Journal of the Ramanujan Mathematical Society

Volume 38, Issue 2, June 2023 pp. 157–176.

Tiling proofs of Jacobi triple product and Rogers-Ramanujan identities

Authors: Alok Shukla
Author institution: Mathematical and Physical Sciences Division, School of Arts and Sciences, Ahmedabad University, India.

Summary: We use the method of tiling to give elementary combinatorial proofs of some celebrated q-series identities, such as Jacobi triple product identity, Rogers-Ramanujan identities, and some identities of Rogers.We give a tiling proof of the q-binomial theorem and a tiling interpretation of the q-binomial coefficients. A new generalized k-product q-series identity is also obtained by employing the 'tiling-method', wherein the generating function of the set of all possible tilings of a rectangular board is computed in two different ways to obtain the desired q-series identity. Several new recursive q-series identities were also established. The 'tiling-method' holds promise for giving an aesthetically pleasing approach to prove old and new q-series identities.

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