Asymptotics and weighted estimates of Meixner polynomials orthogonal on the grid $\{0,\delta,2\delta,\dots\}$

I. I. Sharapudinov

Matematičeskie zametki, Tome 62 (1997) no. 4, pp. 603-616

Résumé

Suppose that $0\delta\le1$, $N=1/\delta$ and $0\le\alpha$ is an integer. For the classical Meixner polynomials $\mathfrak M_{n,N}^\alpha(x)$ orthonormal on the gird $\{0,\delta,2\delta,\dots\}$ with weight $\rho(x)=(1-e^{-\delta})^\alpha\times\Gamma(Nx+\alpha+1)/\Gamma(Nx+1)$, the following asymptotic formula is obtained: $$ \mathfrak M_{n,N}^\alpha(z)=\Lambda_n^\alpha(z)+v_{n,N}^\alpha(z). $$ The remainder $v_{n,N}^\alpha(z)$ for $n\le\lambda N$ satisfies the estimate $$ |v_{n,N}^\alpha(z)|^2\le c(\alpha,\lambda)\delta \sum_{k=0}^n|\Lambda_k^\alpha(z)|^2, $$ where $\Lambda_k^\alpha(x)$ are the Laguerre orthonormal polynomials. As a consequence, a weighted estimate, for the Meixner polynomial $\mathfrak M_{n,N}^\alpha(x)$ on the semiaxis $[0,\infty)$ is obtained.