Rogers-Szegő polynomials and Hall-Littlewood symmetric functions (Q2509286)

Three classical identities for Schur functions present \(\sum s_{\lambda}(x)\) where the sum runs on all partitions, all even partitions, and all partitions conjugate to even partitions, in terms of products with denominators \(1-x_i\), \(1-x_i^2\) and \(1-x_ix_j\), \(i<j\). These identities have analogues and generalizations for Hall-Littlewood symmetric functions, due to Macdonald and Kawanaka. In the paper under review the author presents a new identity which incorporates both Rogers-Szegő polynomials and Hall-Littlewood symmetric functions. As a consequence, the author obtains uniformly the identities of Macdonald and Kawanaka. Some of the main results are considered from the point of view of \(\lambda\)-rings and are restated in terms of plethysms, in the spirit of the recent work of Lascoux.