Monotonicity of zeros of Jacobi–Sobolev type orthogonal polynomials

Consider the inner product〈p,q〉=Γ(α+β+2)2α+β+1Γ(α+1)Γ(β+1)∫−11p(x)q(x)(1−x)α(1+x)βdx+Mp(1)q(1)+Np′(1)q′(1)+M˜p(−1)q(−1)+N˜p′(−1)q′(−1) where α,β>−1α,β>−1 and M,N,M˜,N˜⩾0. If μ=(M,N,M˜,N˜), we denote by xn,kμ(α,β), k=1,…,nk=1,…,n, the zeros of the n  -th polynomial Pn(α,β,μ)(x), orthogonal with respect to the above inner product. We investigate the location, interlacing properties, asymptotics and monotonicity of xn,kμ(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 pn(x)=hn(x)+cgn(x)pn(x)=hn(x)+cgn(x) as functions of c.