The $C_\ell $ Rogers–Selberg Identity

Abstract

In this paper the classical Rogers–Selberg identity is extended to the setting of multiple basic hypergeometric series very well poised on symplectic $C_\ell $ groups. The $C_\ell $ Rogers–Selberg identity is deduced from the $C_\ell $q-Whipple transformation. Schur functions and q-Kostka polynomials are then used to simplify the balanced “sum side” of these identities. A study of several special limiting cases then provides an elegant generalization of how the classical Rogers–Selberg identity is simplified termwise in the standard analytical proofs of the Rogers–Ramanujan identities. For $C_2 $, explicit Rogers–Ramanujan-type identities are obtained. One of these gives a new expansion, involving q-Kostka polynomials, of the product side of one of Bressoud’s $(\bmod 6)$ Rogers–Ramanujan identities. It is also found that a similar analysis applied to the $C_\ell $ terminating ${}_6 \phi _5 $ summation theorem leads to a $C_\ell $ extension of Sylvester’s identity. Special limiting $C_2 $ and $C_3 $ cases ...