Monotonicity of zeros of Jacobi-Sobolev type orthogonal polynomials

Abstract

Consider the inner product< p, q > = Gamma(alpha + beta + 2)/2(alpha+beta+1) Gamma (alpha + 1)Gamma(beta +1) integral(t)(-t) p(x)q(x)(alpha) (1 + x)(beta) dx+ Mp(1)q(1)+ Np'(1)q'(1) + 1 (M) over tildep(-1)q(-1)+ (N) over tildep'(-1)q'(-1)where alpha, beta > -1 and M,N,(M) over tilde,(N) over tilde >= 0. If mu = (M,N,(M) over tilde,(N) over tilde), we denote by x(n,k)(mu)(alpha,beta), k =1,...n, the zeros of the n-th polynomial P(n)((alpha,beta,mu)) (x), orthogonal with respect to the above inner product. We investigate the location, interlacing properties, asymptotics and monotonicity of x(n,k)(mu)(alpha,beta) with respect to the parameters M, N,(M) over tilde,(N) over tilde in two important cases, when either i = N = 0 or N = 0. The results are obtained through careful analysis of the behavior and the asymptotics of the zeros of polynomials of the form p,,(x)= hn(x) + cgn(x) as functions of(C) 2010 IMACS. Published by Elsevier BA/. All rights reserved.