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Asymptotic expansions of the generalized Bessel polynomials. (English) Zbl 0880.41027
Summary: In this paper, we investigate the asymptotic behavior of the generalized Bessel polynomials $y_n (z;a)$. Let $z= \alpha/(n+1)$. We first derive infinite asymptotic expansions for $y_n (z;a)$ when $\alpha$ lies in various regions of the complex plane, except when $\alpha$ is near $\pm i$. Then we construct uniform asymptotic expansions for $y_n (z;a)$ in neighborhoods of $\alpha= \pm i$. These expansions involve the Airy function and its derivative. Finally, we use these approximations to study the asymptotic behavior of the zeros of $y_n (z;a)$ near $\alpha =i$.

41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
33C45Orthogonal polynomials and functions of hypergeometric type
Full Text: DOI
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