proceedings of the american mathematical society Volume 88. Number 3. July 1983 A MATRIX INVERSE D. M. BRESSOUD1 Abstract. George Andrews has demonstrated that the Bailey transform is equiva- lent to the inversion of an infinite-dimensional matrix whose entires are rational functions in q. We generalize this inversion by introducing an extra parameter which brings much greater symmetry. Let A = {Ank)^k=0 be an infinite-dimensional lower triangular matrix; k > « implies that A„k = 0. We say that A has an inverse, written A'] = [A~n\], if n 2d "-nkAkm — onm, k = m for all nonnegative « and m, m < «. The inversion of such matrices when the entries are rational functions in q plays an important role in ^-series identities. Andrews [1] has shown that the Bailey transform [2, 3] used to prove and generalize the Rogers-Ramanujan identities is equivalent to the following matrix inversion. Let B={BHk)Zk=0 where B 1 (q)n-k(a<l)n+k' (*)» = n?=o(l - aqi), (a)m = (a)x/(aqm)x. Then R"1 = {Rj} where (\-aq2")(a)k+n(-\)n-kq^ (\-a)(q)n.k More recently, Gessel and Stanton proved a number of ^-series identities using this same inversion (Theorem 1.2 of [5]). Andrews' inversion is a special case of a far more appealing result. Theorem. Let D = [Dnk(a, b))^k=Q where n( .A_(l-aq2k)(b)k+„(ba-l)n-k(ba-l)k Dnk(a,b)- {l-a){aq)k+Áq)n_k Then D- = [Dnk(b, a))~k=Q. Received by the editors October 28, 1982. 1980 Mathematics Subject Classification. Primary 33A30;Secondary 05A17, 10A45, 05A15. 'Partially supported by National Science Foundation and Sloan Foundation. Research done at University of Wisconsin, Madison. ©1983 American Mathematical Society 0002-9939/82/0000-1316/$01.75 446 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use