Ratio and Plancherel-Rotach asymptotics for Meixner-Sobolev orthogonal polynomials
The authors study the monic Meixner-Sobolev polynomials \(\{Q_n(x)\}\), orthogonal with respect to the inner product (derived from the Pascal distribution): \[ (p,q)_S=(p,q)+\lambda (\Delta p,\Delta q) = \sum_{i=0}^{\infty} \{p(i)q(i)+\lambda\Delta p(i)\Delta q(i)\}{\mu^i(\gamma)_i\over i!}, \] where \(\gamma>0\), \(0<\mu<1\), \(\lambda\geq 0\), \(\Delta\) the forward difference operator with \((\Delta f)(x)=f(x+1)-f(x)\) and \((\gamma)_n\) the Pochhammer symbol (ascending factorial). For \(\lambda=0\) we have the ordinary Meixner polynomials \(M_n^{(\gamma,\mu)}(x)\). The main results are: \(\bullet\) recurrence relations and bounds for \({\widetilde k}_n=\|Q_n\|^2_S\), \(\bullet\) the limit behaviour of \({\widetilde k}_n/k_n^{(\gamma,\mu)}\) for \(n\rightarrow\infty\;(k_n^{(\gamma,\mu)}= \|M_n^{(\gamma,\mu)}\|^2)\), \(\bullet\) the relative asymptotics for \(Q_n(x)/M_n^{(\gamma,\mu)}(x)\) with \(x\in\mathbb{C}\setminus [0,\infty)\), \(\bullet\) that the contracted zeroes \(\{x_{i,n}/n\}_{i=1}^n\) of \(Q_n\) accumulate on \([0,(1+\sqrt{\mu})^2/(1-\mu)\), \(\bullet\) Plancherel-Rotach asymptotics for \[ Q_n(nx)/\left\{\sqrt{k_n^{(\gamma,\mu)}}\prod_{i=1}^n \varphi\left({nx_i-b_ i) \over 2a_i}\right)\right\} \] where \(a_i=\sqrt{\mu i(i+\gamma-1)}/(1-\mu), b_i=\{(1+\mu)i+\gamma\mu\}/(1-\mu)\) and \(\varphi(x)=x+\sqrt{x^2-1}\) with \(\sqrt{x^2-1}>0\) for \(x>1\). A nicely written paper.
scientific article; zbMATH DE number 1437653
scientific article