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## Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

2020 | 74 | 1 |
Tytuł artykułu

### On the complex q-Appell polynomials

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EN
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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 $$C_{\nu,q}(x,y)$$ and $$S_{\nu,q}(x,y)$$ 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.
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2020
online
2020-10-20
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Bibliografia
• Brinck, I., Persson, A., Elementar teori for analytiska funktioner (Swedish) (Elementary theory for analytic functions), Lund, 1979.
• Ernst, T., A Comprehensive Treatment of q-calculus, Birkhauser, Basel, 2012.
• Ernst T., A new semantics for special functions, to appear.
• Kim, T., Ryoo, C. S., Some identities for Euler and Bernoulli polynomials and their zeros, Axioms 7 (3), 56 (2018), pp. 19.
• Kim, D., A note on the degenerate type of complex Appell polynomials, Symmetry 11 (11), 1339 (2019), pp. 14.
• Range, R., Holomorphic Functions and Integral Representations in Several Complex Variables, Springer-Verlag, New York, 1986.
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