Formulas involving sums of powers, special numbers and polynomials arising from p-adic integrals, trigonometric and generating functions

Abstract

The formula for the sums of powers of positive integers, given by Faulhaber in 1631, is proven by using trigonometric identities and some properties of the Bernoulli polynomials. Using trigonometric functions identities and generating functions for some well-known special numbers and polynomials, many novel formulas and relations including alternating sums of powers of positive integers, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the Fubini numbers, the Stirling numbers, the tangent numbers are also given. Moreover, by applying the Riemann integral and p-adic integrals involving the fermionic p-adic integral and the Volkenborn integral, some new identities and combinatorial sums related to the aforementioned numbers and polynomials are derived. Furthermore, we serve up some revealing and historical remarks and observations on the results of this paper.

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

“…(cf. [30,31]; see also [26][27][28][29]). Kucukoglu and Simsek [32] defined a new sequence of special numbers β n (k) by means of the following generating function:…”

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:…”

Extended Degenerate r-Central Factorial Numbers of the Second Kind and Extended Degenerate r-Central Bell Polynomials

“…As general references on polynomials, functions, combinatorics and probability, one may refer to [2,5,11,13,33,34]. In addition, the reader refers to [1,4,[6][7][8][9]14,17] for some related special numbers and polynomials. For the rest of this section, we recall the facts that are needed throughout this paper.…”