ON THE UBIQUITY OF THE LÉVY INTEGRAL — ITS RELATIONSHIP WITH THE GENERALIZED EULER-JACOBI SERIES AND THEIR ASYMPTOTICS BEYOND ALL ORDERS

We present here an overview of the history, applications and important properties of a function which we refer to as the Lévy integral. For certain values of its characteristic parameter, the Lévy integral defines the symmetric Lévy stable probability density function. As we discuss however, the Lévy integral has applications to a number of other fields besides probability, including random matrix theory, number theory and asymptotics beyond all orders. We exhibit a direct relationship between the Lévy integral and a number theoretic series which we refer to as the generalized Euler-Jacobi series. The complete asymptotic expansions for all natural values of its parameter are presented, and in particular it is pointed out that the intricate exponentially small series become dominant for certain parameter values.

Asymptotics beyond all orders and Stokes lines in nonlinear differential-difference equations

A technique for calculating exponentially small terms beyond all orders in singularly perturbed difference equations is presented. The approach is based on the application of a WKBJ-type ansatz to the late terms in the naive asymptotic expansion and the identification of Stokes lines, and is closely related to the well-known Stokes line smoothing phenomenon in linear ordinary differential equations. The method is illustrated by application to examples and the results extended to time-dependent differential-difference problems.