Degenerate binomial coefficients and degenerate hypergeometric functions
Taekyun Kim, Dae San Kim, Hyunseok Lee, Jongkyum Kwon
Abstract
Abstract In this paper, we investigate degenerate versions of the generalized p th 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 λ -binomial coefficients and λ -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 λ -hypergeometric numbers of order p and Apostol type λ -hypergeometric numbers of order p , of which the latter reduce in a limiting case to the generalized p th order Franel numbers.