q-analogues of Lagrange inversion and q-Sheffer sequences

Eulerian differential operators are used to obtain q-analogues of the sequence of powers {dollar}1,f(u),f(u)sp2,...,{dollar} where f(u) is a formal power series with f(0) = 0, {dollar}fspprime{dollar}(0) {dollar}ne{dollar} 0. A composition rule for such sequences is proved which gives rise to a systematic method for obtaining q-series identities. The method places into a common setting two of the existing approaches to q-Lagrange inversion. Transformations of basic hypergeometric series are obtained by means of q-Lagrange inversion, leading to new Rogers-Ramanujan type identities. q-Analogues of polynomials of binomial type and Sheffer sequences arise naturally in this context, and their properties are developed. The technique of insertion of z is introduced in order to obtain q-analogue of the binomial convolution satisfied by polynomials of binomial type. A new proof of the q-Saalschutz identity is obtained by means of insertion of z, and a bibasic extension of this identity is derived. A combinatorial proof is given of a q-Lagrange inversion theorem due to A. M. Garsia.