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

journal of mathematical analysis and applications

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