Degenerate binomial coefficients and degenerate hypergeometric functions

Abstract

In this paper, we investigate degenerate versions of the generalized pth order Franel numbers which are certain finite sums involving powers of binomial coefficients. In more detail, we introduce degenerate generalized hypergeometric functions and study degenerate hypergeometric numbers of order p. These numbers involve powers of lambda-binomial coefficients and lambda-falling sequence, and can be represented by means of the degenerate generalized hypergeometric functions. We derive some explicit expressions and combinatorial identities for those numbers. We also consider several related special numbers like lambda-hypergeometric numbers of order p and Apostol type lambda-hypergeometric numbers of order p, of which the latter reduce in a limiting case to the generalized pth order Franel numbers.