Differential equations for generalized Jacobi polynomials

Abstract

We look for differential equations of the form M ∑ i=0 ∞ a i(x)y (i)(x)+N ∑ i=0 ∞ b i(x)y (i)(x)+MN ∑ i=0 ∞ c i(x)y (i)(x) +(1−x 2)y″(x)+[β−α−(α+β+2)x]y′(x)+n(n+α+β+1)y(x)=0 satisfied by the generalized Jacobi polynomials { P n α, β, M, N ( x)} n=0 ∞ which are orthogonal on the interval [−1,1] with respect to the weight function Γ(α+β+2) 2 α+β+1Γ(α+1)Γ(β+1) (1−x) α(1+x) β+Mδ(x+1)+Nδ(x−1), where α>−1, β>−1, M⩾0 and N⩾0. We give explicit representations for the coefficients { a i ( x)} i=0 ∞, { b i ( x)} i=0 ∞ and { c i ( x)} i=0 ∞ and we show that this differential equation is uniquely determined. For M 2+ N 2>0 the order of this differential equation is infinite, except for α∈{0,1,2,…} or β∈{0,1,2,…}. Moreover, the order equals 2β+4 if M>0, N=0 and β∈{0,1,2,…}, 2α+4 if M=0, N>0 and α∈{0,1,2,…}, 2α+2β+6 if M>0, N>0 and α,β∈{0,1,2,…}.