The Rogers-Ramanujan identities, the finite general linear groups, and the Hall-Littlewood polynomials

We connect Gordon’s generalization of the Rogers-Ramanujan identities with the Hall-Littlewood polynomials and with generating functions which arise in a probabilistic setting in the finite general linear groups. This yields a Rogers-Ramanujan type product formula for the n → ∞ n → ∞ probability that an element of G L ( n , q ) GL(n,q) or M a t ( n , q ) Mat(n,q) is semisimple.