Derivation of computational formulas for certain class of finite sums: Approach to generating functions arising from p$$ p $$‐adic integrals and special functions

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