Differential equations for generalized Jacobi polynomials

摘要

We look for differential equations of the form M∑i=0∞ai(x)y(i)(x)+N∑i=0∞bi(x)y(i)(x)+MN∑i=0∞ci(x)y(i)(x)+(1−x2)y″(x)+[β−α−(α+β+2)x]y′(x)+n(n+α+β+1)y(x)=0satisfied by the generalized Jacobi polynomials {Pnα,β,M,N(x)}n=0∞ which are orthogonal on the interval [−1,1] with respect to the weight functionΓ(α+β+2)2α+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 {ai(x)}i=0∞, {bi(x)}i=0∞ and {ci(x)}i=0∞ and we show that this differential equation is uniquely determined. For M2+N2>0 the order of this differential equation is infinite, except for α∈{0,1,2,…} or β∈{0,1,2,…}. Moreover, the order equals2β+4ifM>0,N=0andβ∈{0,1,2,…},2α+4ifM=0,N>0andα∈{0,1,2,…},2α+2β+6ifM>0,N>0andα,β∈{0,1,2,…}.