On Generating Functions for Parametrically Generalized Polynomials Involving Combinatorial, Bernoulli and Euler Polynomials and Numbers

The aim of this paper is to give generating functions for parametrically generalized polynomials that are related to the combinatorial numbers, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the cosine-Bernoulli polynomials, the sine-Bernoulli polynomials, the cosine-Euler polynomials, and the sine-Euler polynomials. We investigate some properties of these generating functions. By applying Euler's formula to these generating functions, we derive many new and interesting formulas and relations related to these special polynomials and numbers mentioned as above. Some special cases of the results obtained in this article are examined. With this special case, detailed comments and comparisons with previously available results are also provided. Furthermore, we come up with open questions about interpolation functions for these polynomials. The main results of this paper highlight the existing symmetry between numbers and polynomials in a more general framework. These include Bernouilli, Euler, and Catalan polynomials.

Automated summary

This paper investigates the generating functions of parametrically generalized polynomials related to combinatorial numbers, Bernoulli polynomials and numbers, Euler polynomials and numbers, cosine-Bernoulli polynomials, sine-Bernoulli polynomials, cosine-Euler polynomials, and sine-Euler polynomials. By applying Euler's formula, the paper derives new formulas and relations for these special polynomials and numbers. The paper also examines special cases and provides comparisons with previous results. Additionally, open questions about interpolation functions for these polynomials are addressed. The main results of the paper highlight the symmetry between numbers and polynomials in a more general framework, including Bernoulli, Euler, and Catalan polynomials.