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Rogers-Ramanujan type identities and Nil-DAHA
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In the theory of the Nil-DAHA Fourier transform, the inner products of q-Hermite polynomials for the measure function multiplied by a level one theta function are the key. They are used to 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 Nahm's conjecture is discussed. Some of our q-series can be identified with known ones, but their interpretation seems new. Using that the q-Hermite polynomials are closely related to the Demazure level one characters in the twisted case (Sanderson, Ion), we outline a connection of our formulas to the level one integrable Kac-Moody modules and the coset theory. Several instances of the level-rank duality are provided.
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$A_1^{(1)}$-Grounded partitions at levels $1$ and $2$, Part I: bijections
Grounded partitions at level 2 of type A_1^(1) are in size- and length-preserving bijection with odd overpartitions (ground b) and with partitions whose even parts are distinct (ground a).
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