Wong, R.; Zhao, Y.-Q. Smoothing of Stokes’s discontinuity for the generalized Bessel function. II. (English) Zbl 0974.41021 Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 455, No. 1988, 3065-3084 (1999). [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. Reviewer: N.M.Temme (Amsterdam) Cited in 4 Documents MSC: 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\) Keywords:asymptotic expansions; generalized Bessel functions; Stokes’ phenomenon; hyperasymptotics PDFBibTeX XMLCite \textit{R. Wong} and \textit{Y. Q. Zhao}, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 455, No. 1988, 3065--3084 (1999; Zbl 0974.41021) Full Text: DOI Digital Library of Mathematical Functions: §10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions