The Riemann-Hilbert problem and special functions

Abstract

A generic scheme based on modern soliton theory is proposed for the description of classical special functions of mathematical physics. To this end, the special function is considered as isomonodromic deformation of some linear ODE with rational coefficients. This ODE plays a role of one of the two equations of the Lax pair. Thus the ODE, or difference equation satisfied by the special function is integrable, i.e. it has commuting integrals of motions, the invariant submanifolds and the corresponding action‐angle variables.Moreover, an integration scheme for the Lax pair involves a matrix Riemann‐Hilbert problem which provides an integral representation of the special function. Examples of relevant Riemann‐Hilbert problems are given for hypergeometric functions and orthogonal polynomials. We discuss a way how to get the non‐Abelian generalizations of the Riemann‐Hilbert problems, leading to “nonlinear” special functions, such as Painlevé transcendents.