ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 16, Number 2, Spring 1986 ANOTHER FAMILY OF ^-LAGRANGE INVERSION FORMULAS IRA GESSEL AND DENNIS STANTON ABSTRACT. A ^-analog of Lagrange inversion is stated for (xj (1 — x r ) b ). Applications to basic hypergeometric series, identities of the Rogers-Ramanujan type, and orthogonal polynomials are given. 1. Introduction. The generalized Lagrange inversion problem is: given CO (1.1) G k (x) = 2V,*= 0,1,..., n=k for some lower triangular non-singular matrix B nh and a formal power series CO (1-2) Ax) = Zf n x", n=Q find constants a k such that CO (1.3) f{x) = J^a k G k {x). It is clear that (1-4) L = S B nk a„. Thus to find a k it is sufficient to find the inverse matrix B^}\ (1.5) a k = JlB;}f,. The usual Lagrange inversion formula takes G k (x) = y k , where y(x) is a formal power series in x such that y(0) = 0 and y'(0) ^ 0. In a recent paper [10] we gave a ^-analog of B nk , B^}, and G k (x) for G^(x) = x k /(\ — x) a+ib+1)k . In this paper we similarly find a ^-Lagrange inversion formula for a ^-analog of G k (x) = x k j{\ — x r ) a+ib+1)k for r = 1, 2, • • •. Our main theorem is stated as Theorem 2.3. Just as in [10], * This work was partially supported by NSF grants MCS 8105188 and MCS 8300872. Received by the editors on September 20, 1984 Copyright © 1986 Rocky Mountain Mathematics Consortium 373