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.