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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 (n) -(-) n r=0 p α r z r u (r) =0(n>p0),

where α r (r=0,1,...,p-1) are arbitrary constants with α p =1 and α 0 0. He gives the following particular solution of this equation

U n,p (-z)= k=0 p (-n p/n z) k k! r=1 p Γk+β r n(n>p0),

where the parameters -β r are the zeros of the polynomial of degree P given by

α 0 + r=1 p α r k=0 r-1 (x-k)= r=1 p (x+β r )·

The main results of the paper are the following asymptotic expansions U n,p (-z) for large |z|, namely

U n,p (-z)H(z)in|argz|<1 2π1+p n,
U n,p (-z)H(z)+E(ze Fiπ )in|arg(-z)|<π1-p n,

where n>p0 and the upper or lower sign is chosen according as |argz|>0 or |argz|<0,

E(z)=(2π) 1 2p K -1 2 z 1/K n θ exp(Kz 1/K ) K=0 d K (Kz 1/K ) -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=n-p n, θ=1 n r=1 p β r -1 2p, H(z)=n r=1 p (n p/n z) -β r S n,p (β r ;z); provided no two of the β r either coincide or differ by an integer multiple of n and

S n,p (β r ;z)= K=0 (-1) K K!Γ(nK+β r ) j=1 p ' Γβ j -β r n-Kn p/n z -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:
34M99Differential equations in the complex domain