Differential equations for generalized Jacobi polynomials

[摘要] We look for differential equations of the form M Sigma (infinity)(i=0) a(i)(x)y((i))(x) + N Sigma (infinity)(i=0) b(i)(x)y((i)) + MN Sigma (infinity)(i=0) c(i)(x)y((i))(x) + (1 - x(2))y(x) + [beta - alpha - (alpha + beta + 2)x]y' (x) + n(n + alpha + beta + 1)y(x) = 0 satisfied by the generalized Jacobi polynomials {P-n(alpha,beta ,M,N)(x)}(n=0)(infinity) which are orthogonal on the interval [-1, 1] with respect to the weight function Gamma(alpha + beta + 2)/2(alpha+beta +1)Gamma(alpha + 1)Gamma(beta + 1) (1-alpha)(alpha)(1 + x)(beta) + M delta (x + 1) + N delta (x - 1), where alpha > -1, beta > -1, M greater than or equal to 0 and N greater than or equal to 0. We give explicit representations for the coefficients {alpha (i)(x)}(i=0)(infinity), {b(i)(x)}(i=0)(infinity) and {c(i)(x)}(i=0)(infinity) 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 alpha epsilon {0,1,2,...} or beta epsilon {0,1,2,...}. Moreover, the order equals {2 beta + 4 if M > 0, N = 0 and beta epsilon {0,1,2,...}, 2 alpha + 4 if M = 0, N > 0 and alpha epsilon {0,1,2,...}, 2 alpha + 2 beta + 6 if M > 0, N > 0 and alpha, beta epsilon {0,1,2,...}. (C) 2000 Elsevier Science B.V. All rights reserved. MSC: 33C45; 34A35.

[发布日期] 2000-12-30

[关键词] differential equations;generalized Jacobi polynomials