Riesz fractional derivatives of a function, Dα xf(x) (also called Riesz potentials), are defined as fractional powers of the Laplacian. Asymptotic expansions for large x are computed for the Riesz fractional derivatives of the Airy function of the first kind, Ai(x), and the Scorer function, Gi(x). Reduction formulas are provided that allow one to express Riesz potentials of products of Airy fun...

An analytical expression of an Airy-Gauss beam passing through a fractional Fourier transform (FRFT) system is derived. The normalized intensity distribution, phase distribution, centre of gravity, effective beam size, linear momentum, and kurtosis parameter of the Airy-Gauss beam are demonstrated in FRFT plane, respectively. The influence of the fractional order p on the normalized intensity d...

In a celebrated paper, Dyson shows that the spectrum of a n× n randomHermitian matrix, diffusing according to an Ornstein-Uhlenbeck process, evolves as n non-colliding Brownian motions held together by a drift term. The universal edge and bulk scalings for Hermitian random matrices, applied to the Dyson process, lead to the Airy and Sine processes. In particular, the Airy process is a continuou...

We compute Stokes matrices for generalised Airy equations and prove that they are regular unipotent (up to multiplication with the formal monodromy). This class of differential was defined by Katz includes classical equation. In addition, it which not rigid. Our approach is based on topological computation enhanced Fourier-Sato transform a perverse sheaf due D'Agnolo, Hien, Morando Sabbah.