Δ-Sobolev orthogonal polynomials of Meixner type: asymptotics and limit relation

摘要

Let {Qn(x)}n be the sequence of monic polynomials orthogonal with respect to the Sobolev-type inner product〈p(x),r(x)〉S=〈u0,p(x)r(x)〉+λ〈u1,(Δp)(x)(Δr)(x)〉,where λ⩾0, (Δf)(x)=f(x+1)-f(x) denotes the forward difference operator and (u0,u1) is a Δ-coherent pair of positive-definite linear functionals being u1 the Meixner linear functional. In this paper, relative asymptotics for the {Qn(x)}n sequence with respect to Meixner polynomials on compact subsets of C⧹[0,+∞) is obtained. This relative asymptotics is also given for the scaled polynomials. In both cases, we deduce the same asymptotics as we have for the self-Δ-coherent pair, that is, when u0=u1 is the Meixner linear functional. Furthermore, we establish a limit relation between these orthogonal polynomials and the Laguerre–Sobolev orthogonal polynomials which is analogous to the one existing between Meixner and Laguerre polynomials in the Askey scheme.