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 whose solutions, in general, involve compound asymptotic expansion with more than one dominant and subdominant series. The author considers the following -th order differential equation
where are arbitrary constants with and . He gives the following particular solution of this equation
where the parameters are the zeros of the polynomial of degree given by
The main results of the paper are the following asymptotic expansions for large namely
where and the upper or lower sign is chosen according as or ,
where the coefficients are independent of [see the author and A. D. Wood, Asymptotics of high order differential equations (1986; Zbl 0644.34052)], , , ; provided no two of the either coincide or differ by an integer multiple of and
with the prime denoting the omission of the term corresponding to . Finally the author gives two special cases when and when , to demonstrate the importance of his generalized results.