Branching rules for symmetric functions and \(\mathfrak{sl}_n\) basic hypergeometric series. (English) Zbl 1227.05261
Summary: A one-parameter rational function generalisation \(R_\lambda (X;b)\) 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 \(R_\lambda (X;b)\). Our main motivation for studying \(R_\lambda (X;b)\) is that it leads to a new class of \(\mathfrak{sl}_n\) basic hypergeometric series, generalising the well-known basic hypergeometric series with Macdonald polynomial argument. For these new series we prove \(\mathfrak{sl}_n\) 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.
MSC:
| 05E05 | Symmetric functions and generalizations |
| 33D52 | Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.) |
| 33D67 | Basic hypergeometric functions associated with root systems |
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