Rogers–Ramanujan type identities and Nil-DAHA

Abstract

Using the DAHA-Fourier transform of q-Hermite polynomials multiplied by level-one theta functions, we obtain expansions of products of any number of such theta functions in terms of the q-Hermite polynomials. An ample family of modular functions satisfying Rogers–Ramanujan type identities for arbitrary (reduced, twisted) affine root systems is obtained as an application. A relation to Rogers dilogarithm and Nahmʼs conjecture is discussed. The q-Hermite polynomials are closely related to the Demazure level-one characters in the twisted case (Sanderson, Ion), which connects our formulas to tensor products of level-one integrable Kac–Moody modules, their coset theory and the level-rank duality.