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.


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