  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.
