Branching rules for symmetric functions and [formula omitted] basic hypergeometric series

Abstract

A one-parameter rational function generalisation Rλ(X;b) of the symmetric Macdonald polynomial and interpolation Macdonald polynomial is studied from the point of view of branching rules. We establish a Pieri formula, evaluation symmetry, principal specialisation formula and q-difference equation for Rλ(X;b). Our main motivation for studying Rλ(X;b) is that it leads to a new class of sln basic hypergeometric series, generalising the well-known basic hypergeometric series with Macdonald polynomial argument. For these new series we prove sln analogues of the q-Gauss and q-Kummer–Thomae–Whipple formulas. In a special limit, one of our results implies an elegant binomial formula for Jack polynomials, different to that of Kaneko, Lassalle, Okounkov and Olshanski.