‐differential equations for ‐classical polynomials and ‐Jacobi–Stirling numbers

Abstract

ABSTRACT. We introduce, characterise and provide a combinatorial interpretation for the so-called q-Jacobi-Stirling numbers. This study is motivated by their key role in the (reciprocal) expansion of any power of a second order qdifferential operator having the q-classical polynomials as eigenfunctions in terms of other even order operators, which we explicitly construct in this work. The results here obtained can be viewed as the q-version of those given by Everitt et al. and by the first author, whilst the combinatorics of this new set of numbers is a q-version of the Jacobi-Stirling numbers given by Gelineau and the second author.