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

Abstract

The origin of this study is based on not only explicit formulas of finite sums involving higher powers of binomial coefficients, but also explicit evaluations of generating functions for this sums. It should be emphasized that this study contains both new results and literature surveys about some of the related results that have existed so far. With the aid of hypergeometric function, generating functions for a new family of the combinatorial numbers, related to finite sums of powers of binomial coefficients, are constructed. By using these generating functions, a number of new identities have been obtained and at the same time previously well-known formulas and identities have been generalized. Moreover, on this occasion, we identify new families of polynomials including some families of well-known numbers such as Bernoulli numbers, Euler numbers, Stirling numbers, Franel numbers, Catalan numbers, Changhee numbers, Daehee numbers and the others, and also for the polynomials such as the Legendre polynomials, Michael Vowe polynomial, the Mirimanoff polynomial, Golombek type polynomials, and the others. We also give both Riemann and $p$-adic integral representations of these polynomials. Finally, we give combinatorial interpretations of these new families of numbers, polynomials and finite sums of the powers of binomial coefficients. We also give open questions for ordinary generating functions for these numbers.

36 pages

References in corpus (2)

• $q$-Bernoulli Numbers and Polynomials Associated with Multiple $q$-Zeta Functions and Basic $L$-series
• New families of special numbers for computing negative order Euler numbers

Cited by in corpus (3)

• New integral formulas and identities involving special numbers and functions derived from certain class of special combinatorial sums
• Derivation of Computational Formulas for certain class of finite sums: Approach to Generating functions arising from $p$-adic integrals and special functions
• Degenerate hypergeometric functions and degenerate hypergeometric numbers of order p