The WKB method used in the number theory

Some generating functions for so-called multiple zeta values are directly related with some solutions to linear differential equations with meromorphic coefficients. For example, the function f₃(x) = 1 - ζ(3)x³ + ζ(3,3)x⁶ - ...  is equal to the value at t = 1 of the hypergeometric function F(-x, εx, ε^-1x; 1, 1; t) (where ε = e^iπ/3). Together with M. Zakrzewski we applied asymptotic analysis as x→∞ (of WKB type) for the hypergeometric equations related with ζ(2) and ζ(3). We proved that the corresponding generating functions satisfy linear differential equations of Bessel type.