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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 $$u\sp{(n)}-(- )\sp n\sum\sp p\sb{r=0}\alpha\sb rz\sp ru\sp{(r)}=0\quad (n>p\ge 0),$$ where $\alpha\sb r$ $(r=0,1,\ldots,p-1)$ are arbitrary constants with $\alpha\sb p=1$ and $\alpha\sb 0\ne 0$. He gives the following particular solution of this equation $$U\sb{n,p}(-z)=\sum\sp p\sb{k=0}{(- n\sp{p/n}z)\sp k\over k!}\prod\sp p\sb{r=1}\Gamma\left({k+\beta\sb r\over n}\right) \quad (n>p\ge 0),$$ where the parameters $-\beta\sb r$ are the zeros of the polynomial of degree $P$ given by $$\alpha\sb 0+\sum\sp p\sb{r=1}\alpha\sb r\prod\sp{r-1}\sb{k=0}(x-k)=\prod\sp p\sb{r=1}(x+\beta\sb r). $$ The main results of the paper are the following asymptotic expansions $U\sb{n,p}(-z)$ for large $\vert z\vert,$ namely $$U\sb{n,p}(-z)\sim H(z)\text{ in } \vert\arg z\vert<{1\over 2}\pi\left(1+{p\over n}\right),$$ $$U\sb{n,p}(-z)\sim H(z)+E(ze\sp{Fi\pi})\text{ in } \vert\arg(-z)\vert<\pi\left(1-{p\over n}\right),$$ where $n>p\ge 0$ and the upper or lower sign is chosen according as $\vert\arg z\vert>0$ or $\vert\arg z\vert<0$, $$E(z)=(2\pi)\sp{{1\over 2}p}K\sp{-{1\over 2}}\left({z\sp{1/K}\over n}\right)\sp \theta\exp(Kz\sp{1/K})\sum\sp \infty\sb{K=0}d\sb K(Kz\sp{1/K})\sp{-K},$$ where the coefficients $d\sb K$ are independent of $z$ [see the author and {\it A. D. Wood}, Asymptotics of high order differential equations (1986; Zbl 0644.34052)], $K={n-p\over n}$, $\theta={1\over n}\sum\sp p\sb{r=1}\beta\sb r-{1\over 2}p$, $H(z)=n\sum\sp p\sb{r=1}(n\sp{p/n}z)\sp{-\beta\sb r}$ $S\sb{n,p}(\beta\sb r;z)$; provided no two of the $\beta\sb r$ either coincide or differ by an integer multiple of $n$ and $$S\sb{n,p}(\beta\sb r;z)=\sum\sp \infty\sb{K=0}{(-1)\sp K\over K!}\Gamma(nK+\beta\sb r){\prod\sp p\sb{j=1}}' \Gamma\left({\beta\sb j-\beta\sb r\over n}- K\right)\left(n\sp{p/n}z\right)\sp{-nK},$$ 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.

34M99Differential equations in the complex domain
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