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

Abstract

We present a new family ${\left\{S_n(x;q)\right\}}_{n\ge 0}$ 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.

12 pages