Asymptotics and weighted estimates of meixner polynomials orthogonal on the gird {0, δ, 2δ, ...}

Abstract

Suppose that 0<δ≤1,N=1/δ, and α, ga≥0, is an integer. For the classical Meixner polynomials$\mathfrak{M}_{n,N}^\alpha (x)$ orthonormal on the gird {0, δ, 2δ, ...} with weight ρ(x)=(1-e ^−δ)^αг(Nx+α+ 1)/г(Nx+1), the following asymptotic formula is obtained:$\mathfrak{M}_{n,N}^\alpha (z) = \Lambda _n^\alpha (z) + v_{n,N}^\alpha (z)$. The remainderv ^α _n,N (z) forn≤λN satisfies the estimate

$$|v_{n,N}^\alpha (z)|^2 \leqslant c(\alpha ,\lambda )\delta \sum\limits_{k = 0}^n {|\Lambda _k^\alpha (z)|^2 ,} $$

where Λ ^α _k (x) are the Laguerre orthonormal polynomials. As a consequence, a weighted estimate, for the Meixner polynomial$\mathfrak{M}_{n,N}^\alpha (x)$ on the semiaxis [0, ∞) is obtained.