q-Pascal and q-Wronskian Matrices with Implications to q-Appell Polynomials

Abstract EN

We introduce a q-deformation of the Yang and Youn matrix approach for Appell polynomials.

This will lead to a powerful machinery for producing new and old formulas for q-Appell polynomials, and in particular for q-Bernoulli and q-Euler polynomials.

Furthermore, the q-L-polynomial, anticipated by Ward, can be expressed as a sum of products of q-Bernoulli and q-Euler polynomials.

The pseudo q-Appell polynomials, which are first presented in this paper, enable multiple q-analogues of the Yang and Youn formulas.

The generalized q-Pascal functional matrix, the q-Wronskian vector of a function, and the vector of q-Appell polynomials together with the q-deformed ϵ matrix multiplication from the authors recent article are the main ingredients in the process.

Beyond these results, we give a characterization of q-Appell numbers, improving on Al-Salam 1967.

Finally, we find a q-difference equation for the q-Appell polynomial of degree n.