Uniform asymptotic expansions of the Pollaczek polynomials
The Pollaczek polynomials \(P_n(x; a,b)\) can be defined by the generating function \[ (1- we^{i\theta})^{-{1\over 2}+ ih\theta}(1- we^{-i\theta})^{{1\over 1}- ih(\theta)}, \] \(x= \cos\theta\), \(\theta\in (0,\pi)\) and \(h(\theta)= {\cos\theta+ b\over 2\sin \theta}\), \(a> \pm b\). Two uniform asymptotic expansions are obtained valid in a respective interval depending of \(n\) and a parameter \(\delta< \sqrt{a+b}\) for \(\{P_n(x; a,b)\}\). One of them contain a turning point and all possible zeros of \(P_n(\cos\theta)\) in \(\theta\in (0,{\pi\over 2})\).
scientific article; zbMATH DE number 5015166