Extensions of several famous combinatorial identities via hypergeometric functions

Arjun Kumar Rathie; Feng Qi; Dongkyu Lim

Extensions of several famous combinatorial identities via hypergeometric functions

General Mathematics 2026-08-03 v1

Authors: Arjun Kumar Rathie , Feng Qi , Dongkyu Lim

Abstract

The objective of this paper is to develop an extension of Kummer's second theorem and to establish generalized forms of four classical combinatorial identities---Knuth's old sum (also known as Reed--Dawson's combinatorial identity), Riordan's combinatorial identity, Gould's combinatorial identity, and Touchard's combinatorial identity---using a hypergeometric-series approach. Several new identities also arise as special cases of our main results.

Keywords

combinatorial identities hypergeometric functions Kummer's theorem Knuth's old sum supergeometric series

Cite

@article{arxiv.2608.05192,
  title  = {Extensions of several famous combinatorial identities via hypergeometric functions},
  author = {Arjun Kumar Rathie and Feng Qi and Dongkyu Lim},
  journal= {arXiv preprint arXiv:2608.05192},
  year   = {2026}
}

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16 pages