Addition formula for q-disk polynomials

Abstract

The purpose of this paper is to present an addition formula for so-called $q$-disk polynomials, using some quantum group theory. This result is a $q$-analogue of a result which was proved around 1970 by ${\breve{\text S}}$apiro [S] and Koornwinder [Koo1,2] independently. They considered the homogeneous space $U(n)/U(n-1)$, were $U(n)$ denotes the group of unitary transformations of the vector space $\C^n$, and identified the corresponding zonal spherical functions as disk polynomials, in the terminology of [Koo2]. These are orthogonal polynomials in two variables whose orthogonality measure is supported by the closed unit disk in the complex plane, and which can be expressed in terms of Jacobi polynomials $P_n^{(\a,\b)} (x)$ for certain integer values of the parameters $\a$ and $\b$. The associated spher ical funtions were shown to be also expressible in terms of disk polynomials, and from this an addition formula was proved for (positive) integer values of $\a$ and $\b$. By an easy argument this identity was then extended to all complex values of $\a$ (and from this Koornwinder could even derive an addition formula for general Jacobi polynomials). And in fact the line of arguing and the results in this paper will be very similar to (part of) the ones in [Koo2].