AN INTEGRAL IDENTITY WITH APPLICATIONS IN ORTHOGONAL POLYNOMIALS

For lambda = (lambda(1), ..., lambda(d)) with lambda(i) > 0, it is proved that Pi(d)(i=1) 1/(1 - rx(i))(lambda i) = Gamma(vertical bar lambda vertical bar)/Pi(d)(i-1) Gamma(lambda(i)) integral(tau d) 1/(1 - r < x, u >)(vertical bar lambda vertical bar) Pi(d)(i=1) u(i)(lambda i-1) du, where tau(d) is the simplex in homogeneous coordinates of 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.