  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.
