On an extension of generalized coherent pairs of orthogonal polynomials: the classical case

TL;DR

This paper extends the concept of generalized coherent pairs of orthogonal polynomials, unifying various cases and deriving new recurrence relations and modifications when classical functionals are involved.

Contribution

It introduces a broader framework for extended coherent pairs, including symmetric cases, and provides explicit formulas for recurrence coefficients and rational modifications.

Findings

Unified extended coherent pairs framework

Explicit recurrence coefficients for new polynomial systems

Rational modifications of classical moment functionals

Abstract

Given two quasi-definite moment functionals, the corresponding orthogonal polynomial systems satisfy an algebraic differential relation(called an extended coherent pair). We study generalizing extended coherent pairs that unify extended coherent pairs and extended symmetric coherent pairs and find the related coefficients. When one of the moment functionals is (strongly) classical, we find another orthogonal polynomial system to find three-term recurrence coefficients. Moreover, we determine the companion moment functional as a rational modification of the classical one.

Equations

$${\varphi }_{\lambda }(f,g):=⟨u_0,fg⟩+\lambda ⟨u_1,f^{\prime }{g}^{\prime }⟩(\lambda \ne 0),$$

$$R_n(x)$$

$$\text{(resp., }{R}_n(x)$$

$$R_n(x)=Q_n(x)-{\sigma }_n{Q}_{n-1}(x)-{\tau }_{n-1}{Q}_{n-2}(x),n\ge 0,$$

$$\begin{gathered} P_{n+1}(x)-{\sigma }_n{P}_n(x)-{\tau }_{n-1}{P}_{n-1}(x) \\ =S_{n+1}^{(\lambda )}(x)-{\mu }_n{S}_n^{(\lambda )}(x)-{\theta }_{n-1}{S}_{n-1}^{(\lambda )}(x),n\ge 0, \end{gathered}$$

$$P_{n+1}(x)-{\sigma }_n{P}_n(x)=S_{n+1}^{(\lambda )}(x)-{\mu }_n{S}_n^{(\lambda )}(x),n\ge 0.$$

$$R_n(x)-d_n{R}_{n-1}(x)=\frac{1}{n+1}{P}_{n+1}^{\prime }(x)-\frac{{\sigma }_n}{n}{P}_n^{\prime }(x),n\ge 1,$$

$$P_{n+1}(x)-{\tau }_{n-1}{P}_{n-1}(x)=S_{n+1}^{(\lambda )}(x)-{\theta }_{n-1}{S}_{n-1}^{(\lambda )}(x),n\ge 0,$$

$$R_n(x)-e_{n-1}{R}_{n-2}(x)=\frac{1}{n+1}{P}_{n+1}^{\prime }(x)-\frac{{\tau }_{n-1}}{n-1}{P}_{n-1}^{\prime }(x),n\ge 2,$$

$$\sum _{i=0}^N{r}_{i,n}{R}_{n-i}(x)=\sum _{i=0}^M{p}_{i,n}{P}_{n+1-i}^{\prime },n\ge 0,$$

$$\sum _{i=0}^N{r}_{i,n}{R}_{n-i}^{(k)}(x)=\sum _{i=0}^M{p}_{i,n}{P}_{n+1-i}^{(m)},n\ge 0,$$

$${\Delta }_n(u):=\det [a_{i+j}{]}_{i,j=0}^n\ne 0,({\Delta }_n(u)>0,\text{respectively}),n\ge 0.$$

$$⟨u,P_m{P}_n⟩=p_n{\delta }_{mn},m,n\ge 0,$$

$$P_{n+1}(x)=(x-b_n)P_n(x)-c_n{P}_{n-1}(x),n\ge 0,$$

$$⟨u^{\prime },p⟩$$

$$⟨\varphi u,p⟩$$

$$⟨(x-c{)}^{-1}u,p⟩$$

$$(\phi u{)}^{\prime }=\psi u,$$

$$s:=\min \max (\mathrm{deg}(\phi )-2,\mathrm{deg}(\psi )-1)$$

$$(\phi u{)}^{\prime }=\psi u,(\phi w{)}^{\prime }=(\psi -{\phi }^{\prime })w,\text{ and }\phi w=u.$$

$$R_n(x)=Q_n(x)-{\sigma }_{n-1}{Q}_{n-1}(x),n\ge 0,$$

$$R_n(x)=Q_n(x)-{\tau }_{n-2}{Q}_{n-2}(x),n\ge 0,$$

$$R_n(x)=Q_n(x)-{\sigma }_{n-1}{Q}_{n-1}(x)-{\tau }_{n-2}{Q}_{n-2}(x),n\ge 0,$$

$$R_n(x)-d_{n-1}{R}_{n-1}(x)=Q_n(x)-{\sigma }_{n-1}{Q}_{n-1}(x),n\ge 0,$$

$$R_n(x)-e_{n-2}{R}_{n-2}(x)=Q_n(x)-{\tau }_{n-2}{Q}_{n-2}(x),n\ge 0,$$

$$p_n:=⟨u_0,P_n^2⟩,r_n:=⟨u_1,R_n^2⟩,s_n:={\varphi }_{\lambda }(S_n^{(\lambda )},S_n^{(\lambda )}),n\ge 0.$$

$$\begin{gathered} P_{n+1}(x)-{\tilde{\sigma }}_n{P}_n(x)-{\tilde{\tau }}_{n-1}{P}_{n-1}(x) \\ =S_{n+1}^{(\lambda )}(x)-{\mu }_n{S}_n^{(\lambda )}(x)-{\theta }_{n-1}{S}_{n-1}^{(\lambda )}(x),n\ge 0, \end{gathered}$$

$$\{\begin{array}{rl}{\mu }_n{s}_n& ={\tilde{\sigma }}_n{p}_n+{\tilde{\tau }}_{n-1}({\mu }_{n-1}-{\tilde{\sigma }}_{n-1})p_{n-1},n\ge 1,\\ {\theta }_n{s}_n& ={\tilde{\tau }}_n{p}_n,n\ge 1.\end{array}$$

$$\begin{gathered} (n+1)R_n(x)-{\tilde{d}}_n{R}_{n-1}(x)-{\tilde{e}}_{n-1}{R}_{n-2}(x) \\ =P_{n+1}^{\prime }(x)-{\tilde{\sigma }}_n{P}_n^{\prime }(x)-{\tilde{\tau }}_{n-1}{P}_{n-1}^{\prime }(x),n\ge 0, \end{gathered}$$

$$\begin{array}{rl}{\tilde{d}}_n=& \frac{{\mu }_n{s}_n-{\tilde{\sigma }}_n{p}_n+({\tilde{\sigma }}_{n-1}-{\mu }_{n-1}){\tilde{\tau }}_{n-1}{p}_{n-1}}{n\lambda r_{n-1}}\\ & +\frac{r_{n-2}{\tilde{e}}_{n-1}\{{\tilde{d}}_{n-1}-(n-1){\mu }_{n-1}\}}{n{r}_{n-1}},n\ge 2,\\ {\tilde{d}}_1=& \frac{{\mu }_1{s}_1-{\tilde{\sigma }}_1{p}_1+({\tilde{\sigma }}_0-{\mu }_0){\tilde{\tau }}_0{p}_0}{\lambda r_0},\end{array}$$

$${\tilde{e}}_{n-1}=\frac{{\theta }_{n-1}{s}_{n-1}-{\tilde{\tau }}_{n-1}{p}_{n-1}}{(n-1)\lambda r_{n-2}},n\ge 2.$$

$$\begin{gathered} {\varphi }_{\lambda }(S_{n+1}^{(\lambda )}-{\mu }_n{S}_n^{(\lambda )}-{\theta }_{n-1}{S}_{n-1}^{(\lambda )},x^k) \\ =\lambda k⟨u_1,(P_{n+1}^{\prime }-{\tilde{\sigma }}_n{P}_n^{\prime }-{\tilde{\tau }}_{n-1}{P}_{n-1}^{\prime })x^{k-1}⟩=0,0\le k\le n-2, \end{gathered}$$

Full text

††journal: Indagationes Mathematicae

\affiliation

[inst1]organization=ARIST,addressline=85 Wolpyeongbuk-ro, city=Seo-gu, postcode=35213, state=Daejeon, country=Korea

\affiliation

[inst2]organization=Dept. of Intelligent System Engineering,addressline=Cheju Halla University, 38 Halladaehak-ro, city=Jeju-si, postcode=63092, state=Jeju Special Self-Governing Province, country=Korea

\affiliation

[inst3]organization=Dept. of Mathematics,addressline=University of Utah Asia Campus, 119-3 Songdo Moonwha-ro, city=Yeonsu-Gu, postcode=21985, state=Incheon, country=Korea

1 Introduction

Concerning polynomials orthogonal with respect to a Sobolev inner product

[TABLE]

where $u_{0}$ and $u_{1}$ are quasi-definite moment functionals, Iserles et al.[8] introduced the concept of coherency and symmetric coherency for the pair of moment functionals $\{u_{0},u_{1}\}$. $\{u_{0},u_{1}\}$ is a coherent pair(resp. symmetrically coherent pair) if there are nonzero constants $\sigma_{n}$(resp. $\tau_{n}$) such that

[TABLE]

where $\{P_{n}(x)\}_{n=0}^{\infty}$ and $\{R_{n}(x)\}_{n=0}^{\infty}$ are the monic orthogonal polynomials with respect to $u_{0}$ and $u_{1}$, respectively, and in [20], all coherent pairs are classified.

After the work by Iserles et al.[8], the coherency was studied in various ways. One is concerned with generalizing the coherency. In [13], generalized coherency, which unifies both coherency and symmetric coherency, was studied. Generalized coherency is given by the relation,

[TABLE]

where $Q_{n}(x)=\frac{1}{n+1}P^{\prime}_{n+1}(x),~{}n\geq 0$, $Q_{-1}(x)=Q_{-2}(x)=0$, and $\sigma_{0}=\tau_{-1}=\tau_{0}=0$.

In this case, the monic orthogonal polynomials $\{S_{n}^{(\lambda)}(x)\}_{n=0}^{\infty}$ with respect to $\phi_{\lambda}(\cdot,\cdot)$ and $\{P_{n}(x)\}_{n=0}^{\infty}$ satisfy the relation

[TABLE]

where $\tau_{-1}=\theta_{-1}=P_{-1}(x)=S_{-1}^{(\lambda)}(x)=0$, and $\mu_{n}$ and $\theta_{n}$ are evaluated.

Another approach for generalizing coherency has occurred. Extended coherency was introduced and studied in [11] when $\{P_{n}(x)\}_{n=0}^{\infty}$ and $\{S_{n}^{(\lambda)}(x)\}_{n=0}^{\infty}$ have the relation

[TABLE]

Note that this relation is the 2-term case of (1.3), and $\{P_{n}(x)\}_{n=0}^{\infty}$ and $\{R_{n}(x)\}_{n=0}^{\infty}$ satisfy the relation

[TABLE]

for some $d_{n}$. We call this extended coherency.

In [7], extended symmetric coherency was studied. When $\{P_{n}(x)\}_{n=0}^{\infty}$ and $\{S_{n}^{(\lambda)}(x)\}_{n=0}^{\infty}$ have the relation

[TABLE]

note that this relation is the symmetric case of (1.3), and then $\{P_{n}(x)\}_{n=0}^{\infty}$ and $\{R_{n}(x)\}_{n=0}^{\infty}$ satisfy the relation

[TABLE]

for some $e_{n}$. We call this extended symmetric coherency.

The concept of coherency is based on Sobolev orthogonality and is applicable for Fourier-Sobolev expansions. This brought a lot of attention in the area of coherency and the research of extending coherency has been studied in many different directions.

In [3, 17], the concept of coherent pairs of measures was extended from the real line to Jordan arcs and curves, focusing on the coherent pairs supported on the unit circle.

In [10, 22], $(M,N)-$coherency was introduced as extending terms of polynomials. $(M,N)-$coherency has the relation

[TABLE]

with the conditions $r_{N,n}\neq 0$ if $n\geq N$ and $p_{M,n}\neq 0$ if $n\geq M$, and the conventions $r_{i,n}=0$ if $n<i\leq N$ and $p_{i,n}=0$ if $n<i\leq M$.

In [4, 9, 16], coherency was extended to the relation of higher-order differentials of the corresponding polynomials. Combining with $(M,N)-$coherency, $(M,N)-$coherency of order $(m,k)$ has the relation

[TABLE]

where $r_{i,n}$ and $p_{j,n}$ are complex numbers with $r_{N,n}\neq 0$ if $n\geq N$ and $p_{M,n}\neq 0$ if $n\geq M$, and $r_{i,n}=p_{i,n}=0$ if $i>n$.

In [1], linearly related orthogonal polynomials and their functionals are studied, and which can be applied to the linear relation of two polynomials. In [6], companion of linear functional in Sobolev inner product is studied.

In [18], Marcellán and Xu provided a survey of Sobolev orthogonal polynomials, including coherent pairs and generalizing coherent pairs.

In this work, we will generalize extended coherency that unifies extended coherency and extended symmetric coherency, that is, in the case that $\{S_{n}^{(\lambda)}(x)\}_{n=0}^{\infty}$ and $\{P_{n}(x)\}_{n=0}^{\infty}$ have the relation (1.3), we have the result that unifies the result of extended coherency and that of extended symmetric coherency.

In section 3, we introduce generalized extended coherency and find the related coefficients. In section 4, when one of the moment functionals is (strongly) classical, we find the three-term recurrence relation coefficients of the monic orthogonal polynomial system relative to the other moment functional. We find the companion moment functional if one is given.

2 Preliminaries

Let $\mathbb{P}$ be the linear space of all polynomials in one variable with complex coefficients. We denote the degree of a polynomial $P(x)$ by $\deg(P)$ with the convention that $\deg(0)=-1$. A polynomial system(PS) is a sequence of polynomials $\{P_{n}(x)\}_{n=0}^{\infty}$ with $\deg(P_{n})=n,~{}n\geq 0$. For convenience, let $P_{n}(x)=0$ if $n<0$, and we denote $P_{n}$ instead of $P_{n}(x)$ for simplicity.

A linear functional $u$ on $\mathbb{P}$ is called a moment functional, and we denote its action on a polynomial $\phi(x)$ by $\langle u,\phi\rangle$. We say that a moment functional $u$ is quasi-definite(positive-definite, respectively) if its moments $a_{n}:=\langle u,x^{n}\rangle,~{}n\geq 0$, satisfy the Hamburger condition

[TABLE]

Definition 2.1**.**

A PS $\{P_{n}(x)\}_{n=0}^{\infty}$ is said to be an orthogonal polynomial system(OPS) if there is a linear functional $u$ on $\mathbb{P}$ such that

[TABLE]

where $p_{n}$ are non-zero constants.

In this case, we call $\{P_{n}(x)\}_{n=0}^{\infty}$ an OPS relative to $u$, and $u$ is said to be an orthogonalizing moment functional of $\{P_{n}(x)\}_{n=0}^{\infty}$.

A linear functional $u$ is quasi-definite if and only if there is an OPS $\{P_{n}(x)\}_{n=0}^{\infty}$ relative to $u$ (see [5]).

Moreover, in this case, each $P_{n}(x)$ is uniquely determined up to a non-zero constant factor.

A polynomial $P_{n}(x)$ is monic if the leading coefficient of $P_{n}(x)$ is equal to $1$. We call $\{P_{n}(x)\}_{n=0}^{\infty}$ an MOPS(monic OPS) if all polynomials in $\{P_{n}(x)\}_{n=0}^{\infty}$ are monic.

It is well-known([5]) that such sequences of monic polynomials satisfy three-term recurrence relations

[TABLE]

where $P_{-1}(x)=0$, $P_{0}(x)=1$, $c_{n}\neq 0$ for $n\geq 0$, and $c_{0}=\langle u,1\rangle$.

For a moment functional $u$, a polynomial $\phi(x)$, and a constant $c$, we define moment functionals $u^{\prime},~{}\phi u~{}$, and $(x-c)^{-1}u$ by

[TABLE]

Definition 2.2**.**

([19]) A quasi-definite moment functional $u$ is said to be semiclassical if $u$ satisfies

[TABLE]

for some polynomials $\varphi(x)$ and $\psi(x)$ with $(\varphi,\psi)\neq(0,0)$. We then have $\deg(\varphi)\geq 0$ and $\deg(\psi)\geq 1$. The corresponding OPS is called a semiclassical OPS.

For a semiclassical moment functional $u$,

[TABLE]

the class number of $u$, where the minimum is taken over all pairs $(\varphi,\psi)\neq(0,0)$ of polynomials satisfying (2.2). In particular, a semiclassical moment functional of class [math] is called a classical moment functional.

It is well-known that there are essentially four distinct classical OPS’s, up to a linear change of variable([2, 12]), for each case, we denote the corresponding orthogonalizing moment functional by $u$ :

• (i)

Hermite polynomials $\{H_{n}(x)\}_{n=0}^{\infty}$: $\varphi(x)=1$, $\psi(x)=-2x$, $u=u_{H}$;

• (ii)

Laguerre polynomials $\{L_{n}^{(\alpha)}(x)\}_{n=0}^{\infty}$: $\varphi(x)=x$, $\psi(x)=-x+\alpha+1~{}(\alpha\notin\{-1,-2,\cdots\}$),$u=u_{L}^{(\alpha)}$;

• (iii)

Bessel polynomials $\{B_{n}^{(\alpha)}(x)\}_{n=0}^{\infty}$: $\varphi(x)=x^{2}$, $\psi(x)=(\alpha+2)x+2$ ($\alpha\notin\{-2,-3,$ $\cdots\}$), $u=u_{B}^{(\alpha)}$;

• (iv)

Jacobi polynomials $\{P_{n}^{(\alpha,\beta)}(x)\}_{n=0}^{\infty}$: $\varphi(x)=1-x^{2}$, $\psi(x)=\beta-\alpha-(\alpha+\beta+2)x$ ($\alpha,\beta,\alpha+\beta+1\notin\{-1,-2,\cdots\}$), $u=u_{J}^{(\alpha,\beta)}$.

We say a quasi-definite moment functional $u$ with MOPS $\{P_{n}(x)\}_{n=0}^{\infty}$ to be strongly classical(see [13]) if there is another MOPS $\{T_{n}(x)\}_{n=0}^{\infty}$ relative to $w$ such that $P_{n}(x)=\frac{1}{n+1}T^{\prime}_{n+1}(x),~{}n\geq 0$. Then $u$ and $w$ must be classical moment functionals of the same type satisfying

[TABLE]

Classical moment functionals $u_{J}^{(\alpha,\beta)}$ $(\alpha,\beta,\alpha+\beta\neq 0,-1,-2,\cdots)$, $u_{B}^{(\alpha)}$ $(\alpha\neq 0,-1,$ $-2,\cdots)$, $u_{L}^{(\alpha)}~{}(\alpha\neq 0,-1,-2,\cdots)$, and $u_{H}$ are strongly classical.

Definition 2.3**.**

([13]) Let $u_{0}$ and $u_{1}$ be quasi-definite moment functionals with corresponding MOPS $\{P_{n}(x)\}_{n=0}^{\infty}$ and $\{R_{n}(x)\}_{n=0}^{\infty}$, respectively. Let $Q_{n}(x)=\frac{1}{n+1}P^{\prime}_{n+1}(x),$ $n\geq 0.$

• (i)

$\{u_{0},u_{1}\}$ is a coherent pair if there exist complex numbers $\sigma_{n}$, $n\geq 1$, such that

[TABLE]

where $\sigma_{-1}=0$.

• (ii)

Assume $u_{0}$ and $u_{1}$ are symmetric moment functionals. $\{u_{0},u_{1}\}$ is a symmetrically coherent pair if there exist complex numbers $\tau_{n}$, $n\geq 1$, such that

[TABLE]

where $\tau_{-1}=\tau_{-2}=0$.

• (iii)

$\{u_{0},u_{1}\}$ is a generalized coherent pair if there exist complex numbers $\sigma_{n}$ and $\tau_{n}$, $n\geq 1$, such that

[TABLE]

where $\sigma_{-1}=\tau_{-1}=\tau_{-2}=0$.

Definition 2.4**.**

([7, 11]) Let $u_{0}$ and $u_{1}$ be quasi-definite moment functionals with corresponding MOPS $\{P_{n}(x)\}_{n=0}^{\infty}$ and $\{R_{n}(x)\}_{n=0}^{\infty}$, respectively. Let $Q_{n}(x)=\frac{1}{n+1}P^{\prime}_{n+1}(x),$ $n\geq 0.$

• (i)

$\{u_{0},u_{1}\}$ is an extended (2-term) coherent pair if there exist complex numbers $d_{n}$ and $\sigma_{n}$, $n\geq 0$, such that

[TABLE]

where $d_{-1}=\sigma_{-1}=0$.

• (ii)

Assume $u_{0}$ and $u_{1}$ are symmetric moment functionals. $\{u_{0},u_{1}\}$ is an extended symmetrically coherent pair if there exist complex numbers $e_{n}$ and $\tau_{n}$, $n\geq 0$, such that

[TABLE]

where $e_{-1}=e_{-2}=\tau_{-1}=\tau_{-2}=0$.

3 Extended Generalized Coherent Pairs

Let $u_{0}$ and $u_{1}$ be two quasi-definite moment functionals with corresponding MOPS’s $\{P_{n}(x)\}_{n=0}^{\infty}$ and $\{R_{n}(x)\}_{n=0}^{\infty}$, respectively. Let $\{S_{n}^{(\lambda)}(x)\}_{n=0}^{\infty}$ be the MOPS relative to the Sobolev inner product (1.1). We define

[TABLE]

When $\{u_{0},u_{1}\}$ is a generalized coherent pair, $\{P_{n}(x)\}_{n=0}^{\infty}$ and $\{R_{n}(x)\}_{n=0}^{\infty}$ satisfy the relation in Definition 2.3 (iii).

In this case, $\{S_{n}^{(\lambda)}(x)\}_{n=0}^{\infty}$ and $\{P_{n}(x)\}_{n=0}^{\infty}$ satisfy the relation (see [13]), for some $\tilde{\sigma}_{n},\tilde{\tau}_{n},\mu_{n},$ and $\theta_{n},$ $n\geq 0$

[TABLE]

where $\theta_{-1}=\tilde{\tau}_{-1}=0$ and $\mu_{n}$ and $\theta_{n}$ satisfy the relations

[TABLE]

Now we consider the inverse case. We assume that $\{S_{n}^{(\lambda)}(x)\}_{n=0}^{\infty}$ and $\{P_{n}(x)\}_{n=0}^{\infty}$ satisfy the relation (3.1).

Theorem 3.1**.**

Assume $\{P_{n}(x)\}_{n=0}^{\infty}$ and $\{S_{n}^{(\lambda)}(x)\}_{n=0}^{\infty}$ satisfy the relation (3.1), then $\{P_{n}(x)\}_{n=0}^{\infty}$ and $\{R_{n}(x)\}_{n=0}^{\infty}$ satisfy the relation

[TABLE]

for some $\tilde{d}_{n}$ and $\tilde{e}_{n}$, $n\geq 1$ with $\tilde{d}_{0}=\tilde{e}_{0}=\tilde{e}_{-1}=0$ and

[TABLE]

*Proof.  *By the orthogonality of $\phi_{\lambda}(\cdot,\cdot)$, we have

[TABLE]

or equivalently,

[TABLE]

so we have (3.3) for some $\tilde{d}_{n}$ and $\tilde{e}_{n}$, $n\geq 1$ with $\tilde{d}_{0}=\tilde{e}_{0}=\tilde{e}_{-1}=0$. To find $\tilde{e}_{n}$, by using the orthogonality of $\phi_{\lambda}(\cdot,\cdot)$, we have

[TABLE]

so we have

[TABLE]

This gives (3.5). To find $\tilde{d}_{n}$, by using the orthogonality of $\phi_{\lambda}(\cdot,\cdot)$, we have, for $n\geq 1$,

[TABLE]

Since, from (3.1),

[TABLE]

we have the first part of (3.6) as

[TABLE]

Since from (3.1) and (3.3),

[TABLE]

the second part of (3.6) becomes

[TABLE]

Combining (3.7) and (3.8) with (3.6), we have

[TABLE]

Hence we have $\tilde{d}_{n}$, $n\geq 1$, in (3.4). ∎

Now we define extended generalized coherent pair.

Definition 3.1**.**

Let $u_{0}$ and $u_{1}$ be quasi-definite moment functionals with corresponding MOPS’s $\{P_{n}(x)\}_{n=0}^{\infty}$ and $\{R_{n}(x)\}_{n=0}^{\infty}$, respectively. Let $Q_{n}(x)=\frac{1}{n+1}P^{\prime}_{n+1}(x),$ $n\geq 0.$

$\{u_{0},u_{1}\}$ is an extended generalized coherent pair if there exist complex numbers $d_{n}$, $e_{n}$, $\sigma_{n},$ and $\tau_{n},$ $n\geq 0$ such that

[TABLE]

where $d_{-1}=e_{-1}=e_{-2}=\sigma_{-1}=\tau_{-1}=\tau_{-2}=0$.

In particular, $\{u_{0},u_{1}\}$ is an extended 3-term coherent pair if $e_{n}\tau_{n}\neq 0$, $n\geq 1$.

Notice that if $e_{n}=0,$ $n\geq 1$, then extended generalized coherency is reduced to a simpler case such as extended coherency or $\{R_{n}(x)\}_{n=0}^{\infty}=\{Q_{n}(x)\}_{n=0}^{\infty}$.

This definition of an extended generalized coherent pair is equivalent to (3.3).

Remark 3.1*.*

The following show that Theorem 3.1 covers the extended coherent pair and the extended symmetric coherent pair.

• (i)

If $\tilde{\tau}_{n}=0$ and $\theta_{n}=0$ for $n\geq 0$ in (3.1), then $\{u_{0},u_{1}\}$ is an extended coherent pair.

• (ii)

Assume $u_{0}$ and $u_{1}$ are symmetric moment functionals. If $\tilde{\sigma}_{n}=0$ and $\mu_{n}=0$ for $n\geq 0$ in (3.1), then $\{u_{0},u_{1}\}$ is an extended symmetric coherent pair.

Theorem 3.2**.**

$\{u_{0},u_{1}\}$* is a generalized coherent pair if and only if the associated MOPS’s $\{S_{n}^{(\lambda)}(x)\}_{n=0}^{\infty}$ and $\{P_{n}(x)\}_{n=0}^{\infty}$ satisfy the relation in (3.1) along with (3.2).*

*Proof.  *Assume that the MOPS’s $\{S_{n}^{(\lambda)}(x)\}_{n=0}^{\infty}$ and $\{P_{n}(x)\}_{n=0}^{\infty}$ associated with the functionals $\{u_{0},u_{1}\}$ satisfy (3.1) and (3.2). Then from (3.4) and (3.5), (3.2) implies that $\tilde{d}_{n}=0$ and $\tilde{e}_{n}=0$ for $n\geq 1$ in (3.3). The proof of the converse can be found in [13]. ∎

From this theorem, notice that extended generalized coherency is an extension of generalized coherency.

4 Extended 3-term Coherent Pairs: The Classical Case

In this section, let $\{u_{0},u_{1}\}$ be an extended generalized coherent pair. And let $\{P_{n}(x)\}_{n=0}^{\infty}$ and $\{R_{n}(x)\}_{n=0}^{\infty}$ be the MOPS’s relative to quasi-definite moment functionals $u_{0}$ and $u_{1}$, respectively, and satisfy extended 3-term coherency in (3.9) with $\{Q_{n}(x):=\frac{P^{\prime}_{n+1}(x)}{n+1}\}_{n=0}^{\infty}$.

We also assume that one of the moment functionals is classical, that is, the first case is when $u_{0}$ is classical, and the second case is when $u_{1}$ is strongly classical.

For the first case, consider that $u_{0}$ is classical with $(\varphi u_{0})^{\prime}=\psi u_{0}$. Then, it is well-known([5]) that $\{Q_{n}(x):=\frac{P^{\prime}_{n+1}(x)}{n+1}\}_{n=0}^{\infty}$ is also a classical MOPS relative to $u=\varphi u_{0}$.

By acting $u$ and $u_{1}$ on (3.9), we have

[TABLE]

Theorem 4.1**.**

Let $\{u_{0},u_{1}\}$ be an extended generalized coherent pair, and let $u_{0}$ be a classical moment functional(or $u_{1}$ be a strongly classical one). Then

1. (i)

if $\sigma_{0}=d_{0}$ and $\tau_{0}=e_{0}$ (or equivalently, $\sigma_{1}=d_{1}$ and $\tau_{1}=e_{1}$), then $\{R_{n}(x)\}_{n=0}^{\infty}=\{Q_{n}(x)\}_{n=0}^{\infty}$ and $\sigma_{n}=d_{n}$ and $\tau_{n}=e_{n}$ for $n\geq 1$; 2. (ii)

if $e_{n}=\tau_{n}=0$, $n\geq 0$ (i.e., extended coherent pair) and $\sigma_{0}=d_{0}$, then $\{R_{n}(x)\}_{n=0}^{\infty}=\{Q_{n}(x)\}_{n=0}^{\infty}$ and $\sigma_{n}=d_{n}$; 3. (iii)

if $\sigma_{n}=d_{n}=0$, $n\geq 0$ (i.e., extended symmetric coherent pair) and $\tau_{0}=e_{0}$, then $\{R_{n}(x)\}_{n=0}^{\infty}=\{Q_{n}(x)\}_{n=0}^{\infty}$ and $\tau_{n}=e_{n}$ for $n\geq 1$.

*Proof.  *(i) From (4.1) and (4.2), the given conditions $\sigma_{0}=d_{0}$ and $\tau_{0}=e_{0}$ give $\langle u,R_{n}\rangle=0$ and $\langle u_{1},Q_{n}\rangle=0$, $n\geq 1$, inductively. Thus $\{R_{n}(x)\}_{n=0}^{\infty}=\{Q_{n}(x)\}_{n=0}^{\infty}$ and consequently $\sigma_{n}=d_{n}$ and $\tau_{n}=e_{n}$ for $n\geq 1$. In the same way, (ii) and (iii) (see [11]) hold. ∎

In the following, we assume that $\{u_{0},u_{1}\}$ is an extended 3-term coherent pair in (3.9) with $e_{n}\tau_{n}\neq 0$, $n\geq 1$. And we classify the 3-term extended coherent pairs when one of the moment functionals is classical or strongly classical. We find the coefficient relations between two corresponding MOPS’s relative to the functionals.

We assume that $\{Q_{n}(x)\}_{n=0}^{\infty}$ and $\{R_{n}(x)\}_{n=0}^{\infty}$ satisfy the following three-term recurrence relation, for some $b_{n},c_{n},\beta_{n},\gamma_{n},$ $n\geq 0$,

[TABLE]

Now, we find the relations between coefficients, that is, $\{b_{n},c_{n}\}$ for $\{Q_{n}(x)\}_{n=0}^{\infty}$, $\{\beta_{n},\gamma_{n}\}$ for $\{R_{n}(x)\}_{n=0}^{\infty}$, $\{\sigma_{n},\tau_{n}\}$ and $\{d_{n},e_{n}\}$ for the extended generalized coherent relation.

By multiplying (3.9) by $x$, then using (4.3) and (4.4), we have

[TABLE]

By applying (3.9) to (4.5) and then applying $R_{n-3}$ in (3.9) with $n$ replaced by $n-1$, we have

[TABLE]

where

[TABLE]

with $B_{0}=E_{0}=C_{0}=C_{1}=F_{0}=F_{1}=G_{0}=G_{1}=G_{2}=0$.

From (4.6), since the leading coefficient of both sides are the same, we have $A_{n}=D_{n}$, $n\geq 0$.

Case I: $A_{n}=D_{n}=0$, $n\geq 0$.

Then (4.6) becomes

[TABLE]

Since the leading coefficients of both sides are the same, we have $B_{n}=E_{n}$, $n\geq 1$.

Case I-1: $B_{n}=E_{n}=0$, $n\geq 1$.

Then (4.8) becomes

[TABLE]

Since the leading coefficients of both sides are the same, we have $C_{n}=F_{n}$, $n\geq 2$.

Case I-1-(i): $C_{n}=F_{n}=0$, $n\geq 2$.

Then $G_{n}=0$, $n\geq 3$. Hence we have from (4.7)

[TABLE]

Case I-1-(ii): $C_{n}=F_{n}\neq 0$, $n\geq 2$.

Then (4.9) is to be

[TABLE]

Hence with (3.9), $G_{n}=0$, $n\geq 3$. This implies $\{R_{n}(x)\}_{n=0}^{\infty}=\{Q_{n}(x)\}_{n=0}^{\infty}$.

Case I-2: $B_{n}=E_{n}\neq 0$, $n\geq 1$.

Then (4.8) is to be

[TABLE]

Hence with (3.9) this is reduced to trivial case or 2-term extended coherency, not the extended 3-term case.

Case II: $A_{n}=D_{n}\neq 0$, $n\geq 0$.

Then by (3.9), (4.6) is to be

[TABLE]

Since the leading coefficients of both side are the same, we have

[TABLE]

Case II-1: $K_{n}=0$, $n\geq 1$.

Then (4.11) is to be

[TABLE]

Hence,

[TABLE]

Case II-1-(i): $X_{n}=0$, $n\geq 2$.

Then $G_{n}=0$, $n\geq 3$. Hence we have from (4.7)

[TABLE]

Case II-1-(ii): $X_{n}\neq 0$, $n\geq 2$.

Then (4.12) is to be

[TABLE]

Then with (3.9), $G_{n}=0$, $n\geq 3$. This implies $\{R_{n}(x)\}_{n=0}^{\infty}=\{Q_{n}(x)\}_{n=0}^{\infty}$.

Case II-2: $K_{n}\neq 0$, $n\geq 1$.

Then (4.11) is to be

[TABLE]

Then with (3.9), this is not the extended 3-term case. ∎

For the second case, we define $\{T_{n}(x)\}_{n=0}^{\infty}$ as $R_{n}(x)=\frac{T^{\prime}_{n+1}(x)}{n+1}$, then we integrate (3.9). Hence we have

[TABLE]

where $e_{-1}=\tau_{-1}=0$. For $\{T_{n}(x)\}_{n=0}^{\infty}$ to be classical, we assume $u_{1}$ is strongly classical. Then this case is the same as the first case with different notations. So we have the same result as the first case: when $u_{0}$ is classical.

So far, we find the relations concerned with the coefficients of extended generalized coherent pairs and the three-term recurrence relations of the corresponding MOPS when one of the moment functional is classical (or strongly classical).

In this classification, there are four pairs of coefficients, that is, $\{d_{n},e_{n}\}$ and $\{\sigma_{n},\tau_{n}\}$ ($n\geq 0$) for extended 3-term coherent pairs in (3.9) and $\{b_{n},c_{n}\}$, $\{\beta_{n},\gamma_{n}\}$ ($n\geq 0$) for the three-term recurrence coefficients of $\{Q_{n}(x)\}_{n=0}^{\infty}$ and $\{R_{n}(x)\}_{n=0}^{\infty}$, respectively in (4.3) and (4.4). To find other coefficient relations for four pairs when some of the pairs are given, it is enough to give only one pair.

In the above classification, there are two cases which are the extended 3-term coherency.

For the first case Case I-1-(i), (4.10) gives

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Now, we consider four cases that one of the four pairs is given, then we find coefficient relations for the other three pairs as follows.

Case 1 $\{\sigma_{n},\tau_{n}\}$: For the first case, when $\{\sigma_{n},\tau_{n}\}$ are given, then from (4.16) and (4.18), we have

[TABLE]

Hence, we have $b_{n}$, $n\geq 0$ from (4.22), and also we have $c_{n}$, $n\geq 1$ from (4.22) and (4.23) recursively. Next, from (4.17), (4.19), and (4.21), we have

[TABLE]

Hence, from (4.15) and (4.24)—(4.26), we find $\beta_{n}$, $\gamma_{n}$, $d_{n}$, and $e_{n}$, $n\geq 1$, recursively with initial condition $d_{0}$ and $e_{0}$ (or equivalently $\beta_{0}$ and $\gamma_{1}$) in the order of $[\beta_{n},\gamma_{n+1},e_{n+1},d_{n+1}]$, $n\geq 1$.

Case 2 $\{b_{n},c_{n}\}$: Now, consider the second case by assuming that only $\{b_{n},c_{n}\}$ are given, then (4.16) implies

[TABLE]

we get $\sigma_{n}$, $n\geq 1$ with initial condition $\sigma_{0}$. (4.18) and (4.21) imply

[TABLE]

we have $\tau_{n}$, $n\geq 0$. Then, in the same way as the first case, we get $\beta_{n}$, $\gamma_{n}$, $d_{n}$, and $e_{n}$, $n\geq 1$, recursively with initial condition $d_{0}$ and $e_{0}$ (or equivalently $\beta_{0}$ and $\gamma_{1}$) in the order of $[\beta_{n},\gamma_{n+1},e_{n+1},d_{n+1}]$, $n\geq 1$.

Case 3 $\{\beta_{n},\gamma_{n}\}$: For third case, we assume that only $\{\beta_{n}$, $\gamma_{n}\}$ are given, then (4.15) implies

[TABLE]

hence we get $d_{n}$, $n\geq 1$ with initial condition $d_{0}$. And (4.17) implies

[TABLE]

hence we have $e_{n}$, $n\geq 1$ with the initial $e_{0}$.

Then, in the same way as the first case, we get $b_{n}$, $c_{n}$, $\sigma_{n}$, and $\tau_{n}$, $n\geq 1$, recursively with initial condition $d_{0}$ and $e_{0}$ (or equivalently $\beta_{0}$ and $\gamma_{1}$) in the order of $[b_{n},c_{n+1},\tau_{n+1},\sigma_{n+1}]$, $n\geq 1$.

Case 4 $\{d_{n},e_{n}\}$: Lastly, we assume $d_{n},d_{n}$ are given, then we find from (4.15) $\beta_{n}$, $n\geq 0$, from (4.17) we find $\gamma_{n}$, $n\geq 1$.

[TABLE]

Then, in the same way as the first case, we get $b_{n}$, $c_{n}$, $\sigma_{n}$, and $\tau_{n}$, $n\geq 1$, recursively with initial condition $\sigma_{0}$ and $\tau_{0}$ in the order of $[b_{n},c_{n+1},\tau_{n+1},\sigma_{n+1}]$, $n\geq 1$.

For the second case Case II-1-(i), we have from (4.13)

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

In this case, (4.32)—(4.37) are related to each of the two pairs, (4.33) and (4.35) are related to the first two pairs $\{\beta_{n},\gamma_{n}\}$ and $\{d_{n},e_{n}\}$ and (4.34) and (4.36) are related to the second two pairs $\{b_{n},c_{n}\}$ and $\{\sigma_{n},\tau_{n}\}$. Thus to solve four pairs of coefficients, we need at least two pairs of coefficients, of which one must be given from the first two pairs, and the other is given from the second two pairs.

From these classifications, we conclude with the following theorem.

Theorem 4.2**.**

Let $v$ and $u$ be moment functionals with the corresponding MOPS’s $\{R_{n}(x)\}_{n=0}^{\infty}$ and $\{Q_{n}(x)\}_{n=0}^{\infty}$, respectively, are related by (3.9) with $e_{n}\tau_{n}\neq 0$, $n\geq 0$. And $\{Q_{n}(x)\}_{n=0}^{\infty}$ and $\{R_{n}(x)\}_{n=0}^{\infty}$ satisfy the three-term recurrence relations (4.3) and (4.4), respectively.

Then we have the relation (4.6), $A_{n}=D_{n}$, $n\geq 0$, and $G_{n}=0$, $n\geq 3$, that is,

[TABLE]

[TABLE]

Moreover, there are only two cases for extended 3-term coherent pairs.

• (I)

When $A_{n}=D_{n}=0,$ $n\geq 0$, $B_{n}=E_{n}=0,$ $n\geq 1$, and $C_{n}=F_{n}=0,$ $n\geq 2$ in (4.6), the coefficients for the above four pairs have the relations (4.15)—(4.21).

And for getting four pairs of coefficients, it is enough that only one of the pairs is given, then we find other three pairs of coefficients relations as Case 1 — Case 4

• (II)

When $A_{n}=D_{n}\neq 0,$ $n\geq 0$, ${B_{n}}+A_{n}d_{n-1}={E_{n}}+A_{n}\sigma_{n-1}=0,$ $n\geq 1$, and ${C_{n}}-A_{n}e_{n-2}={F_{n}}-A_{n}\tau_{n-2}=0,~{}n\geq 2,$ the coefficients for the above four pairs have the relations (4.32)—(4.37).

And for getting four pairs of coefficients, it is enough that only two of the pairs are given, of which one is one of $\{\beta_{n},\gamma_{n}\}$ and $\{d_{n},e_{n}\}$ and the other is one of $\{b_{n},c_{n}\}$ and $\{\sigma_{n},\tau_{n}\}$, then we find other two pairs of coefficients relations from (4.32)—(4.37).

Now, we study the companion moment functional of the extended 3-term coherent pairs when $u_{0}$ is classical or when $u_{1}$ is strongly classical. When $u_{1}$ is strong classical, we have the same result as when $u_{0}$ is classical. Hence we state the following:

Theorem 4.3**.**

Let $v$ and $u$ be moment functionals with the corresponding MOPS’s $\{R_{n}(x)\}_{n=0}^{\infty}$ and $\{Q_{n}(x)\}_{n=0}^{\infty}$, respectively, related by (3.9). Then there exist polynomials $A(x)$ and $D(x)$ of degree 2, satisfying

[TABLE]

*Proof.  *Let $A(x)$ and $D(x)$ be polynomials of degree 2. We define $M_{k}:=\langle A(x)v,R_{k}(x)\rangle$, $k\geq 0$.

By using (3.9), we have

[TABLE]

and

[TABLE]

Since $M_{k}=0$, $k\geq 3$, only $M_{0}$, $M_{1}$, and $M_{2}$ remain.

To satisfy (4.39) inductively, $\langle A(x)v,Q_{3}(x)\rangle=\langle A(x)v,Q_{4}(x)\rangle=0$ hold in (4.40).

Since $\deg(A(x))=2$, we have $M_{2}=\langle A(x)v,R_{2}(x)\rangle\neq 0$. Hence we can solve $M_{0}$ and $M_{1}$ from (4.40) (iv) and (v). Then these $M_{0}$ and $M_{1}$ make (4.40) (iii) $\frac{{e}_{2}M_{2}}{\tau_{2}}\neq 0$. Thus, we have a polynomial $D(x)$ of degree 2, satisfying $A(x)v=D(x)u$ and (4.40). ∎

Remark 4.1*.*

(i) When a strongly classical moment functional $v$ with MOPS $\{Q_{n}(x)\}_{n=0}^{\infty}$ and a classical moment functional $u_{0}$ with MOPS $\{P_{n}(x)\}_{n=0}^{\infty}$ satisfy $Q_{n}(x)=\frac{1}{n+1}P^{\prime}_{n+1}(x)$, then $(\varphi u)^{\prime}=\psi u$ and $\varphi u_{0}=u$. Hence Theorem 4.3 implies, if $\{u_{0},u_{1}\}$ is an extended 3-term coherent pair, then

[TABLE]

and $2\leq\deg(D(x)\varphi)\leq 4$, in detail

[TABLE]

(ii) When $u_{0}$ and $u_{1}$ are a symmetric coherent pair, then Theorem 4.3 gives the result of Theorem 3.5 in [7].

When only $\{b_{n},c_{n}\}$ are given in Case 2, we describe the algorithm in order to obtain $\sigma_{n}$, $\tau_{n}$, $\beta_{n}$, $\gamma_{n}$, $d_{n}$, and $e_{n}$, $n\geq 1$. Note that $\sigma_{n}$ and $\tau_{n}$, $n\geq 1$, can be obtained without concerning other parameters. So we construct the algorithm with two parts.

1. STEP I

FROM (4.27) FINDING $\{\sigma_{n}\}$ AND FROM (4.28) FINDING$\{\tau_{n}\}$

• STARTING INITIAL VALUES:

[TABLE]

• STEP I-1

FROM (4.27) WE GET $\sigma_{n}$ RECURSIVELY BY

[TABLE]

• STEP I-2

FROM (4.28) WE GET $\tau_{n}$ RECURSIVELY BY

[TABLE]

[TABLE]

[TABLE] 2. STEP II

FINDING $\{\beta_{n}\}$, $\{\gamma_{n}\}$, $\{d_{n}\}$, AND $\{e_{n}\}$

• STARTING INITIAL VALUES:

[TABLE]

• STEP II-1

FROM (4.15) AND (4.24) FINDING

[TABLE]

• STEP II-2

FROM (4.26) AND (4.25), WE FIND THE FOLLOWING ONE BY ONE

[TABLE]

• STEP II-3

FROM (4.10), (4.24), (4.26), AND (4.25), WE FIND THE FOLLOWING ONE BY ONE

[TABLE]

[TABLE]

• STEP II-$n$

FROM (4.10), (4.24), (4.26), AND (4.25), WE FIND THE FOLLOWING ONE BY ONE

[TABLE]

Example 4.1**.**

Let $u_{0}=u_{J}^{(1,0)}$ be a Jacobi moment functional, that is,

[TABLE]

Then $u=\varphi u_{0}=(1-x^{2})u_{0}$ and the corresponding OPS is $\{P^{(2,1)}_{n}(x)\}_{n=0}^{\infty}$. So from [21], the MOPS $\{Q_{n}(x)\}_{n=0}^{\infty}$ relative to $u$ is

[TABLE]

and $\{Q_{n}(x)\}_{n=0}^{\infty}$ satisfy the following three-term recurrence relation

[TABLE]

[TABLE]

• (i)

To find coefficients for four pairs, this is Case 2: $\{b_{n},c_{n}\}$. From STEP I of the above algorithm and (4.27) and (4.28), we can find the parameters $\sigma_{n}$ and $\tau_{n}$ with initial $\sigma_{0}$ of recurrence relation in (3.9) as the following

[TABLE]

[TABLE]

[TABLE]

Also, we can find from STEP II of the above algorithm $\beta_{n}$ and $\gamma_{n}$ which are the three-term recurrence relation coefficients of $\{R_{n}(x)\}_{n=0}^{\infty}$ in (4.3) and $d_{n}$ and $e_{n}$ in (3.9) the coefficients of extended 3-term coherent pairs, recursively.

• (ii)

We assume $A(x)=(1-x)^{2}$ and $D(x)=(1+x)^{2}$ in (4.38) with $v=u_{1}$ under the same conditions as in (i), then

[TABLE]

where $M=\langle u_{1},1\rangle=\gamma_{0}$ and $N=\langle u_{1},1-x\rangle=\gamma_{0}(1-\beta_{0})$ are constants.(see Lemma2.4(iii) in [13] and [15]) Therefore, $u_{1}$ is a perturbed Jacobi moment functional.

• (iii)

We assume $A(x)=(1-x)(1+x)$ and $D(x)=(1-x)(1+x)$ in (4.38) with $v=u_{1}$ under the same conditions as in (i), then

[TABLE]

where $M=\frac{1}{2}\langle u_{1},1-x\rangle=\frac{1}{2}\gamma_{0}(1-\beta_{0})$ and $N=-\frac{1}{2}\langle u_{1},1+x\rangle=-\frac{1}{2}\gamma_{0}(1+\beta_{0})$ are constants.(see Lemma2.4(ii) in [13] and [15]) Therefore, $u_{1}$ is a perturbed Jacobi moment functional(see [14]).

We have different companion moment functionals in (ii) and (iii). This is because of different initial values $\sigma_{0}$. ∎

Example 4.2**.**

Let $u_{1}=u_{L}^{(2)}$ be a Laguerre moment functional, that is,

[TABLE]

Then the corresponding OPS is $\{L^{(2)}_{n}(x)\}_{n=0}^{\infty}$. So from [21], the MOPS $\{R_{n}(x)\}_{n=0}^{\infty}$ relative to $u_{1}$ is

[TABLE]

and $\{Q_{n}(x)\}_{n=0}^{\infty}$ satisfy the following three-term recurrence relation

[TABLE]

[TABLE]

• (i)

To find the coefficients for four pairs, this case is Case 3: $\{\beta_{n},\gamma_{n}\}$. From (4.24)—(4.26), we can find the parameters $d_{n}$ and $e_{n}$ with initial $d_{0}$ and $e_{0}$ as the following

[TABLE]

[TABLE]

[TABLE]

Also, we can find from STEP II of the above algorithm $\beta_{n}$ and $\gamma_{n}$ which are the three-term recurrence relation coefficients of $\{R_{n}(x)\}_{n=0}^{\infty}$ in (4.3) and $\sigma_{n}$ and $\tau_{n}$ in (3.9) the coefficients of extended 3-term coherent pairs, recursively.

• (ii)

We assume $A(x)=x^{2}$ and $D(x)=(1+x)^{2}$ in (4.38) with $v=u_{1}$ under the same conditions as in (i), then

[TABLE]

where $M=\langle u,1\rangle=c_{0}$ and $N=\langle u,1+x\rangle=c_{0}(1+b_{0})$ are constants.(see Lemma2.4(iii) in [13] and [15]) Moreover, the companion of $u_{1}$ is $\phi u_{0}=u$, that is $u_{0}=\frac{1}{x}u+K\delta(x)$, where $K=\langle u_{0},1\rangle$.(see Lemma2.4(i) in [13] and [15])

Therefore, $u$ and $u_{0}$ are different from the perturbed Laguerre moment functionals. ∎