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
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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
20d0d21b0a2cd2" title="Click to view the MathML source" alt="Click to view the MathML source">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
An or
U(n+1) basic hypergeometric series in Theorem 1.49 of [S.C. Milne, An elementary proof of the Macdonald identities for
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, Adv. Math. 57 (1985) 34–70], as rewritten in Lemma 7.3 on p. 163 of [S.C. Milne, Balanced
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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
An extension of Ramanujan's
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summation with applications to multilateral
An series, Rocky Mountain J. Math. 32 (2002) 759–792].