An integral identity with applications in orthogonal polynomials

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 Gegenbuer 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.