The annihilation operator for certain family of q-Hermite Sobolev-type orthogonal polynomials

TL;DR

This paper introduces a new family of Sobolev-type orthogonal polynomials related to q-Hermite polynomials, providing connection formulas and an annihilation operator, expanding the understanding of their algebraic and analytical properties.

Contribution

The paper develops a new family of Sobolev-type orthogonal polynomials associated with q-Hermite polynomials and derives their annihilation operator, a novel contribution to the theory of q-orthogonal polynomials.

Findings

Defined a new family of Sobolev-type orthogonal polynomials

Derived connection formulas for these polynomials

Established the annihilation operator for the family

Abstract

We present a new family $\left\{ S_{n}(x;q)\right\} _{n\geq 0}$ of monic polynomials in $x$, orthogonal with respect to a Sobolev-type inner product related to the $q$-Hermite I orthogonal polynomials, involving a first-order $q$-derivative on a mass-point $\alpha \in \mathbb{R}$ located out of the corresponding orthogonality interval $[-1,1]$, for some fixed real number $q \in (0, 1)$. We present connection formulas, and the annihilation operator for this non-standard orthogonal polynomial family.