Ratio and Plancherel–Rotach asymptotics for Meixner–Sobolev orthogonal polynomials

We study the analytic properties of the monic Meixner–Sobolev polynomials {Qn} orthogonal with respect to the inner product involving differences(p,q)S=∑i=0∞[p(i)q(i)+λΔp(i)Δq(i)]μi(γ)ii!, γ>0, 0<μ<1,where λ⩾0, Δ is the forward difference operator (Δf(x)=f(x+1)−f(x)) and (γ)n denotes the Pochhammer symbol. Relative asymptotics for Meixner–Sobolev polynomials with respect to Meixner polynomials is obtained. This relative asymptotics is also given for the scaled polynomials. Moreover, a zero distribution for the scaled Meixner–Sobolev polynomials and Plancherel–Rotach asymptotics for {Qn} are deduced.