On parameter derivatives of a family of polynomials in two variables

The purpose of the present paper is to give the parameter derivative representations of the form∂Pn,k(λ;x,y)∂λ=∑m=0n-1∑j=0mdn,j,mPm,j(λ;x,y)+∑j=0ken,j,kPn,j(λ;x,y)for a family of orthogonal polynomials of variables x and y, with λ being a parameter and 0⩽k⩽n;n,k=0,1,2,…. First, we shall present the representations of the parameter derivatives of the generalized Gegenbauer polynomials Cn(λ,μ)(x) with the help of the parameter derivatives of the classical Jacobi polynomials Pn(α,β)(x), i.e. ∂∂αPn(α,β)(x) and ∂∂βPn(α,β)(x). Then, by using these derivatives, we investigate the parameter derivatives for two-variable analogues of the generalized Gegenbauer polynomials. Furthermore, we discuss orthogonality properties of the parametric derivatives of these polynomials.