Monotonicity of zeros of Jacobi-Sobolev type orthogonal polynomials

摘要

Consider the inner productrn,q>=Γ(α+β+2)/2~(α+β+1)Γ(α+1)Γ(β+1) ~1∫_(-1)p(x)q(x)(1-x)α(1+x)βdx+Mp(1)q(1)+Np'(1)q'(1)+Mp(-1)q(-1)+Np'(-1)q'(-1)rnwhere α,β ＞ -1 and M,N,M,N ≥ 0. If μ = (M, N, M, N), we denote by x_(n,k)~μ(α,β),rnk = 1,...,n, the zeros of the n-th polynomial P_n~(α,β,μ)(x), orthogonal with respect to the above inner product. We investigate the location, interlacing properties, asymptotics and monotonicity of x_(n,k)~μ(α,β) with respect to the parameters M,N,M,N in two important cases, when either M = N = 0 or N = N = 0. The results are obtained through careful analysis of the behavior and the asymptotics of the zeros of polynomials of the form P_n(x) = h_n(x) + cg_n(x) as functions of c.