A study of two new integral representations for Gauss’s hypergeometric function with applications

In this paper, first we present two new integral representations of the Gauss hypergeometric function in the form of double integrals obtained recently by us Chammam et al. (Bull Belgian Math Soc 31(3), 336–340, 2024). Two specific cases of our representations allow for an alternative proofs of the well-known Gauss summation theorem and Watson summation theorem. Additionally, as an application, we introduce a new class of double integrals and provide interesting integral representations for Catalan–Qi numbers and Fuss–Catalan–Qi numbers.