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Given a system of
meromorphic differential equations, an important problem which has attracted
increasing attention in recent years is the computation of the Stokes’ multipliers.
Only in few special cases one can explicitly compute these matrices in terms of
known higher transcendental functions in the data of the equation. In general, the
non-trivial entries in the Stokes’ multipliers appear to be “new” transcendental
functions in the data of the equation whose analytic resp. singular behavior should
be made as clear as possible—for example, for questions of stability of numerical
computations, information on the nature of singularities will certainly be of
importance. For the so-called hypergeometric system, fixing the formal invariants, the
Stokes’ multipliers are entire functions in the rest of the data. Generally, an
analogous result on the analytic dependence of Stokes’ multipliers for a family of
“iso-formal” equations has been obtained by Babbitt and Varadarajan, using more
powerful methods.
Roughly speaking, this paper is devoted to a question converse to the one
above: Prescribing a Stokes’ phenomenon, can one construct a family of
equations analytic in the Stokes’ multipliers? In principle, this problem is
solved positively by results of Birkhoff and, more elegantly, Sibuya on the
freedom of the Stokes’ multipliers. However, in light of the Birkhoff-Turrittin
Reduction Theorem, one should better ask the following (harder) question:
Can we construct equations whose coefficient matrix is a polynomial in the
independent variable and depends analytically on the Stokes’ multipliers? In
case of dimension n = 2, one can see that the entries in such a polynomial
equation are multi-valued meromorphic functions of the (two) parameters in the
multipliers. In the present paper, this is shown to be true in general. Moreover, we
will explicitly find the branch points of these functions and show how to
calculate their power series expansion about points of analyticity. In contrast
to the situation of n = 2, we do not know the location and order of their
poles.
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