*(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

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

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

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

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

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)], $K=\frac{n-p}{n}$, $\theta =\frac{1}{n}{\sum}_{r=1}^{p}{\beta}_{r}-\frac{1}{2}p$, $H\left(z\right)=n{\sum}_{r=1}^{p}{\left({n}^{p/n}z\right)}^{-{\beta}_{r}}$ ${S}_{n,p}({\beta}_{r};z)$; provided no two of the ${\beta}_{r}$ either coincide or differ by an integer multiple of $n$ and

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 |