Computational and implementational analysis of generating functions for higher order combinatorial numbers and polynomials attached to Dirichlet characters

Abstract

The main purpose of this paper is to give computational and implementational analysis of generating functions for some extensions of combinatorial numbers and polynomials attached to Dirichlet characters. By using generating function methods and p‐adic q‐integral techniques, it is also aimed to derive some computational formulas for these numbers and polynomials. By implementing both the derived computational formulas and the constructed generating functions in Mathematica, we present some tables and plots for these numbers and polynomials, and their generating functions to illustrate the effects of their parameters for some special cases. Moreover, by making an observation on some results of this paper, we derive some novel computational formulas for the finite sums that contain the Dirichlet characters and falling factorials. We also assess some cases of these special finite sums. In the sequel, we pose some open problems regarding these special finite sums. Apart from these findings, we also give not only Fourier series expansion but also asymptotic estimates for the mentioned combinatorial numbers. In addition, we give some Raabe‐type multiplication formulas for the mentioned combinatorial polynomials. Finally, we give an observation on decompositions of the generating functions attached to any group homomorphism.

Convergence Properties of a Kantorovich Type of Szász Operators Involving Negative Order Genocchi Polynomials

“…In analytic number theory, the generating functions method has an important place because this method provides to construct many useful and significant results, identities, and theorems for special polynomials and numbers (Simsek, 2008;2012;2013;2017;2018;Kucukoglu et al, 2019;Kucukoglu, 2022;Kilar & Simsek, 2020). The following is a definition of the Genocchi polynomials' generating function:…”