Uniform asymptotic expansions for Meixner polynomials (Q1381499)

Meixner polynomials \(m_n(x;\beta,c)\) are considered for large values of \(n\). Two uniform expansions of \(m_n(n\alpha;\beta,c)\) are given, in terms of parabolic cylinder functions, one holding uniformly for \(\alpha\in [\varepsilon,1+a]\) and the other one for \(\alpha\in [1-b,M]\), where \(\varepsilon, a\) and \(b\) are small positive numbers and \(M<\infty\). The results are obtained by steepest descent methods and include five asymptotic formulas given earlier by W. M. Y. Goh.