Variants of the Rogers-Ramanujan identities
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 \[ \begin{aligned} \sum_{n=0}^{\infty} {q^{n^2} (qf)^{2n} \over (q;q)_n} =\;& {(f^4q^5;q)_{\infty} \over (f^4q^5,f^6q^{10};q^5)_{\infty} (f^2q^3;q)_{\infty}}\\ & \times\;\sum_{n=0}^{\infty} { (1-f^6q^{10n+5})(f^6q^5,f^4q^{10};q^5)_n (f^2;q)_{5n} \over (1-f^6q^5)(q^5,f^2;q^5)_n (f^4q^6;q)_{5n} }\\ & \qquad \times q^{5n(n-1)/2} (-f^4q^{10})^n. \end{aligned} \] {}.
scientific article; zbMATH DE number 1420095
scientific article