Orthogonal polynomials on the real line corresponding to a perturbed chain sequence

TL;DR

This paper investigates how perturbations of chain sequences influence orthogonal polynomials on the real line, revealing connections to measure transformations and illustrating with Laguerre polynomials.

Contribution

It introduces a specific perturbation of chain sequences affecting orthogonal polynomials on [0,∞) and links these to measure transformations, with applications to Laguerre polynomials.

Findings

Perturbations relate to transformations of symmetric measures.

The study reveals kernel polynomial consequences of chain sequence disturbances.

Application to generalized Laguerre polynomials demonstrates the theory.

Abstract

In recent years, chain sequences and their perturbations have played a significant role in characterising the orthogonal polynomials both on the real line as well as on the unit circle. In this note, a particular disturbance of the chain sequence related to orthogonal polynomials having their true interval of orthogonality as a subset of $[0,\infty)$ is studied leading to an important consequence related to the kernel polynomials. Such perturbations are shown to be related to transformations of symmetric measures. An illustration using the generalized Laguerre polynomials is also provided.

Equations

$$p_n(x)=c_{n,n}{x}^n+c_{n,n-1}{x}^{n-1}+\cdots +c_{n,1}x+c_{n,0},c_{n,n}>0,$$

$$\mathcal{L}[p_n(t)p_m(t)]={\delta }_{m,n},m,n\ge 0.$$

$$a_{n+1}{p}_{n+1}(x)=(x-b_{n+1})p_n(x)-a_n{p}_{n-1}(x),n\ge 0,$$

$$a_n=\frac{c_{n-1,n-1}}{c_{n,n}}\text{mbox}and{b}_{n+1}=\frac{c_{n,n-1}}{c_{n,n}}-\frac{c_{n+1,n}}{c_{n+1,n+1}},n\ge 1.$$

$${\mathcal{P}}_{n+1}(x)=(x-b_{n+1}){\mathcal{P}}_n(x)-a_n^2{\mathcal{P}}_{n-1}(x),n\ge 0,$$

$$\displaystyle J=\left(\begin{array}[]{ccccc}b_{1}&1&0&0&\cdots\\ a_{1}^{2}&b_{2}&1&0&\cdots\\ 0&a_{2}^{2}&b_{3}&1&\cdots\\ 0&0&a_{3}^{2}&b_{4}&\cdots\\ \vdots&\vdots&\vdots&\vdots&\ddots\\ \end{array}\right)$$

$$z_{n+1}=(x-b_{n+1})z_n-a_n^2{z}_{n-1},n\ge 1,$$

$$\mathcal{S}(x)={\int }_{\mathbb{E}}\frac{d\mu (t)}{x-t},$$

$$\displaystyle\int_{\mathbb{E}}\dfrac{d\mu(t)}{x-t}=\dfrac{1}{x-b_{1}}\begin{array}[]{cc}\\ $-$\end{array}\dfrac{a_{1}^{2}}{x-b_{2}}\begin{array}[]{cc}\\ $-$\end{array}\dfrac{a_{2}^{2}}{x-b_{3}}\begin{array}[]{cc}\\ $-$\end{array}\ldots$$

$$\mathcal{S}(x)=\frac{{\mu }_0}{x}+\frac{{\mu }_1}{x^2}+\frac{{\mu }_2}{x^3}+\cdots$$

$$\displaystyle\mathcal{F}_{n}(x)=\dfrac{1}{1}\begin{array}[]{cc}\\ $-$\end{array}\dfrac{d_{1}(x)}{1}\begin{array}[]{cc}\\ $-$\end{array}\dfrac{d_{2}(x)}{1}\begin{array}[]{cc}\\ $-$\end{array}\ldots\begin{array}[]{cc}\\ $-$\end{array}\dfrac{d_{n-1}(x)}{1}$$

$${\mathcal{F}}_n(x)=(x-b_1)\frac{{\mathcal{P}}_n^{(1)}(x)}{{\mathcal{P}}_n(x)}\text{mbox}and{d}_n(x)=\frac{a_n^2}{(x-b_n)(x-b_{n+1})}.$$

$$d_n(t)=\frac{{\mathcal{P}}_n(t)}{(t-b_n){\mathcal{P}}_{n-1}(t)}\left[1-\frac{{\mathcal{P}}_{n+1}(t)}{(t-b_{n+1}){\mathcal{P}}_n(t)}\right],n\ge 1.$$

$$M_k=\sup \{g_k:\{g_n\}\text{mbox}anyotherparametersequenceof\{d_n\}\}$$

$${\mathcal{S}}_n(x)=x{\mathcal{S}}_{n-1}(x)-{\gamma }_n^{\varphi }{\mathcal{S}}_{n-2}(x),n\ge 1,$$

$$4k{\gamma }_{n+1}^{{\varphi }_1}=(l_n-1)(l_{n-1}+1)\text{mbox}and4k{\gamma }_{n+1}^{{\varphi }_2}=(l_n-1)(l_{n+1}+1)$$

$${\int }_0^{\infty }{t}^{-n+s}{\mathcal{B}}_n(t)d\psi (t)=0,0\le s\le n-1,$$

$${\mathcal{B}}_{n+1}(t)=(t-{\beta }_{n+1}^{\psi }){\mathcal{B}}_n(t)-{\alpha }_{n+1}^{\psi }t{\mathcal{B}}_{n-1}(t),n\ge 1,$$

$$\frac{d\psi (t)}{\sqrt{t}}=\frac{d\psi ({\beta }^2/t)}{\beta /\sqrt{t}},t\in ({\beta }^2/b,b).$$

$${\alpha }_{n+1}^{{\psi }_2}=\beta (l_n-1)(l_{n+1}+1),n\ge 1.$$

$${\alpha }_{n+1}^{{\psi }_1}=\beta (l_n-1)(l_{n-1}+1),n\ge 1.$$

$$d{\psi }_1(t)=A_1\frac{t}{t+\beta }d{\varphi }_1(x(t)),d{\psi }_2(t)=A_2\frac{t}{t+\beta }d{\varphi }_2(x(t)),$$

$$l_1$$

$$l_2$$

$$\{{\gamma }_1^{{\varphi }_2},{\gamma }_2^{{\varphi }_2},{\gamma }_3^{{\varphi }_2},{\gamma }_4^{{\varphi }_2},\cdots \}=\{{\gamma }_2^{{\varphi }_1},{\gamma }_1^{{\varphi }_1},{\gamma }_4^{{\varphi }_1},{\gamma }_3^{{\varphi }_1},\cdots \}$$

$${\omega }_n(t)=\frac{a_n^2}{(t-b_n)(t-b_{n+1})},n\ge 1,$$

$$(1-g_{n-1})g_n=\frac{a_n^2}{b_n{b}_{n+1}}={\omega }_n(0),n\ge 1.$$

$$b_{n+1}={\gamma }_{2n+1}+{\gamma }_{2n+2},n\ge 0\text{mbox}and{a}_n^2={\gamma }_{2n}{\gamma }_{2n+1},n\ge 1.$$

$${\tilde{\omega }}_n(0)=\frac{{\gamma }_{2n-1}{\gamma }_{2n+2}}{b_n{b}_{n+1}}=(1-k_{n-1})k_n,n\ge 2,$$

$${\hat{\vartheta }}_n(0)=(1-k_{n-1}^{\prime })k_n^{\prime }=\frac{{\gamma }_{2n-1}{\gamma }_{2n+2}}{b_n{b}_{n+1}},n\ge 1.$$

$${\mathcal{S}}_{2n}(x)={\mathcal{P}}_n(x^2),{\mathcal{S}}_{2n+1}(x)=x{\mathcal{K}}_n(x^2).$$

Full text

1. Preliminaries

Let $\mathcal{L}$ be a moment functional defined on the vector space of polynomials with real coefficients. The quantities $\mu_{k}=\mathcal{L}[x^{k}]$, $k\geq 0$ are called the moments of order $k$ and are used to construct the Hankel matrix $H=(\mu_{i+j})_{i,j=0}^{\infty}$. In case the principal submatrices $H_{n}$ are non-singular, $\mathcal{L}$ is said to be quasi-definite and there always exists a sequence of polynomials $\{p_{n}(x)\}_{n=0}^{\infty}$, where for $n\geq 0$,

[TABLE]

satisfying the property,

[TABLE]

The sequence $\{p_{n}(x)\}_{n=0}^{\infty}$ is said to be orthonormal with respect to $\mathcal{L}$.

Following the notation used in [16, Page 12], the sequence $\{p_{n}(x)\}_{n=0}^{\infty}$ satisfies the three term recurrence relation (the so called Favard’s Theorem),

[TABLE]

where $p_{-1}(x)$ and $p_{0}(x)$ are pre-defined. Further, $b_{n}\in\mathbb{R}$ and $a_{n}>0$, $n\geq 1$ and are related to the coefficients of $p_{n}(x)$ as [10],

[TABLE]

Moreover, though $a_{0}$ is arbitrary, it is usually taken to be equal to $\mathcal{L}[1]=\mu_{0}>0$ for convenience. It may be noted that the recurrence coefficients $\{b_{n+1}\}$ and $\{a_{n}\}$ satisfy the above conditions if and only if $\mathcal{L}$ is positive definite, that is, if $\mathcal{L}[r(x)]>0$ for every polynomial $r(x)$ such that $r(x)>0$ for all $x\in\mathbb{R}$.

In many cases, the polynomials $p_{n}(x)$ are normalized to make their leading coefficient equal to one. These monic orthogonal polynomials denoted as $\{\mathcal{P}_{n}(x)\}_{n=0}^{\infty}$ and defined by $c_{n,n}\mathcal{P}_{n}(x)=p_{n}(x)$, $n\geq 0$, satisfy the recurrence relation

[TABLE]

with $\mathcal{P}_{-1}(x)=0$ and $\mathcal{P}_{0}(x)=1$. In the sequel, these will be called an orthogonal polynomials sequence (OPS). The system (1.2) is usually written in the more precise form: $x\mathbb{\bar{P}}=J\mathbb{\bar{P}}$, where $\mathbb{\bar{P}}=[\mathcal{P}_{0}(x)\,\,\mathcal{P}_{1}(x)\,\,\mathcal{P}_{2}(x)\,\,\cdots]^{T}$ and

[TABLE]

is called the monic Jacobi matrix associated with the orthogonal polynomials generated by (1.2). The polynomials defined as $\mathcal{L}[(\mathcal{P}_{n}(y)-\mathcal{P}_{n}(t))/(y-t)]$, $n\geq 0$, are called the associated polynomials associated with $\mu$ and constitute a second solution of the recurrence relation

[TABLE]

with the modified initial conditions $z_{0}=0$ and $z_{1}=1$. Following the notation used in [9], these polynomials will be denoted as $\mathcal{P}^{(1)}_{n}(x)$, $n\geq 0$. By induction, it can be seen that $\mathcal{P}^{(1)}_{n}(x)$, $n\geq 1$ is a monic polynomial of degree $n-1$.

It is interesting to note that if $\mathcal{L}$ is positive definite, it can be used to define an inner product on the space of polynomials. In such a case, there exists a positive measure $\mu(t)$ supported on a subset $\mathbb{E}$ of the real line $\mathbb{R}$ with $\mathbb{\bar{E}}$ denoting the smallest closed interval containing this support. The linear functional now has the representation $\mathcal{L}[t^{k}]=\int_{\mathbb{E}}t^{k}d\mu(t)$. Further, the Stieltjes transform of the measure

[TABLE]

yields the Jacobi continued fraction expansion

[TABLE]

where the $\{a_{n}^{2}\}$ and $\{b_{n}\}$ are the same coefficients appearing in (1.2). The Stieltjes function $\mathcal{S}(x)$ plays a very fundamental role in the theory of orthogonal polynomials on the real line. It admits the power series expansion at infinity

[TABLE]

and hence acts as the generating function for the moments $\mu_{k}^{\prime}s$ associated with $d\mu(t)$. Further, the $n^{th}$ convergent of the continued fraction in the right hand side of (1.15) is easily seen to be $\mathcal{P}^{(1)}_{n}(x)/\mathcal{P}_{n}(x)$ and hence as $n\rightarrow\infty$, $\mathcal{P}^{(1)}_{n}(x)/\mathcal{P}_{n}(x)$ converges uniformly to $\frac{1}{\mu_{0}}\mathcal{S}(x)$ on compact subsets of $\mathbb{\bar{C}}\setminus\mathbb{\bar{E}}$ [4]. Here $\mathbb{\bar{C}}$ is the extended complex plane.

An equivalence transformation of the $n^{th}$ convergent of the continued fraction in (1.15) yields [4]

[TABLE]

where

[TABLE]

The quantities $d_{n}(t)$ can also be obtained, [9, Page 110], from the recurrence relation (1.2) as

[TABLE]

Such structures, called chain sequences have been studied in [21], followed by a systematic treatment in [9] (see also [11, Section 7.2]), particularly in the context of orthogonal polynomials on the real line.

Formally stated, a sequence $\{d_{n}\}_{n=1}^{\infty}$ is called a positive chain sequence if it can be expressed in terms of another sequence $\{g_{n}\}_{n=0}^{\infty}$ as $d_{n}=(1-g_{n-1})g_{n}$, $n\geq 1$. Here, $\{g_{n}\}_{n=0}^{\infty}$ called a parameter sequence of the chain sequence $\{d_{n}\}_{n=1}^{\infty}$ is such that $0\leq g_{0}<1$ and $0<g_{n}<1$, $n\geq 1$. The parameter sequence is in general not unique, and in such cases it is interesting to study the bounds for each parameter $g_{n}$, $n\geq 0$. The minimal parameter sequence $\{m_{n}\}_{n=0}^{\infty}$ of the positive chain sequence $\{d_{n}\}$ is defined as $m_{0}=0$, $0<m_{n}<1$, with $d_{n}=(1-m_{n-1})m_{n}$, $n\geq 1$. Thus, for instance, the minimal parameter sequence of the chain sequence $\{d_{n}(t)\}_{n=1}^{\infty}$ as defined in (1.16) is given by $m_{n}(t)=[1-\mathcal{P}_{n+1}(t)/(t-b_{n+1})\mathcal{P}_{n}(t)]$, $n\geq 0$. At the other end, there is the maximal parameter sequence $\{M_{n}\}_{n=0}^{\infty}$, whose parameters are defined as,

[TABLE]

where $\sup{}$ denotes the supremum of the set. It is known that $m_{k}\leq g_{k}\leq M_{k}$, $k\geq 0$, for any parameter sequence $\{g_{n}\}$ of $\{d_{n}\}$. In case, $M_{0}=m_{0}=0$, the parameter sequence is unique and $\{d_{n}\}$ is called a single parameter positive chain sequence, abbreviated as SPPCS in the sequel.

2. Symmetric Orthogonal Polynomials

A positive measure $d\phi(x)$ is called symmetric if it satisfies $d\phi(-x)=-d\phi(x)$. The monic polynomials $\mathcal{S}_{n}(x)$, $n\geq 0$, which are orthogonal with respect to $d\phi(x)$ satisfy

[TABLE]

with $\mathcal{S}_{-1}(x)=0$ and $\mathcal{S}_{0}(x)=1$.

In a series of research articles [3, 17, 19, 18], many properties of symmetric orthogonal polynomials have been discussed in the context of certain orthogonal Laurent polynomials. Particularly, if $d\phi_{1}(x)$ and $d\phi_{2}(x)$ are two different symmetric distributions related by the Christoffel transformation as $d\phi_{2}(x)=(1+kx^{2})d\phi_{1}(x)$, where $k$ is positive real constant, then

Theorem 2.1**.**

[19, Theorem 1]* Associated with $d\phi_{1}(x)$ and $d\phi_{2}(x)$ there exists a sequence of positive numbers $\{l_{n}\}$, with $l_{0}=1$ and $l_{n}>1$ for $n\geq 1$, such that*

[TABLE]

This result was established using the transformation $t(x)=(\sqrt{\alpha x^{2}+\beta}+\sqrt{\alpha}x)^{2}$ for $x\in(-\infty,\infty)$ and the monic polynomials $\{\mathcal{B}_{n}(t)\}_{n=0}^{\infty}$, uniquely defined by

[TABLE]

where $d\psi(t)$ is strong distribution on $(0,\infty)$. Note that if $x=d$ corresponds to $t=b$, the transformation maps $(-d,d)$ to $(\beta^{2}/b,b)\subseteq(0,\infty)$. For any strong distribution $d\psi(t)$ on $(0,\infty)$, the monic polynomials $\{\mathcal{B}_{n}(t)\}_{n=0}^{\infty}$ satisfy

[TABLE]

with $\mathcal{B}_{0}(t)=1$ and $\mathcal{B}_{1}(t)=t-\beta_{1}^{\psi}$. Further, the unique coefficients $\beta_{n+1}^{\psi}$, $n\geq 0$ and $\alpha_{n+1}^{\psi}$, $n\geq 1$ are all positive real numbers.

Two kinds of strong distributions play important role in this analysis. The first is the ScS$(a,b)$ distribution and the second is the S$\bar{c}$S$(a,b)$ distribution. A strong distribution $d\psi(t)$ with its support inside $(\beta^{2}/b,b)$ is called a ScS$(\beta^{2}/b,b)$ distribution if [19]

[TABLE]

A strong distribution $d\psi(t)$ with its support inside $(\beta^{2}/b,b)$ is called a S$\bar{c}$S$(\beta^{2}/b,b)$ distribution if $d\psi(t)=-d\psi(\beta^{2}/b)$, $t\in(\beta^{2}/b,b)$ [19]. Then it is known that $d\psi_{2}(t)$ is a ScS$(\beta^{2}/b,b)$ distribution if and only if $(t+\beta)d\psi_{0}(t)$ is a S$\bar{c}$S$(\beta^{2}/b,b)$ distribution. Further, if $l_{n}^{2}-1=\alpha^{\psi_{0}}_{n+1}/\beta_{n}^{\psi_{0}}$, then

[TABLE]

Conversely, $d\psi_{0}(t)$ is a S$\bar{c}$S$(\beta^{2}/b,b)$ distribution if and only if $\frac{t+\beta}{t}d\psi_{1}(t)$ is a ScS$(\beta^{2}/b,b)$ distribution and in this case

[TABLE]

Then choosing

[TABLE]

with $k=\alpha/\beta$ and appropriate choices of $A_{1}$ and $A_{2}$, the relations (2.1) are obtained.

In this note, we consider the particular case when the sequence $\{l_{n}\}$ satisfy $l_{1}\neq l_{2}$ and

[TABLE]

This sequence can be obtained from appropriate conditions imposed on the coefficients $\beta^{\psi}_{n+1}$ and $\alpha^{\psi}_{n+1}$, for instance, choosing the coefficients such that they satisfy $\frac{\alpha_{2}^{\psi_{0}}}{\beta_{1}^{\psi_{0}}}=\frac{\alpha_{5}^{\psi_{0}}}{\beta_{4}^{\psi_{0}}}$ and so on. Consequently, the coefficients associated with the Christoffel transformation mentioned above satisfy

[TABLE]

Thus it can be seen that transformations of the symmetric measures are related to transformations of strong distributions on $(0,\infty)$ which possess special properties like symmetry. In this note, we study this particular case of the transformation from the point of view of a perturbation of the chain sequences associated with polynomial sequences orthogonal on $(0,\infty)$ and some related consequences. We also provide an illustration using the Laguerre polynomials in the last section.

3. Perturbed chain sequences

The theory of chain sequences has been used to study many properties of a given orthogonal polynomial system on the real line, for instance, its true interval of orthogonality, which is the smallest closed interval that contains all the zeros of all the polynomials of such a system. Denoting the true interval of orthogonality of $\{\mathcal{P}_{n}(x)\}$ satisfying (1.2) as $[\xi,\eta]$ and

[TABLE]

the following are known to be equivalent, [11, Corollary 7.2.4],

1. (i)

$[\xi,\eta]$ is contained in $(a,b)$, 2. (ii)

$b_{n+1}\in(a,b)$ for $n\geq 0$ and both $\{\omega_{n}(a)\}$ and $\{\omega_{n}(b)\}$ are chain sequences.

We note that here $t\in(-\infty,\infty)\setminus\mathbb{\bar{E}}$. In particular, the true interval of orthogonality is a subset of $[0,\infty]$ if and only if $b_{n+1}>0$ for $n\geq 0$ and there are numbers $g_{n}$ such that $0\leq g_{0}<1$, $0<g_{n}<1$, $n\geq 1$, satisfying,

[TABLE]

As in [9], the sequence $\{g_{n}\}_{n=0}^{\infty}$ is constructed using another sequence $\{\gamma_{n}\}_{n=1}^{\infty}$, where $\gamma_{1}\geq 0$ and $\gamma_{n}>0$, $n\geq 2$. For details, the reader is referred to [9, Chapter 1, Theorems 9.1, 9.2], where it is shown that

[TABLE]

and the parameters are given by $g_{n}=\gamma_{2n+1}/b_{n+1}$, $n\geq 0$. Hence when $\gamma_{1}=0$, we obtain the minimal parameter sequence $\{m_{n}\}_{n=0}^{\infty}$.

Associated with the chain sequence $\{\omega_{n}(0)\}$, another sequence $\{\tilde{\omega}_{n}(0)\}$ arises in a very natural way. Defining $\tilde{\omega}_{1}(0)=(1-k_{0})k_{1}=\gamma_{4}/b_{2}$ and

[TABLE]

it can be seen that $\{\tilde{\omega}_{n}\}_{n=1}^{\infty}$ becomes a chain sequence with the minimal parameter sequence $\{k_{n}\}_{n=0}^{\infty}$ where $k_{0}=0$ and $k_{n}=1-g_{n}$, $n\geq 1$. Hence, we give the following definition which was given earlier in [2] with reference to orthogonal polynomials on the unit circle.

Definition 3.1**.**

Suppose $\{d_{n}\}_{n=1}^{\infty}$ is a chain sequence with $\{m_{n}\}_{n=0}^{\infty}$ as its minimal parameter sequence. Let $\{k_{n}\}_{n=0}^{\infty}$ be another sequence given by $k_{0}=0$ and $k_{n}=1-m_{n}$ for $n\geq 1$. Then the chain sequence $\{a_{n}\}_{n=1}^{\infty}$ having $\{k_{n}\}_{n=0}^{\infty}$ as its minimal parameter sequence is called as complementary chain sequence of $\{d_{n}\}_{n=1}^{\infty}$.

If $\gamma_{1}>0$, a non-minimal parameter sequence $\{g_{n}\}_{n=0}^{\infty}$ is obtained for the chain sequence $\{\omega_{n}(0)\}_{n=1}^{\infty}$. In this case, the associated chain sequence $\{\hat{\vartheta}_{n}(0)\}_{n=1}^{\infty}$, is defined as

[TABLE]

where $k_{n}^{\prime}=1-g_{n}$ for $n\geq 0$.

Definition 3.2**.**

Suppose $\{d_{n}\}_{n=1}^{\infty}$ is a chain sequence with $\{g_{n}\}_{n=0}^{\infty}$ as its non-minimal parameter sequence. Let $\{k_{n}^{\prime}\}_{n=0}^{\infty}$ be another sequence given by $k_{n}^{\prime}=1-g_{n}$ for $n\geq 0$. Then the chain sequence $\{a_{n}\}_{n=1}^{\infty}$ having $\{k_{n}^{\prime}\}_{n=0}^{\infty}$ as its parameter sequence is called as generalised complementary chain sequence of $\{d_{n}\}_{n=1}^{\infty}$.

It may be noted from the above two definitions that for a fixed chain sequence, while its complementary chain sequence is unique, its generalised complementary chain sequence need not be unique. In fact, a chain sequence will have as many generalised complementary chain sequences as its non-minimal parameter sequences. Naturally, the complementary chain sequence and all the generalised complementary chain sequences will coincide only for a SPPCS.

We would like to mention that the chain sequences $\{\tilde{\omega}_{n}(0)\}_{n=1}^{\infty}$ and $\{\hat{\vartheta}_{n}(0)\}_{n=1}^{\infty}$ have definite sources in the theory of orthogonal polynomials on the real line. To see this, first note that the symmetric polynomials $\{\mathcal{S}_{n}(x)\}$ satisfy the property $\mathcal{S}_{n}(-x)=(-1)^{n}\mathcal{S}_{n}(x)$, $n\geq 1$, which implies the existence of two OPS $\{\mathcal{P}_{n}(x)\}_{n=1}^{\infty}$ and $\{\mathcal{K}_{n}(x)\}_{n=1}^{\infty}$ such that

[TABLE]

It is interesting to note that $\{\mathcal{K}_{n}(x)\}=\{\mathcal{K}_{n}(0;x)\}$ is the sequence of kernel polynomial corresponding to $\mathcal{P}_{n}(x)$ based at the origin and abbreviated as KOPS in the sequel.

Further, if the polynomials $\{\mathcal{R}_{n}^{(i)}(x)\}$, $i=1,2$, satisfy the recurrence relation,

[TABLE]

with $\mathcal{R}_{-1}^{(i)}(x)=0$ and $\mathcal{R}_{0}^{(i)}(x)=1$ then,

1. (i)

$\mathcal{R}_{n}^{(1)}(x)\equiv\mathcal{P}_{n}(x)$, $n\geq 1$, if and only if

[TABLE] 2. (ii)

$\mathcal{R}_{n}^{(2)}(x)\equiv\mathcal{K}_{n}(x)$, $n\geq 1$, if and only if

[TABLE]

With these notations, the parameter sequences can be denoted as $m_{n}=\gamma_{2n+1}/b_{n+1}^{(1)}$ and $g_{n}=\gamma_{2n+1}/b_{n+1}^{(1)}$, $n\geq 0$. Further, denoting $\tilde{a}_{n}^{2}=\gamma_{2n-1}\gamma_{2n+2}$, $n\geq 1$, the following theorem shows that the polynomials $\{\tilde{\mathcal{P}}_{n}(x)\}$ and $\{\hat{\mathcal{P}}_{n}(x)\}$ associated respectively, with the complementary chain sequence $\{\tilde{\omega}_{n}(0)\}$ and the generalised complementary chain sequence $\{\hat{\vartheta}_{n}(0)\}$, can be attributed to a particular perturbation of the recurrence coefficients of the polynomials $\{S_{n}(x)\}$.

Remark 3.1**.**

Such perturbations of the recurrence coefficients as well as of the Stieltjes function have been studied deeply. The reader is referred to [7, 10, 22] for some details. In most of the cases only a single modification or a finite composition of modifications is considered. In this note however, all the recurrence coefficients are perturbed.

Theorem 3.1**.**

Let the symmetric polynomials $\{\tilde{S}_{n}(x)\}_{n=0}^{\infty}$ satisfy

[TABLE]

with $\tilde{S}_{-1}(x)=0$, $\tilde{S}_{0}(x)=1$ and where, for $n\geq 1$,

[TABLE]

Then, with $\gamma_{1}\neq 0$, $\{\tilde{\mathcal{P}}_{n}(x)\}_{n=0}^{\infty}$, where $\tilde{S}_{2n}(x)=\tilde{\mathcal{P}}_{n}(x^{2})$, satisfy,

[TABLE]

with the initial conditions $\tilde{\mathcal{P}}_{0}(x)=1$ and $\tilde{\mathcal{P}}_{1}(x)=(x-\gamma_{1})$.

Proof.

First note that, the perturbation (3.5) implies that the sequence of coefficients $\{\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4},\cdots\}$ is replaced by $\{\gamma_{2},\gamma_{1},\gamma_{4},\gamma_{3},\cdots\}$. That is, $\{\gamma_{2k-1},\gamma_{2k}\}$ are pair-wise interchanged to $\{\gamma_{2k},\gamma_{2k-1}\}$, $k\geq 1$. Then, for $n=2m$,

[TABLE]

which implies,

[TABLE]

or equivalently,

[TABLE]

Similarly, for $n=2m+1$,

[TABLE]

which implies,

[TABLE]

or equivalently,

[TABLE]

Using (3.7) and (3.8), it is easy to find the three term recurrence relations for $\tilde{\mathcal{P}}_{n}(x)$ and $\tilde{\mathcal{K}}_{n}(x)$. For this, first $\tilde{\mathcal{P}}_{n}(x)$ is eliminated. From (3.8), it can be seen that,

[TABLE]

Using this in (3.7), gives,

[TABLE]

with $\tilde{\mathcal{K}}_{-1}(x)=0$ and (using (3.8)) $\tilde{\mathcal{K}}_{0}(x)=1$. Similarly, (3.7) gives

[TABLE]

Using this in (3.8) yields,

[TABLE]

with the initial conditions $\tilde{\mathcal{P}}_{0}(x)=1$ and (using (3.7)) $\tilde{\mathcal{P}}_{1}(x)=x-\gamma_{1}$, thus proving the theorem. ∎

Corollary 3.1**.**

Consider the OPS $\{\hat{P}_{n}(x)\}_{n=0}^{\infty}$ satisfying (3.10) but for $m\geq 0$. Then $\{\hat{P}_{n}(x)\}_{n=0}^{\infty}$ is associated with the generalised complementary chain sequence $\{\hat{\vartheta}_{n}(0)\}_{n=1}^{\infty}$.

Proof.

From the recurrence relation

[TABLE]

with $\hat{P}_{-1}(x)=0$ and $\hat{P}_{0}(x)=1$, the chain sequence is given by

[TABLE]

with the parameter sequence $\{k^{\prime}_{n}\}_{n=0}^{\infty}$= $\{\gamma_{2n+2}/(\gamma_{2n+1}\gamma_{2n+2})\}_{n=0}^{\infty}$. The result now follows since $k^{\prime}_{n}=1-g_{n}$, $n\geq 0$. ∎

The OPS $\{\tilde{\mathcal{P}}_{n}(x)\}_{n=1}^{\infty}$ can be seen to be co-recursive with respect to the OPS $\{\hat{\mathcal{P}}_{n}(x)\}_{n=1}^{\infty}$ arising from the initial conditions $\tilde{\mathcal{P}}_{0}(x)=1$ and $\tilde{\mathcal{P}}_{1}(x)=\hat{\mathcal{P}}_{1}(x)+\gamma_{2}$. The co-recursive polynomials have been investigated in the past; see for example, [8], and later [12] in which the structure and spectrum of the generalised co-recursive polynomials have been studied.

Further, from (3.10), the associated chain sequence is $\{\tilde{a}_{n}^{2}/b_{n}^{(1)}b_{n+1}^{(1)}\}_{n=1}^{\infty}$ with the first few terms as

[TABLE]

Proceeding as above, we obtain the minimal parameter sequence $\{k_{n}\}_{n=0}^{\infty}$ where $k_{0}=0$ and $k_{n}=\gamma_{2n+2}/b_{n+1}^{(1)}=1-g_{n}$, $n\geq 1$. which shows that the OPS $\{\tilde{P}_{n}(x)\}_{n=0}^{\infty}$ is associated with the complementary chain sequence $\tilde{\omega}_{n}(0)\}_{n=1}^{\infty}$.

Viewing the generalised complementary chain sequences as perturbations of the minimal parameters or simply a transformation of the original chain sequence, we give an important consequence of Theorem 3.1.

Corollary 3.2**.**

The kernel polynomial system $\{\mathcal{K}_{n}(x)\}$ remains invariant under generalised complementary chain sequence if the sequence $\{\gamma_{n}\}_{n=1}^{\infty}$ satisfies,

[TABLE]

Proof.

The proof follows from a comparison of (3.9) and the expressions for $b_{n+1}^{(2)}$ and $a_{n}^{(2)}$. ∎

Corollary 3.2 is important because it is known [9, Ex. 7.2, p. 39], that the relation between the monic orthogonal polynomials and the kernel polynomials is not unique. That is, for fixed $t\in\mathbb{R}$, though $\{\mathcal{P}_{n}(x)\}$ will lead to a unique kernel polynomial system $\{\mathcal{K}_{n}(t;x)\}$, there are infinite number of other monic orthogonal polynomial systems which has the same $\{\mathcal{K}_{n}(t;x)\}$ as their kernel polynomial system. Hence generalised complementary chain sequences can be used to construct two orthogonal polynomials systems having the same kernel polynomial systems.

The following theorem unifies the recurrence relations for the polynomials and the associated kernel polynomials for both the chain sequence as well as its generalised complementary chain sequence.

Theorem 3.2**.**

Consider the recurrence relation,

[TABLE]

with $\mathcal{T}_{-1}(x)=0$ and $\mathcal{T}_{0}(x)=1$. Then,

1. (i)

$\mathcal{T}_{n}(x)\equiv\tilde{\mathcal{P}}_{n}(x)(\equiv\hat{\mathcal{P}}_{n}(x))$, $n\geq 1$, if and only if,

[TABLE] 2. (ii)

$\mathcal{T}_{n}(x)\equiv\tilde{\mathcal{K}}_{n}(x)$, $n\geq 1$, if and only if,

[TABLE]

Let the zeros of $\{\mathcal{P}_{n}(x)\}_{n=0}^{\infty}$ and $\{\tilde{\mathcal{P}}_{n}(x)\}_{n=0}^{\infty}$ be denoted as

[TABLE]

For fixed $n$, by interlacing of zeros of $\mathcal{P}_{n}(x)$ and $\tilde{\mathcal{P}}_{n}(x)$ it is understood that $x_{n,j}$ are mutually separated by $\tilde{x}_{n,j}$ for $j=1,2,\cdots,n$. Further, in the present case, it is interesting to note from (3.1) and (3.10), that the sum of the roots of $\mathcal{P}_{n}(x)$ is given by $\gamma_{2}+\gamma_{3}+\cdots+\gamma_{2n}$ while that for $\tilde{\mathcal{P}}_{n}(x)$ is $\gamma_{1}+\gamma_{3}+\cdots+\gamma_{2n}$.

Remark 3.2**.**

It is clear that if $\gamma_{1}=\gamma_{2}$, interlacing of the zeros of $\{\mathcal{P}_{n}(x)\}$ and $\{\tilde{\mathcal{P}}_{n}(x)\}$ can never occur.

For $\gamma_{1}\neq\gamma_{2}$, we have the following result.

Proposition 3.1**.**

For fixed $n$, the zeros $\{x_{n,j}\}_{j=1}^{n}$ and $\{\tilde{x}_{n,j}\}_{j=1}^{n}$ cannot interlace if $(\gamma_{1}-\gamma_{2})$ and $(x_{n,j}-\tilde{x}_{n,j})$ have the same sign for some $j=1,2,\cdots,n$.

Proof.

Suppose $\gamma_{1}>\gamma_{2}$ and $x_{n,j}>\tilde{x}_{n,j}$ for some $j=1,2,\cdots,n$. If the zeros of $\mathcal{P}_{n}(x)$ and $\tilde{\mathcal{P}}_{n}(x)$ interlace, then $\sum_{j=1}^{n}\tilde{x}_{n,j}<\sum_{j=1}^{n}x_{n,j}$ which is a contradiction.

The case $\gamma_{1}<\gamma_{2}$ and $x_{n,j}<\tilde{x}_{n,j}$ can be proved similarly. ∎

The last result in this section shows that while the generalised complementary chain sequence of associated with $\{\hat{\mathcal{P}}_{n}(x)\}$ yields an OPS, that associated with the associated polynomials $\{\hat{\mathcal{P}^{(1)}}_{n}(x)\}$ leads to a KOPS.

Theorem 3.3**.**

Consider the OPS $\{\hat{\mathcal{P}}^{(1)}_{n}(x)\}_{n=0}^{\infty}$. Then the generalised complementary chain sequence associated with $\hat{\mathcal{P}}^{(1)}_{n}(x)$ leads to a KOPS $\{\mathcal{Q}_{n}(x)\}$ satisfying the relation

[TABLE]

with $\mathcal{Q}_{-1}(x)=0$ and $\mathcal{Q}_{0}(x)=1$.

Proof.

It is clear from (3.11) that $\{\hat{\mathcal{P}}^{(1)}_{n}(x)\}_{n=1}^{\infty}$ satisfy

[TABLE]

with $\hat{\mathcal{P}}^{(1)}_{-1}(x)=0$ and $\hat{\mathcal{P}}^{(1)}_{0}(x)=1$. The associated chain sequence is

[TABLE]

with the (non-minimal) parameter sequence $\{\gamma_{2n+4}/(\gamma_{2n+3}+\gamma_{2n+4})\}_{n=0}^{\infty}$. Hence the OPS $\{\mathcal{Q}_{n}(x)\}$ associated with the generalised complementary chain sequence satisfy (3.12).

To prove that $\{\mathcal{Q}_{n}(x)\}$ is a KOPS, consider the polynomials $\{\mathcal{U}_{n}(x)\}_{n=1}^{\infty}$ given by

[TABLE]

The first thing that we require is that $\mathcal{U}_{n+1}(0)=-\gamma_{2n+3}\mathcal{U}_{n}(0)$ so that choosing $\mathcal{U}_{1}(0)=-\gamma_{3}$, we have $\mathcal{U}_{n+1}(0)=(-1)^{n+1}\gamma_{2n+3}\gamma_{2n+1}\cdots\gamma_{5}\gamma_{3}$.

Suppose now that $\{\mathcal{U}_{n}(x)\}_{n=1}^{\infty}$ satisfy the recurrence relation

[TABLE]

with $\mathcal{U}_{0}(x)=1$ and $\mathcal{U}_{1}(x)=x-\gamma_{3}$ and where the coefficients $\{\sigma_{n}\}$ and $\{\eta_{n}\}$ are to be determined. One way for the recurrence relation to hold, is that we must have $\mathcal{U}_{n+1}(0)=-\sigma_{n+1}\mathcal{U}_{n}(0)-\mu_{n}\mathcal{U}_{n-1}(0)$ which implies $\gamma_{2n+1}\sigma_{n+1}-\mu_{n}=\gamma_{2n+3}\gamma_{2n+1}$, $n\geq 1$. One possible choice for $\sigma_{n+1}$ and $\mu_{n}$ satisfying the above equations is

[TABLE]

Since $\mu_{n}>0$ for $n\geq 1$, by Favard’s Theorem [9, Theorem 4.4, p. 21] $\{\mathcal{U}_{n}(x)\}_{n=1}^{\infty}$ becomes a OPS and $\{\mathcal{Q}_{n}(x)\}_{n=1}^{\infty}$ its corresponding KOPS [9, eqn. 7.3, p. 35]. ∎

Corollary 3.3**.**

The following holds

[TABLE]

Proof.

Comparing with the recurrence relation (3.1), it can be observed that $\{\mathcal{U}_{n}(x)\}_{n=1}^{\infty}$ are co-recursive with respect to $\{\mathcal{K}_{n}(x)\}_{n=1}^{\infty}$ and arise due to change in the initial value $\mathcal{U}_{1}(x)=\mathcal{K}_{1}(x)+\gamma_{2}$. Hence $\mathcal{U}_{n}(x)=\mathcal{K}_{n}(x)+\gamma_{2}\mathcal{K}^{(1)}_{n}(x)$, $n\geq 1$ [8], so that

[TABLE]

Since $x\mathcal{K}_{n}(x)=\mathcal{P}_{n+1}(x)+\gamma_{2n+2}\mathcal{P}_{n-1}(x)$, $n\geq 1$, (3.3) follows. ∎

Remark 3.3**.**

The polynomials in the left hand side of (3.3) are quasi-orthogonal of order 2 on $[0,\infty]$. For details regarding quasi-orthogonality, the reader is referred to [5] and the references therein.

4. Concluding remarks

1. (A)

The monic Jacobi matrix of the polynomials $\mathcal{P}_{n}(x)$ and $\tilde{\mathcal{P}}_{n}(x)$ are given respectively by,

[TABLE]

The respective $LU$ decomposition of the above Jacobi matrices are then given by,

[TABLE]

and,

[TABLE]

It may be observed that $L_{\tilde{\mathcal{P}}}$ can be obtained from $U_{\mathcal{P}}$ by deleting the first column of $U_{\mathcal{P}}$ while $U_{\tilde{\mathcal{P}}}$ can be obtained from $L_{\mathcal{P}}$ by adding the first column of $U_{\mathcal{P}}$, with $\gamma_{2}$ replaced by $\gamma_{1}$, as the first column of $L_{\mathcal{P}}$. The Jacobi matrix and its LU decomposition for the polynomial system $\{\hat{\mathcal{P}}_{n}(x)\}_{n=1}^{\infty}$ can be obtained similarly.

The monic Jacobi matrix associated with the three canonical transformations, the Christoffel, the Geronimus and the Uvarov transformations, are found in [6], wherein the procedure is given using the Darboux transformation. Hence, it will be interesting to study the application of Darboux transformation in the case of (generalised) complementary chain sequences. 2. (B)

The results obtained in this note can be applied to the Laguerre polynomials which provide some insights into chain sequences related to these orthogonal polynomials.

Consider the three term recurrence relation satisfied by the generalized Laguerre polynomials $\{L_{n}^{(\alpha)}\}$, [9, Page 154],

[TABLE]

with $\mathcal{R}_{0}^{(1)}(x)=1$ and $\mathcal{R}_{1}^{(1)}(x)=x-(1+\alpha)$ and where $\mathcal{R}_{n}^{(1)}(x)\equiv L_{n}^{(\alpha)}(x)$, $n\geq 1$. Using the notations introduced immediately after (3.1), the associated chain sequence $\{d_{n}\}_{n=1}^{\infty}$ is,

[TABLE]

and as can be easily verified, the minimal parameters are given by, $m_{n}=n/(2n+\alpha+1)$, $n\geq 0$. It is easily seen that $0<m_{n}<1/2$, $n\geq 1$ and hence by [2, Lemma 2.3], the chain sequence complementary to $d_{n}$ is a SPPCS. Moreover, for $-1<\alpha<0$, $m_{n}/(1-m_{n})>n/(n-1)>1$ and hence by Wall’s criteria for SPPCS [21, Theorem 19.3], $\{d_{n}\}$ determines its parameters uniquely. Further, choosing $\gamma_{1}=0$, it is found that $\gamma_{2}=(1+\alpha)$ and $\gamma_{2}\gamma_{3}=1.(1+\alpha)$ implies $\gamma_{3}=1$. Similarly, $\gamma_{3}+\gamma_{4}=\alpha+3$ implies $\gamma_{4}=\alpha+2$. Proceeding further on similar lines, it can be easily proved by induction that $\gamma_{1}=0$, $\gamma_{2n}=n+\alpha$ and $\gamma_{2n+1}=n$, $n\geq 1$. This gives the recurrence relation for the associated kernel polynomials as

[TABLE]

with $\mathcal{R}_{0}^{(2)}(x)=1$ and $\mathcal{R}_{1}^{(2)}(x)=x-(2+\alpha)$. Clearly, as is known, $\mathcal{R}_{n}^{(2)}(x)=L_{n}^{(\alpha+1)}(x)$, $n\geq 1$.

Consider now the polynomials $\{\mathcal{E}_{n}(x)\}_{n=0}^{\infty}$ satisfying the recurrence relation

[TABLE]

with $\mathcal{E}_{0}(x)=1$ and $\mathcal{E}_{1}(x)=x-(\alpha+1)$. From the related chain sequence, we obtain the sequence $\{\gamma_{n}\}_{n=1}^{\infty}$ where $\gamma_{1}=0$, $\gamma_{2n}=n+\alpha$ and $\gamma_{2n+1}=n+1$, $n\geq 1$. The kernel polynomial sequence $\{\mathcal{K}_{n}(x)\}_{n=0}^{\infty}$ associated with $\{\mathcal{E}_{n}(x)\}_{n=0}^{\infty}$ satisfy

[TABLE]

with $\mathcal{K}_{-1}(x)=0$ and $\mathcal{K}_{0}(x)=1$. If we let $\gamma_{1}=1$, the resulting polynomials satisfy

[TABLE]

with $\mathcal{P}_{-1}(x)=0$ and $\mathcal{P}_{0}(x)=1$. From (4.1) it is clear that these polynomials are the associated generalized Laguerre polynomials of order 1 but with $\alpha$ shifted to $\alpha-1$. The polynomial sequence $\{\hat{\mathcal{P}}_{n}(x)\}$ corresponding to the generalized complementary chain sequence satisfy

[TABLE]

with $\hat{\mathcal{P}}_{-1}(x)=0$ and $\hat{\mathcal{P}}_{0}(x)=1$. Comparing with (4.2), we find that $\hat{\mathcal{P}}_{n}(x)\equiv L_{n}^{(\alpha+1)}(x)$, $n\geq 1$.

The (co-recursive) polynomials $\{\tilde{\mathcal{P}}_{n}(x)\}$ corresponding to the complementary chain sequence satisfy the recurrence

[TABLE]

with $\tilde{\mathcal{P}}_{0}(x)=1$ and $\tilde{\mathcal{P}}_{1}(x)=x-1$.

Moreover, since the condition in Corollary 3.2 is satisfied, the kernel polynomials for the OPS $\{\tilde{\mathcal{P}}_{n}(x)\}_{n=0}^{\infty}$ is the same (upto a constant multiple) as that for the OPS $\{\mathcal{E}_{n}(x)\}_{n=0}^{\infty}$. 3. (C)

We recall that the Routh-Romanovski Laguerre polynomials $\{N_{n}^{(p)}(x)\}$ which are analogous to Laguerre polynomials $L_{n}^{(p)}(x)$ are given by

[TABLE]

The sequence $\{N_{n}^{(p)}(x)\}$ are the polynomials studied as one of the three finite classes of hypergeometric orthogonal polynomials [13], see also [14, 15].

Further, the sequence $\{N_{n}^{(p)}(x)\}$ satisfy the three term recurrence [13, eqn. 4.19]

[TABLE]

The above recurrence relation can be normalised to satisfy a recurrence relation of the form (1.2). Hence it will be interesting to see the effect of (generalised) complementary chain sequence on these polynomials. It is important to remark that the results related to $q$-analogue of Routh-Romanovski polynomials on the unit circle are studied in [20].