Derivation of Computational Formulas for certain class of finite sums: Approach to Generating functions arising from $p$-adic integrals and special functions

TL;DR

This paper develops generating functions for special finite sums using $p$-adic integrals and series, leading to new computational formulas, algorithms, and relations among zeta functions and polynomials.

Contribution

It introduces novel generating functions and computational formulas for finite sums involving special numbers and polynomials, connecting them to zeta functions and solving existing open problems.

Findings

New generating functions for finite sums involving special numbers.

A computational algorithm for sums using Bernoulli and Stirling numbers.

Relations among multiple zeta functions, Bernoulli, and Euler polynomials.

Abstract

The aim of this paper is to construct generating functions for some families of special finite sums with the aid of the Newton-Mercator series, hypergeometric series, and $p$-adic integral (the Volkenborn integral). By using these generating functions, their functional equations, and their partial derivative equations, many novel computational formulas involving the special finite sums of (inverse) binomial coefficients, the Bernoulli type polynomials and numbers, Euler polynomials and numbers, the Stirling numbers, the (alternating) harmonic numbers, the Leibnitz polynomials and others. Among these formulas, by considering a computational formula which computes the aforementioned certain class of finite sums with the aid of the Bernoulli numbers and the Stirling numbers of the first kind, we present a computation algorithm and we provide some of their special values. Morover, using the aforementioned special finite sums and combinatorial numbers, we give relations among multiple alternating zeta functions, the Bernoulli polynomials of higher order and the Euler polynomials of higher order. We also give decomposition of the multiple Hurwitz zeta functions with the aid of finite sums. Relationships and comparisons between the main results given in the article and previously known results have been criticized. With the help of the results of this paper, the solution of the problem that Charalambides [8, Exercise 30, p. 273] gave in his book was found and with the help of this solution, we also find very new formulas. In addition, the solutions of some of the problems we have raised in [48] are also given.