The $q$-Binomial Theorem and two Symmetric $q$-Identities

Abstract

We notice two symmetric $q$-identities, which are special cases of the transformations of $_2\phi_1$ series in Gasper and Rahman's book (Basic Hypergeometric Series, Cambridge University Press, 1990, p. 241). In this paper, we give combinatorial proofs of these two identities and the $q$-binomial theorem by using conjugation of $2$-modular diagrams.