Exploring Two New Combinatorial Identities and Their Applications to Catalan–Qi Numbers (European Journal of Pure and Applied Mathematics, 2026)
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.
Keywords: Binomial identities, combinatorial identities, hypergeometric series, Wat- son, Dixon, Whipple summation theorems, terminating hypergeometric series, Catalan numbers, Catalan–Qi numbers, Super Catalan numbers.