Variants of the Rogers-Ramanujan identities
type: Item
instance of: scholarly article
title: Variants of the Rogers-Ramanujan identities
zbMATH Open document ID: 0944.33015
author: Mourad E. H. Ismail
author: Kristina C. Garrett
author: D. Stanton
published in: Advances in Applied Mathematics
publication date: 2000-09-15T00:00:00Z
review text: In 1894, L. J. Rogers published an analysis of what today are called the \(q\)-Hermite and \(q\)-ultraspherical polynomials. Of the many identities he generated in this paper, the best known are the Rogers-Ramanujan identities. The authors of this paper go back to Rogers, simplify his proof by explicitly using the orthogonality of these polynomials, and then go on to generalize his approach, giving new proofs of some known results and many new identities. Among the new results are two quintic transformations that imply the Rogers-Ramanujan identities without the need for the Jacobi triple product identity. To give the flavor, one of these transformations is \[ \begin{aligned} \sum_{n=0}^{\infty} {q^{n^2} (qf)^{2n} \over (q;q)_n} =\;& {(f^4q^5;q)_{\infty} \over (f^4q^5,f^6q^{10};q^5)_{\infty} (f^2q^3;q)_{\infty}}\\ & \times\;\sum_{n=0}^{\infty} { (1-f^6q^{10n+5})(f^6q^5,f^4q^{10};q^5)_n (f^2;q)_{5n} \over (1-f^6q^5)(q^5,f^2;q^5)_n (f^4q^6;q)_{5n} }\\ & \qquad \times q^{5n(n-1)/2} (-f^4q^{10})^n. \end{aligned} \] {}.
Mathematics Subject Classification ID: 33D45
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 11P81
zbMATH DE Number: 1420095
zbMATH Keywords: \(q\)-ultraspherical polynomials
zbMATH Keywords: Rogers-Ramanujan identities
reviewed by: David M. Bressoud
MaRDI profile type: Publication
full work available at URL: aama.1999.0658
OpenAlex ID: W1977204800
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DOI: 10.1006/AAMA.1999.0658
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description: scientific article; zbMATH DE number 1420095