Pieri-Type Formulas for Nonsymmetric Macdonald Polynomials

In symmetric Macdonald polynomial theory, the Pieri formula gives the branching coefficients for the product of the rth elementary symmetric function er(z) and the Macdonald polynomial $P_{\kappa}(z)$. In this paper we give the nonsymmetric analogs for the cases r=1 and r= n−1. We do this by first deducing the decomposition for the product of any nonsymmetric Macdonald polynomial $E_{\eta}(z)$ with zi in terms of nonsymmetric Macdonald polynomials. As a corollary of finding the branching coefficients of $e_1(z)E_{\eta}(z)$ we evaluate the generalized binomial coefficients $(^{\eta}_v)$ associated with the nonsymmetric Macdonald polynomials for $|\eta|=|v|+1$.