A combinatorial proof of an identity of Ramanujan using tilings

摘要

Recently, Little and Sellers proved combinatorially a variety of Rogers-Ramanujan type of identities. They interpreted both sides of the equalities as enumerating the same collection of “weighted tilings”. They tiled an infinite 1 × ∞ board, with squares and dominos. In their articles, the concept of “projection” of tiles was defined. In this article, we use and extend some of their ideas and give a combinatorial proof to the following identity proposed in a survey of Pak $sumlimits_{n = 0}^infty {frac{{(1 + q)(1 + q^3 ) cdots (1 + q^{2n - 1} )}}n{{[(1 - q^2 )(1 - q^4 ) cdots (1 - q^{2n} )]^2 }}q^{n^2 } } = prodlimits_{n = 0}^infty {frac{{1 + q^{2n + 1} }}n{{1 - q^{2n + 2} }}} .$sumlimits_{n = 0}^infty {frac{{(1 + q)(1 + q^3 ) cdots (1 + q^{2n - 1} )}}n{{[(1 - q^2 )(1 - q^4 ) cdots (1 - q^{2n} )]^2 }}q^{n^2 } } = prodlimits_{n = 0}^infty {frac{{1 + q^{2n + 1} }}n{{1 - q^{2n + 2} }}} .