Operator identities involving the bivariate Rogers–Szegö polynomials and their applications to the multiple q-series identities

Abstract

In this paper, we first give several operator identities involving the bivariate Rogers–Szegö polynomials. By applying the technique of parameter augmentation to the multiple q-binomial theorems given by Milne [S.C. Milne, Balanced ϕ 2 3 summation theorems for U ( n ) basic hypergeometric series, Adv. Math. 131 (1997) 93–187], we obtain several new multiple q-series identities involving the bivariate Rogers–Szegö polynomials. These include multiple extensions of Mehler's formula and Rogers's formula. Our U ( n + 1 ) generalizations are quite natural as they are also a direct and immediate consequence of their (often classical) known one-variable cases and Milne's fundamental theorem for A n or U ( n + 1 ) basic hypergeometric series in Theorem 1.49 of [S.C. Milne, An elementary proof of the Macdonald identities for A l ( 1 ) , Adv. Math. 57 (1985) 34–70], as rewritten in Lemma 7.3 on p. 163 of [S.C. Milne, Balanced ϕ 2 3 summation theorems for U ( n ) basic hypergeometric series, Adv. Math. 131 (1997) 93–187] or Corollary 4.4 on pp. 768–769 of [S.C. Milne, M. Schlosser, A new A n extension of Ramanujan's ψ 1 1 summation with applications to multilateral A n series, Rocky Mountain J. Math. 32 (2002) 759–792].