Generating functions for finite sums involving higher powers of binomial coefficients: Analysis of hypergeometric functions including new families of polynomials and numbers

摘要

The aim of this paper is to construct and investigate some of the fundamental generalizations and unifications of new families of polynomials and numbers involving finite sums of higher powers of binomial coefficients and the Franel numbers by means of suitable generating functions and hypergeometric function. We derive several fundamental properties involving the generating functions, formulas, recurrence relations for these polynomials and numbers. These new families of polynomials and numbers are shown to generalize some known special polynomials such as the Legendre polynomials, the Michael Vowe polynomials, the Mirimanoff polynomials, Golombek type polynomials and also the Franel numbers. We give relations between the generalized Franel numbers, the Legendre polynomials, the Bernoulli numbers, the Stirling numbers, the Catalan numbers and other special numbers. Using the Riemann integral and p-adic integrals representations of these new polynomials, we derive combinatorial sums and identities related to sums of powers of binomial coefficients. Moreover, we introduce some revealing and historical remarks and observations on the finite sums of powers of binomial coefficients, special polynomials and numbers. Appropriate connections of identities, formulas, relations and results given in this paper with those in earlier and future studies are pointed out in detail. Special values of explicit formulas of our new numbers give solutions of the open problem 1 raised by Srivastava [58, p. 416, Open Problem 1]. Finally, we pose two open questions related to ordinary generating functions for these new numbers. (C) 2019 Elsevier Inc. All rights reserved.