Expansion formulas for Apostol type $q$-Appell polynomials, and their special cases

Abstract

We present identities of various kinds for generalized $q$-Apostol-Bernoulli
and Apostol-Euler polynomials and power sums, which resemble $q$-analogues of formulas from the 2009 paper by Liu and Wang. These formulas are divided into two types: formulas with only $q$-Apostol-Bernoulli, and only $q$-Apostol-Euler polynomials, or so-called mixed formulas, which contain polynomials of both kinds.
This can be seen as a logical consequence of the fact that the $q$-Appell polynomials form a commutative ring. The functional equations for Ward numbers operating on the $q$-exponential function, as well as symmetry arguments, are essential for many of the proofs.
We conclude by finding multiplication formulas for two $q$-Appell polynomials of general form. This brings us to the $q$-H polynomials, which were discussed in a previous paper.