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Generalized parabolic cylinder functions. (English) Zbl 0753.34041
Summary: In the study of non-selfadjoint elliptic boundary value problems involving an indefinite weight function, there arises a differential equation ${y}^{\text{'}\text{'}}+\left(\nu +{\alpha }^{-1}-{\alpha }^{-2}{z}^{\alpha }\right)y=0$, ${}^{\text{'}}=d/dz$, which is a generalization of the well known equation of Weber. We define a particular solution of this generalized equation and derive some information concerning its asymptotic expansion in certain sectors determined by the independent variable when the independent variable tends to infinity in modulus. We also obtain a series representation of this solution and establish some information concerning the behaviour of the leading coefficients of the series, as this is of importance in applications.

##### MSC:
 34E05 Asymptotic expansions (ODE) 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$ 34M99 Differential equations in the complex domain 34A30 Linear ODE and systems, general