Some transformation formulas associated with Askey–Wilson polynomials and Lassalle's formulas for Macdonald–Koornwinder polynomials

We present a fourfold series expansion representing the Askey–Wilson polynomials. To obtain the result, a sequential use is made of several summation and transformation formulas for the basic hypergeometric series, including the Verma's $q$-extension of the Field and Wimp expansion, Andrews' terminating $q$-analogue of Watson's $_3F_2$ sum, Singh's quadratic transformation. As an application, we present an explicit formula for the Koornwinder polynomial of type $BC_n$ ($n\in\mathbb Z_{>0}$) with one row diagram. When the parameters are specialized, we recover Lassalle's formula for Macdonald polynomials of type $B_n$, $C_n$ and $D_n$ with one row diagram, thereby proving his conjectures.