}(x;-1)caligraphic_G start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; - 1 )
n=6𝑛6n=6italic_n = 6, α=1𝛼1\alpha=1italic_α = 1, β=0.5𝛽0.5\beta=0.5italic_β = 0.5n=5𝑛5n=5italic_n = 5, α=2𝛼2\alpha=2italic_α = 2, β=1𝛽1\beta=1italic_β = 1n=6𝑛6n=6italic_n = 6, α=1𝛼1\alpha=1italic_α = 1, β=0.5𝛽0.5\beta=0.5italic_β = 0.5n=5𝑛5n=5italic_n = 5, α=2𝛼2\alpha=2italic_α = 2, β=1𝛽1\beta=1italic_β = 1
-0.810015-0.728794-0.967813-0.915694
-0.47303-0.325544-0.722321-0.580566
-0.047073-0.147611-0.292037-0.071692
0.3892570.5990350.2166970.477044
0.75567610.6785181
1-1-

A common zero at the end point of the support of measure for the quasi-natured spectrally transformed polynomials (7)
A common zero at the end point of the support of measure for the quasi-natured spectrally transformed polynomials (8)

For α=1𝛼1\alpha=1italic_α = 1 and β=0.5𝛽0.5\beta=0.5italic_β = 0.5, Table 7 and Figure 7 demonstrate that the zeros of 𝒢6Q(x;1)subscriptsuperscript𝒢𝑄6𝑥1\mathcal{G}^{Q}_{6}(x;-1)caligraphic_G start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT ( italic_x ; - 1 ) and 𝒞6Q(x;1)subscriptsuperscript𝒞𝑄6𝑥1\mathcal{C}^{Q}_{6}(x;-1)caligraphic_C start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT ( italic_x ; - 1 ) interlace inside the support of the measure. Similarly, interlacing occurs with parameter α=2𝛼2\alpha=2italic_α = 2 and β=1𝛽1\beta=1italic_β = 1, as shown in Table 7 and Figure 8. In this work, computations of the zeros and graphical representations, showcasing their interlacing properties, are conducted using the Mathematica®superscriptMathematica®\text{Mathematica}^{\text{\textregistered}}Mathematica start_POSTSUPERSCRIPT ® end_POSTSUPERSCRIPT software.

Solution 2. To recover the Jacobi polynomial with parameter (α,β)𝛼𝛽(\alpha,\beta)( italic_α , italic_β ) from the quasi-Geronimus Jacobi polynomial of order one, we find that:

χn=2n(α+n)(2n+α+β1)(2n+α+β2),subscript𝜒𝑛2𝑛𝛼𝑛2𝑛𝛼𝛽12𝑛𝛼𝛽2\displaystyle\chi_{n}=\frac{2n(\alpha+n)}{(2n+\alpha+\beta-1)(2n+\alpha+\beta-%2)},italic_χ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = divide start_ARG 2 italic_n ( italic_α + italic_n ) end_ARG start_ARG ( 2 italic_n + italic_α + italic_β - 1 ) ( 2 italic_n + italic_α + italic_β - 2 ) end_ARG ,(3.4)

also satisfies the equation (3.1). With this χnsubscript𝜒𝑛\chi_{n}italic_χ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, the polynomial 𝒢nQ(x;1)subscriptsuperscript𝒢𝑄𝑛𝑥1\mathcal{G}^{Q}_{n}(x;-1)caligraphic_G start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; - 1 ) becomes orthogonal and recovers the original orthogonal polynomial. Specifically,

𝒢nQ(x;1):=𝒫n(α,β)(x)=𝒫n(α,β1)(x)+2n(α+n)(2n+α+β1)(2n+α+β2)𝒫n1(α,β1)(x).assignsubscriptsuperscript𝒢𝑄𝑛𝑥1subscriptsuperscript𝒫𝛼𝛽𝑛𝑥subscriptsuperscript𝒫𝛼𝛽1𝑛𝑥2𝑛𝛼𝑛2𝑛𝛼𝛽12𝑛𝛼𝛽2subscriptsuperscript𝒫𝛼𝛽1𝑛1𝑥\displaystyle\mathcal{G}^{Q}_{n}(x;-1):=\mathcal{P}^{(\alpha,\beta)}_{n}(x)=%\mathcal{P}^{(\alpha,\beta-1)}_{n}(x)+\frac{2n(\alpha+n)}{(2n+\alpha+\beta-1)(%2n+\alpha+\beta-2)}\mathcal{P}^{(\alpha,\beta-1)}_{n-1}(x).caligraphic_G start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ; - 1 ) := caligraphic_P start_POSTSUPERSCRIPT ( italic_α , italic_β ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ) = caligraphic_P start_POSTSUPERSCRIPT ( italic_α , italic_β - 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ) + divide start_ARG 2 italic_n ( italic_α + italic_n ) end_ARG start_ARG ( 2 italic_n + italic_α + italic_β - 1 ) ( 2 italic_n + italic_α + italic_β - 2 ) end_ARG caligraphic_P start_POSTSUPERSCRIPT ( italic_α , italic_β - 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT ( italic_x ) .(3.5)

The recurrence parameter are given by:

λn+1qg=λn+1,cn+1qg=cn+1.formulae-sequencesubscriptsuperscript𝜆𝑞𝑔𝑛1subscript𝜆𝑛1subscriptsuperscript𝑐𝑞𝑔𝑛1subscript𝑐𝑛1\displaystyle\lambda^{qg}_{n+1}=\lambda_{n+1},c^{qg}_{n+1}=c_{n+1}.italic_λ start_POSTSUPERSCRIPT italic_q italic_g end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT = italic_λ start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT , italic_c start_POSTSUPERSCRIPT italic_q italic_g end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT = italic_c start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT .

4. Conclusion

This work extends the investigations to a broader scope, encompassing Jacobi polynomials 𝒫n(α,β)(x)superscriptsubscript𝒫𝑛𝛼𝛽𝑥\mathcal{P}_{n}^{(\alpha,\beta)}(x)caligraphic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_α , italic_β ) end_POSTSUPERSCRIPT ( italic_x ) under both Christoffel and Geronimus transformations, along with an analysis of the zero properties of the corresponding orthogonal and quasi-spectral polynomials. We observe that at most one zero of the quasi-Christoffel and quasi-Geronimus polynomials lies outside the support of the measure. Upon achieving orthogonality of the quasi-Geronimus and quasi-Christoffel for Jacobi and Laguerre polynomials, we demonstrate that the one zero previously outside the support of the measure now resides on the boundary of the support of the measure. In another scenario, we restore the original Jacobi and Laguerre polynomials by determining the explicit values of parameters responsible for establishing orthogonality of quasi-Geronimus and quasi-Christoffel polynomials of order one. This indicates that, in this particular case, the zeros of the polynomials are located within the support of the measure. We conclude this manuscript by presenting an open problem concerning the behavior of zeros.

Problem 1.

Are there any additional solutions to the nonlinear difference equations outlined in (2.5), (2.2) and (3.1)? Can we characterize the polynomials derived from achieving orthogonality of quasi-spectral polynomials of order one in such a way that one zero lies on the boundary of the support of the measure?

Acknowledgments. The second author acknowledges the support from Project No. NBHM/RP-1/2019 of National Board for Higher Mathematics (NBHM), DAE, Government of India.

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A common zero at the end point of the support of measure for the quasi-natured spectrally transformed polynomials (2024)

FAQs

What is the zero of the polynomial? ›

Zeros of a polynomial can be defined as the points where the polynomial becomes zero as a whole. A polynomial having value zero (0) is called zero polynomial. The degree of a polynomial is the highest power of the variable x.

What is the meaning of zero polynomial? ›

Zero polynomial is any polynomial in which all the variables have a coefficient equal to zero. Therefore, the value of a zero polynomial is 0. Hence, like there are other constants in a polynomial, 0 can also be considered a constant polynomial.

What is the number of zeros in a polynomial? ›

The number of zeros of a polynomial depends on the degree of the polynomial expression y = f(x). For a linear equation in one variable, we have only one root. For a quadratic and cubic polynomial, we have two and three zeros of a polynomial respectively.

What is an example of a zero polynomial function? ›

A zero polynomial is a type of polynomial where the coefficients of the variables are equal to 0. The constant polynomial f(x) = 0. The general form is g(x) = ax + b where a ≠ 0. For example, f(x) = x -4, g(x) = 14x, etc.

What is the real zero of the polynomial? ›

The real zeros of a polynomial are found when setting a polynomial P ( X ) = 0 . The real zeros will come from factoring the polynomial and setting it equal to zero. This cannot include imaginary solutions.

What is a polynomial to the 0 degree called? ›

Types of Polynomials Based on its Degree
DegreePolynomial Name
Degree 0Constant Polynomial
Degree 1Linear Polynomial
Degree 2Quadratic Polynomial
Degree 3Cubic Polynomial
1 more row

How to find a zero of a polynomial? ›

We can use the Factor Theorem to completely factor a polynomial into the product of n factors. Once the polynomial has been completely factored, we can easily determine the zeros of the polynomial. According to the Factor Theorem, k is a zero of f(x) if and only if (x−k) is a factor of f(x).

What do the zeros of a polynomial represent? ›

The zeros of a polynomial p(x) are all the x-values that make the polynomial equal to zero. They are interesting to us for many reasons, one of which is that they tell us about the x-intercepts of the polynomial's graph. We will also see that they are directly related to the factors of the polynomial.

Which polynomial has zeros? ›

i) A linear polynomial has only one zero. ii ) A quadratic polynomial will have two zeroes.

Why is 0 a polynomial function? ›

Like any constant value, the value 0 can be considered as a (constant) polynomial, called the zero polynomial. It has no nonzero terms, and so, strictly speaking, it has no degree either. As such, its degree is usually undefined.

Is zero a constant polynomial? ›

Constant Polynomial

It has no variables, only constants. For example: f(x) = 6, g(x) = -22 , h(y) = 5/2 etc are constant polynomials. In general f(x) = c is a constant polynomial. The constant polynomial 0 or f(x) = 0 is called the zero polynomial.

What are rational zeros of a polynomial? ›

Given a polynomial function f, The rational roots, also called rational zeros, of f are the rational number solutions of the equation f(x) = 0. Solutions that are not rational numbers are called irrational roots or irrational zeros.

Is the zero of a polynomial always 1? ›

The number of zeros of the polynomial depends on the degree of the polynomial equation. Zero of a polynomial can be any real number. Therefore, the statement is false.

Can the zero of a polynomial be 0? ›

Zero of a polynomial is always 0.

Which polynomial degree is zero? ›

The polynomial 0 has no terms at all and is called a zero polynomial. Because the zero polynomial has no non-zero terms, the zero polynomial has no degree.

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