An integral identity with applications in orthogonal polynomials

Abstract:

For $\boldsymbol {\large {\lambda }} = (\lambda _1,\ldots ,\lambda _d)$ with $\lambda _i > 0$, it is proved that \begin{equation*} \prod _{i=1}^d \frac { 1}{(1- r x_i)^{\lambda _i}} = \frac {\Gamma (|\boldsymbol {\large {\lambda }}|)}{\prod _{i=1}^{d} \Gamma (\lambda _i)} \int _{\mathcal {T}^d} \frac {1}{ (1- r \langle x, u \rangle )^{|\boldsymbol {\large {\lambda }}|}} \prod _{i=1}^d u_i^{\lambda _i-1} du, \end{equation*} where $\mathcal {T}^d$ is the simplex in homogeneous coordinates of $\mathbb {R}^d$, from which a new integral relation for Gegenbauer polynomials of different indexes is deduced. The latter result is used to derive closed formulas for reproducing kernels of orthogonal polynomials on the unit cube and on the unit ball.