q-DIFFERENTIAL EQUATIONS FOR q-CLASSICAL POLYNOMIALS AND q-JACOBI-STIRLING NUMBERS

The q-classical polynomials are orthogonal polynomial sequences that are eigenfunctions of a second order q-differential operator of a certain type. We explicitly construct q-differential equations of arbitrary even order fulfilled by these polynomials, while giving explicit expressions for the integer composite powers of the aforementioned second order q-differential operator. The latter is accomplished through the introduction of a new set of numbers, the q-Jacobi Stilring numbers, whose properties along with a combinatorial interpretation are thoroughly worked out. The results here attained are the q-analogue of those given by Everitt et al. and the first author, whilst the combinatorics of this new set of numbers is a q-analogue version of the Jacobi-Stirling numbers given by Gelineau and the second author.