Second structure relation for $q$-semiclassical polynomials of the Hahn Tableau

Hahn型正交性对于预解Delta算子 $\mathscr L=D(I-D)^{-1}$

Hahn-Type Orthogonality for the Resolvent Delta Operator $\mathscr L=D(I-D)^{-1}$

本文研究预解Delta算子下的正交性，证明刚性定理并给出满足Hahn关系的对角族，揭示Delta算子降阶与半经典正交性的结构联系。

Limit relations between $q$-Krall type orthogonal polynomials

In this paper, we consider a natural extension of several results related to Krall-type polynomials introducing a modification of a $q$-classical linear functional via the addition of one or two mass points. The limit relations between the $q$-Krall type modification of big $q$-Jacobi, little $q$-Jacobi, big $q$-Laguerre, and other families of the $q$-Hahn tableau are established.

半经典正交多项式研究补遗

Epilegomena to the study of semiclassical orthogonal polynomials

本文针对Ismail关于Askey-Wilson算子结构关系刻画$\text{D}_q$-经典正交多项式的猜想，给出$\text{D}_q$-半经典正交多项式的两个刻画定理，重建标准导数与Askey-Wilson算子的对称性，构造二阶多项式序列证伪猜想并拓展至Hahn算子。

The Kontorovich-Lebedev transform as a map between $d$-orthogonal polynomials

A slight modification of the Kontorovich-Lebedev transform is an automorphism on the vector space of polynomials. The action of this $KL_α$-transform over certain polynomial sequences will be under discussion, and a special attention will be given the d-orthogonal ones. For instance, the Continuous Dual Hahn polynomials appear as the $KL_α$-transform of a 2-orthogonal sequence of Laguerre type. Finally, all the orthogonal polynomial sequences whose $KL_α$-transform is a $d$-orthogonal sequence will be characterized: they are essencially semiclassical polynomials fulfilling particular condition…

$q$-Classical orthogonal polynomials: A general difference calculus approach

It is well known that the classical families of orthogonal polynomials are characterized as eigenfunctions of a second order linear differential/difference operator. In this paper we present a study of classical orthogonal polynomials in a more general context by using the differential (or difference) calculus and Operator Theory. In such a way we obtain a unified representation of them. Furthermore, some well known results related to the Rodrigues operator are deduced. A more general characterization Theorem that the one given in [1] and [2] for the q-polynomials of the q-Askey and Hahn Table…

Characterization of Orthogonal Polynomials on lattices

We consider two sequences of orthogonal polynomials $(P_n)_{n\geq 0}$ and $(Q_n)_{n\geq 0}$ such that $$ \sum_{j=1} ^{M} a_{j,n}\mathrm{S}_x\mathrm{D}_x ^k P_{k+n-j} (z)=\sum_{j=1} ^{N} b_{j,n}\mathrm{D}_x ^{m} Q_{m+n-j} (z)\;, $$ with $k,m,M,N \in \mathbb{N}$, $a_{j,n}$ and $b_{j,n}$ are sequences of complex numbers, $$2\mathrm{S}_xf(x(s))=(\triangle +2\,\mathrm{I})f(z),~~ \mathrm{D}_xf(x(s))=\frac{\triangle}{\triangle x(s-1/2)}f(z),$$ $z=x(s-1/2)$, $\mathrm{I}$ is the identity operator, $x$ defines a lattice, and $\triangle f(s)=f(s+1)-f(s)$. We show that under some natural conditions, both …