The asymptotics of the kernel functions associated to orthogonal polynomials in several variables on the unit ball

We consider a family of orthogonal polynomials in several variables with respect to a Sobolev-type inner product, obtained from adding a gradient operator of order j, evaluated in a fixed point to a standard inner product. We study explicit relations between the Sobolev-type polynomials and the standard polynomials, among the kernel functions associated to the Sobolev-type polynomials and the kernel functions associated to the standard polynomials. In addition, an example for a particular choice of a classical measure a is an element of R-d is analyzed. Finally, we obtain the asymptotics of the some derivatives of the kernel functions evaluated in some points of the unit ball in d variables.