The purpose of this article is to generalize the ring of q-Appell polynomials to the complex case. The formulas for q-Appell polynomials thus appear again, with similar names, in a purely symmetric way. Since these complex q-Appell polynomials are also q-complex analytic functions, we are able to give a first example of the q-Cauchy-Riemann equations. Similarly, in the spirit of Kim and Ryoo, we can define q-complex Bernoulli and Euler polynomials. Previously, in order to obtain the q-Appell polynomial, we would make a q-addition of the corresponding q-Appell number with x. This is now replaced by a q-addition of the corresponding q-Appell number with two infinite function sequences C_ν,q(x,y) and S_ν,q(x,y) for the real and imaginary part of a new so-called q-complex number appearing in the generating function. Finally, we can prove q-analogues of the Cauchy-Riemann equations.
Ernst, Thomas. On the complex q-Appell polynomials. Annales Universitatis Mariae Curie-Skłodowska. Mathematica, Tome 74 (2020) no. 1. http://geodesic.mathdoc.fr/item/AUM_2020_74_1_a5/
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