Discrete integrable equations and special functions

Abstract

A generic scheme based on the matrix Riemann-Hilbert problem theory is proposed for constructing classical special functions satisfying difference equations. These functions comprise gammaand 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 to 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 for a symmetric group. Mathematics Subject Classification: 33C05, 33C12, 34M55, 34M40, 34E20, 34M60 In work [18], there was considered a scheme for describing classical special functions based on the matrix Riemann-Hilbert problem. It was shown that such functions satisfying ordinary differential equations can be represented in terms of a solution to some Riemann-Hilbert problem, that is, in terms of the problem on recovering an analytic function by its boundary values. In this way, for the corresponding differential equations, there was checked the integrability property treated in the sense of the solutions theory [1], [26]. Such treating of the integrability property as calculating of the values of a function by its global behavior means the presence of an integrable representation for this function. In fact, the method of the Riemann-Hilbert problem demonstrates the equivalency of these two definitions of the integrability [6], [15]. The functions covered by such treating of the integrability are, for instance, hypergeometric and elliptic functions. However, in the handbooks, see, for instance, [7], [14], [27], there are other special functions satisfying no differential equations. Among such functions are Gamma and zeta functions and their generalizations arising in the number theory, combinatorics and the groups representation theory. How one can extend the method of the Riemann-Hilbert problem to these special functions? In the present paper we attempt to answer this question. The key point is that there exists a discrete equation satisfied by special functions. It turns out that these equations can be treated within the scheme of the solitons theory. Namely, for each discrete equation we provide the Lax pair of two linear equations and their compatibility condition is exactly the considered V.Yu. Novokshenov, Discrete integrable equations and special functions. c ○ Novokshenov V.Yu. 2017. The work is financially supported by the grant of Russian Science Foundation (project no. 17-11-01004). Submitted July 1, 2017.