Linear differential equations and multiple zeta values. II. A generalization of the WKB method

For linear differential equations x ( n ) + a 1 x ( n − 1 ) + ⋯ + a n x = 0 (and corresponding linear differential systems) with large complex parameter λ and meromorphic coefficients a j = a j ( t ; λ ) we prove existence of analogues of Stokes matrices for the asymptotic WKB solutions. These matrices may depend on the parameter, but under some natural assumptions such dependence does not take place. We also discuss a generalization of the Hukuhara–Levelt–Turritin theorem about formal reduction of a linear differential system near an irregular singular point t = 0 to a normal form with ramified change of time to the case of systems with large parameter. These results are applied to some hypergeometric equations related with generating functions for multiple zeta values.