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Smoothing of the Stokes phenomenon for high-order differential equations. (English) Zbl 0748.34008

This paper is a sequel of former five papers published by the same author in which the author extends the class of functions for which the smooth transition of a Stokes multiplier across a Stokes line can be established to functions satisfying a certain differential equation of arbitrary order $n$ whose solutions, in general, involve compound asymptotic expansion with more than one dominant and subdominant series. The author considers the following $n$-th order differential equation

${u}^{\left(n\right)}-{\left(-\right)}^{n}\sum _{r=0}^{p}{\alpha }_{r}{z}^{r}{u}^{\left(r\right)}=0\phantom{\rule{1.em}{0ex}}\left(n>p\ge 0\right),$

where ${\alpha }_{r}$ $\left(r=0,1,...,p-1\right)$ are arbitrary constants with ${\alpha }_{p}=1$ and ${\alpha }_{0}\ne 0$. He gives the following particular solution of this equation

${U}_{n,p}\left(-z\right)=\sum _{k=0}^{p}\frac{{\left(-{n}^{p/n}z\right)}^{k}}{k!}\prod _{r=1}^{p}{\Gamma }\left(\frac{k+{\beta }_{r}}{n}\right)\phantom{\rule{1.em}{0ex}}\left(n>p\ge 0\right),$

where the parameters $-{\beta }_{r}$ are the zeros of the polynomial of degree $P$ given by

${\alpha }_{0}+\sum _{r=1}^{p}{\alpha }_{r}\prod _{k=0}^{r-1}\left(x-k\right)=\prod _{r=1}^{p}\left(x+{\beta }_{r}\right)·$

The main results of the paper are the following asymptotic expansions ${U}_{n,p}\left(-z\right)$ for large $|z|,$ namely

${U}_{n,p}\left(-z\right)\sim H\left(z\right)\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}|argz|<\frac{1}{2}\pi \left(1+\frac{p}{n}\right),$
${U}_{n,p}\left(-z\right)\sim H\left(z\right)+E\left(z{e}^{Fi\pi }\right)\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}|arg\left(-z\right)|<\pi \left(1-\frac{p}{n}\right),$

where $n>p\ge 0$ and the upper or lower sign is chosen according as $|argz|>0$ or $|argz|<0$,

$E\left(z\right)={\left(2\pi \right)}^{\frac{1}{2}p}{K}^{-\frac{1}{2}}{\left(\frac{{z}^{1/K}}{n}\right)}^{\theta }exp\left(K{z}^{1/K}\right)\sum _{K=0}^{\infty }{d}_{K}{\left(K{z}^{1/K}\right)}^{-K},$

where the coefficients ${d}_{K}$ are independent of $z$ [see the author and A. D. Wood, Asymptotics of high order differential equations (1986; Zbl 0644.34052)], $K=\frac{n-p}{n}$, $\theta =\frac{1}{n}{\sum }_{r=1}^{p}{\beta }_{r}-\frac{1}{2}p$, $H\left(z\right)=n{\sum }_{r=1}^{p}{\left({n}^{p/n}z\right)}^{-{\beta }_{r}}$ ${S}_{n,p}\left({\beta }_{r};z\right)$; provided no two of the ${\beta }_{r}$ either coincide or differ by an integer multiple of $n$ and

${S}_{n,p}\left({\beta }_{r};z\right)=\sum _{K=0}^{\infty }\frac{{\left(-1\right)}^{K}}{K!}{\Gamma }\left(nK+{\beta }_{r}\right){\prod _{j=1}^{p}}^{\text{'}}{\Gamma }\left(\frac{{\beta }_{j}-{\beta }_{r}}{n}-K\right){\left({n}^{p/n}z\right)}^{-nK},$

with the prime denoting the omission of the term corresponding to $s=r$. Finally the author gives two special cases when $p=0$ and when $p=1$, $n=2$ to demonstrate the importance of his generalized results.

##### MSC:
 34M99 Differential equations in the complex domain