On asymptotics of the Meixner polynomials (Q1277565)

The author states an extension to arbitrary real \(\alpha>-1\) of an asymptotic expression for the Meixner polynomials \(m_n^{\alpha}(x,h)\). The grid used is \(\{0,h,2h,\ldots\}\) and the polynomials are orthonormal with respect to the weight \[ p(x)=(1-e^{-h})^{\alpha +1}e^{-x}{\Gamma(x/h+\alpha +1)\over\Gamma(x/h+1)}. \] The result is \[ m_n^{\alpha}(x,h)=\Lambda_n^{\alpha}(x)+v_n^{\alpha}(x,h),\quad \Lambda_n^{\alpha}(x) =\left({n+\alpha\choose n}\Gamma(\alpha +1)\right)^{-1/2}L_n^{\alpha}(x), \] with \[ | v_n^{\alpha}(x,h)| \leq C(a,\alpha)\left(h\sum_{i=0}^n | \Lambda_i^{\alpha}(x)| ^2\right)^{1/2},\;h\leq a/n, \] where the constant is given by \[ C(a,\alpha)=\left[{a^3e^a\over 576}+\left({5\over 6}a+{\alpha+1\over 2}\right)^2 {a\over \alpha+1}\exp\left(a\left({5\over 3}a+\alpha+7\right)\right)\right]^{1/2}. \] No proofs are given.