Tiling proofs of Jacobi triple product and Rogers-Ramanujan identities

TL;DR

This paper introduces a tiling-based combinatorial approach to prove and interpret classical and new $q$-series identities, including the Jacobi triple product and Rogers-Ramanujan identities, providing elementary and recursive proofs.

Contribution

It develops a novel tiling method to prove and interpret key $q$-series identities, including new generalized $k$-product identities and recursive formulas.

Findings

Elementary tiling proofs of Jacobi and Rogers-Ramanujan identities

A tiling interpretation of $q$-binomial coefficients

New recursive $q$-series identities

Abstract

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.