Degenerate hypergeometric functions and degenerate hypergeometric numbers of order p

TL;DR

This paper introduces degenerate generalized hypergeometric functions and explores associated degenerate hypergeometric numbers of order p, deriving explicit formulas and identities, and relating them to known special numbers.

Contribution

It presents new degenerate hypergeometric functions and defines degenerate hypergeometric numbers of order p, linking them to existing special number sequences and deriving their properties.

Findings

Derived explicit expressions for degenerate hypergeometric numbers.

Established combinatorial identities involving these numbers.

Connected new numbers to classical special numbers like Franel numbers.

Abstract

In this paper, we introduce degenertae generalized hypergeometric functions and study degenerate hypergeometric numbers of order p. These numbers involving of lambda-binomial coefficients and lambda-falling sequence, and can be represented by means of the degenerate generalized hypergeometric functions. we will 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 p-th order Franel numbers.