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Smoothing of Stokes’s discontinuity for the generalized Bessel function. II. (English) Zbl 0974.41021
[For part I see the authors in ibid. 455, No. 1984, 1381-1400 (1999).]
The generalized Bessel function \(\phi(z)=\sum_{n=0}^\infty z^n/[n! \Gamma(\rho n+\beta)]\), is usually defined for \(0<\rho<\infty\) and \(\beta\) real or complex. In an earlier paper the superasymptotics and hyperasymptotics of this functions is considered. In this second part the function is discussed for \( -1<\rho<0\). Saddle point methods are used to derive the asymptotic expansion, with a detailed analysis of the saddle point contours, the Stokes lines and the smoothing of the Stokes discontinuity.

MSC:
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
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