Linear differential equations and multiple zeta-values. III. Zeta(3)

Abstract

We consider the hypergeometric equation (1 − t)∂t∂t∂g + x3g = 0, whose unique analytic solution φ1(t; x) = 1 + O(t) near t = 0 for t = 1 becomes a generating function for multiple zeta values φ1(1; x) = f3(x) = 1 − ζ(3)x3 + ζ(3, 3)x6 − …. We apply the so-called WKB method to study solutions of the hypergeometric equation for large x and we calculate corresponding Stokes matrices. We prove that the function f3(x) near x = ∞ is also expressed via WKB type functions which subject to some Stokes phenomenon. This implies that f3(x) satisfies a sixth order linear differential equation with irregular singularity at infinity.