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 ∂ P n , k ( λ ; x , y ) ∂ λ = ∑ m = 0 n - 1 ∑ j = 0 m d n , j , m P m , j ( λ ; x , y ) + ∑ j = 0 k e n , j , k P n , 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 C n ( λ , μ ) ( x ) with the help of the parameter derivatives of the classical Jacobi polynomials P n ( α , β ) ( x ) , i.e. ∂ ∂ α P n ( α , β ) ( x ) and ∂ ∂ β P n ( 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. [ABSTRACT FROM AUTHOR]