Exploring Two New Combinatorial Identities and Their Applications to Catalan–Qi Numbers

Authors

  • Wathek Chammam Majmaah University
  • Arjun K Rathie 2Department of Mathematics, Vedant College of Engineering & Technology (Rajasthan Technical University), Bundi-323021, Rajasthan, India
  • Mongia Khlifi Department of Mathematics, Faculty of Sciences of Sfax, Sfax University, Sfax Tunisia
  • Muhammad Gulistan Department of Electrical and Computer Engineering, University of Alberta, Canada

DOI:

https://doi.org/10.29020/nybg.ejpam.v19i2.6908

Keywords:

Binomial identities, combinatorial identities, hypergeometric series, Wat- son, Dixon, Whipple summation theorems, terminating hypergeometric series, Catalan numbers, Catalan–Qi numbers, Super Catalan numbers.

Abstract

In this paper, we establish two new and intriguing combinatorial identities, derived using two summation formulas for the terminating 3F2 with unit argument, as introduced by Bailey. Notably, a combinatorial identity previously obtained by Wang and Sun is a special case of our main results. Furthermore, we present two new identities for the Catalan–Qi numbers. As applications, we explore connections to Catalan numbers, Super Catalan numbers, and the Wallis ratio. Moreover, we provide new formulas involving the Touchard’s identity and Koshy’s identity. Finally, we establish some integral representations involving the Catalan numbers.

References

Published

2026-07-28

Issue

Vol. 19 No. 2: (April 2026)

Section

Number Theory

License

Copyright (c) 2026 Wathek Chammam, Arjun K Rathie, Mongia Khlifi, Muhammad Gulistan

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How to Cite

Exploring Two New Combinatorial Identities and Their Applications to Catalan–Qi Numbers. (2026). European Journal of Pure and Applied Mathematics, 19(2), 6908. https://doi.org/10.29020/nybg.ejpam.v19i2.6908