Article
Ufa Mathematical Journal
Volume 9, Number 3, pp. 118-130
Discrete integrable equations and special functions
Novokshenov V.Yu.
DOI:10.13108/2017-9-3-118
Download PDF
Article on MathNetAbstact
A generic scheme based on the matrix Riemann-Hilbert problem
theory is proposed for constructing classical special
functions satisfying difference equations. These functions comprise gamma- and zeta functions, as well as orthogonal polynomials with corresponding recurrence relations. We show that all difference equations are the compatibility conditions of certain Lax pair coming from the Riemann-Hilbert problem. At that, the integral
representations for solutions to the classical Riemann-Hilbert problem on duality of analytic functions on a contour in the complex plane are generalized for the case of discrete measures, that is, for infinite sequences of points in the complex plane. We establish that such generalization allows one to treat a series of nonlinear difference equations integrable in the sense of solitons theory.
The solutions to the mentioned Riemann-Hilbert problems allows us o reproduce analytic properties of classical special functions described in handbooks and to describe a series of new functions pretending to be special. For instance, this is true for difference Painlevé equations. We provide the example of applying a difference second type Painlevé equation to the representation problem of a symmetric group.