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

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 q ‐differential 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.