On the complex q-Appell polynomials

Thomas Ernst

Abstract


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.

Keywords


Complex q-Appell polynomials; q-complex numbers; q-complex Bernoulli and Euler polynomials; q-Cauchy-Riemann equations

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References


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.




DOI: http://dx.doi.org/10.17951/a.2020.74.1.31-43
Data publikacji: 2020-10-20 20:08:02
Data złożenia artykułu: 2020-10-10 21:52:14


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