Another homogeneous \(q\)-difference operator (Q961631)

The Cauchy polynomials \(P_n(x,y) = \prod_{i=0}^n (x - q^i y)\) appear in the homogeneous form of the celebrated Cauchy identity in \(q\)-series. They satisfy a \(q\)-binomial identity (expansion of \(P_n(x,y)\) as a linear combination of all the \(P_k(x,z) P_{n-k}(z,y)\) for any \(z\)) of which some variants and consequences have been proved by Goldman and Rota: two of them are known as the \textit{Goldman-Rota \(q\)-binomial identity} and the \textit{inverse Goldman-Rota \(q\)-binomial identity}. The latter was derived from the former by Goulden and Jackson. In the paper reviewed here, the equivalence of these identities is proved, as well as related formulas in \(q\)-calculus and in combinatorics, notably about the Rogers-Szegö polynomials. The proofs rely on a new \(q\)-difference operator on functions of two variables.