A $q$-summation and the orthogonality relations for the $q$-Hahn polynomials and the big $q$-Jacobi polynomials

Abstract

Using a general $q$-summation formula, we derive a generating function for the $q$-Hahn polynomials, which is used to give a complete proof of the orthogonality relation for the $q$-Hahn polynomials. A new proof of the orthogonality relation for the big $q$-Jacobi polynomials is also given. A simple evaluation of the Nassrallah-Rahman integral is derived by using this summation formula. A new $q$-beta integral formula is established, which includes the Nassrallah-Rahman integral as a special case. The $q$-summation formula also allows us to recover several strange $q$-series identities.