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

Let { Q n ( x ) } n be the sequence of monic polynomials orthogonal with respect to the Sobolev-type inner product 〈 p ( x ) , r ( x ) 〉 S = 〈 u 0 , p ( x ) r ( x ) 〉 + λ 〈 u 1 , ( Δ p ) ( x ) ( Δ r ) ( x ) 〉 , where λ ⩾ 0 , ( Δ f ) ( x ) = f ( x + 1 ) - f ( x ) denotes the forward difference operator and ( u 0 , u 1 ) is a Δ -coherent pair of positive-definite linear functionals being u 1 the Meixner linear functional. In this paper, relative asymptotics for the { Q n ( 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 u 0 = u 1 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.