New Proof of the Wang-Sun Identity with Applications to Super Catalan Numbers (European Journal of Pure and Applied Mathematics, 2026)
In this work, we establish an identity originally due to Wang and Sun by leveraging a classical hypergeometric series identity of Bailey. As an application, we derive new identities for the Catalan numbers and introduce novel identities for the Super Catalan numbers involving the Wallis ratio. Furthermore, we present new formulas related to Touchard's identity and Koshy's identity. Finally, we provide integral representations that stem from the Wang-Sun identity. In this work, we establish an identity originally due to Wang and Sun by leveraging a classical hypergeometric series identity of Bailey. As an application, we derive new identities for the Catalan numbers and introduce novel identities for the Super Catalan numbers involving the Wallis ratio. Furthermore, we present new formulas related to Touchard's identity and Koshy's identity. Finally, we provide integral representations that stem from the Wang-Sun identity.
Keywords: Combinatorial identities, Generalized hypergeometric function, Terminating series, Catalan numbers, Super Catalan numbers.