Geodesic
On the complex q-Appell polynomials
Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 74 (2020) no. 1.

Voir la notice de l'article provenant de la source Library of Science

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.
Keywords: Complex q-Appell polynomials, q-complex numbers, q-complex Bernoulli and Euler polynomials, q-Cauchy-Riemann equations 
@article{AUM_2020_74_1_a5,
     author = {Ernst, Thomas},
     title = {On the complex {q-Appell} polynomials},
     journal = {Annales Universitatis Mariae Curie-Sk{\l}odowska. Mathematica },
     publisher = {mathdoc},
     volume = {74},
     number = {1},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AUM_2020_74_1_a5/}
}
TY  - JOUR
AU  - Ernst, Thomas
TI  - On the complex q-Appell polynomials
JO  - Annales Universitatis Mariae Curie-Skłodowska. Mathematica 
PY  - 2020
VL  - 74
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AUM_2020_74_1_a5/
LA  - en
ID  - AUM_2020_74_1_a5
ER  - 
%0 Journal Article
%A Ernst, Thomas
%T On the complex q-Appell polynomials
%J Annales Universitatis Mariae Curie-Skłodowska. Mathematica 
%D 2020
%V 74
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AUM_2020_74_1_a5/
%G en
%F AUM_2020_74_1_a5
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/

[1] Brinck, I., Persson, A., Elementar teori for analytiska funktioner (Swedish) (Elementary theory for analytic functions), Lund, 1979.

[2] Ernst, T., A Comprehensive Treatment of q-calculus, Birkhauser, Basel, 2012.

[3] Ernst T., A new semantics for special functions, to appear.

[4] Kim, T., Ryoo, C. S., Some identities for Euler and Bernoulli polynomials and their zeros, Axioms 7 (3), 56 (2018), pp. 19.

[5] Kim, D., A note on the degenerate type of complex Appell polynomials, Symmetry 11 (11), 1339 (2019), pp. 14.

[6] Range, R., Holomorphic Functions and Integral Representations in Several Complex Variables, Springer-Verlag, New York, 1986.