Asymptotic Chebyshev coefficients for two functions with very rapidly or very slowly divergent power series about one endpoint
Let \(f(\varepsilon)\) be a divergent power series \(f(\varepsilon)\sim\sum^\infty_{j=0} b_n\varepsilon^n\) with \(b_n=o((n!)^{2G})\) where \(G\) is the ``Gevrey order'' of the asymptotic series. The two cases when \(G\) is infinite or zero are exemplified by functions \[ f_{\text{sub}}(\varepsilon)=\int^\infty_0 dt {\exp(-\exp t)\over (1+\varepsilon t)}\quad\text{and}\quad f_{\text{sup}}(\varepsilon;A)=\sqrt{{A\over \pi}} \int^\infty_0 {\exp(-A\log^2 t)\over 1+\varepsilon t} {dt\over t}. \] Asymptotic approximations for the coefficients of those functions are derived. They are limits with \(r\to0+\) and \(r\to1-\) of well-known asymptotics [see, for instance, \textit{G. Németh}, Mathematical approximation of special functions: ten papers on Chebyshev expansions (Nova Science, New York) (1992), \textit{J. P. Boyd}, Math. Comput. 39, 201-206 (1982; Zbl 0524.41014)]: \[ a^{\text{sub}}_n\sim(-\text{const }n/\log^{1/2}(n))\quad\text{and}\quad a^{\sup}_n=o(\exp(-\text{const }n^r))\quad\text{for any } r>0. \]
scientific article; zbMATH DE number 915067
scientific article