Regular Article: Variants of the Rogers驴Ramanujan Identities

Authors:
Kristina Garrett;Mourad E. H. Ismail;Dennis Stanton
Affiliations:
School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455;Department of Mathematics, University of South Florida, Tampa, Florida, 33620-5700;School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455
Venue:
Advances in Applied Mathematics
Year:
1999

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Abstract

We evaluate several integrals involving generating functions of continuous q-Hermite polynomials in two different ways. The resulting identities give new proofs and generalizations of the Rogers-Ramanujan identities. Two quintic transformations are given, one of which immediately proves the Rogers-Ramanujan identities without the Jacobi triple product identity. Similar techniques lead to new transformations for unilateral and bilateral series. The quintic transformations lead to curious identities involving primitive fifth roots of unity which are then extended to primitive pth roots of unity for odd p.