Monotonicity of zeros of Laguerre–Sobolev-type orthogonal polynomials

Denote by xn,kM,N(α), k=1,…,nk=1,…,n, the zeros of the Laguerre–Sobolev-type polynomials Ln(α,M,N)(x) orthogonal with respect to the inner product〈p,q〉=1Γ(α+1)∫0∞p(x)q(x)xαe−xdx+Mp(0)q(0)+Np′(0)q′(0), where α>−1α>−1, M⩾0M⩾0 and N⩾0N⩾0. We prove that xn,kM,N(α) interlace with the zeros of Laguerre orthogonal polynomials Ln(α)(x) and establish monotonicity with respect to the parameters M and N   of xn,kM,0(α) and xn,k0,N(α). Moreover, we find N0N0 such that xn,nM,N(α)<0 for all N>N0N>N0, where xn,nM,N(α) is the smallest zero of Ln(α,M,N)(x). Further, we present monotonicity and asymptotic relations of certain functions involving xn,kM,0(α) and xn,k0,N(α).