XXVII Workshop on Geometric Methods in Physics 29.06-05.07.2008

Victor Novokshenov

Special Functions and Isomonodromic Deformations

We introduce a new notion, a special function of the isomonodromic type, and show that most of the functions known in applied mathematics and mathematical physics as special functions belong to this type. In this sense, the special function provides isomonodromic deformation of some linear ODE with rational coefficients. This ODE plays a role of one of the two equations of the Lax pair. In its turn, this gives rise to an alternative definition: a matrix Riemann-Hilbert problem with a parameter, entering the conjugation matrix in a manner similar to the soliton theory. Thus the ODE for the special function appears to be integrable in the sense of Lioville-Arnold, i.e., it has the commuting integrals of motions, the invariant submanifolds and the corresponding angle variables. We also show that our definition has not only a conceptual value: many well-known properties of the single-variable special functions can be re-derived from the isomonodromy point of view. The examples of relevant Riemann-Hilbert (RH) problems are given for the Airy, Bessel, gamma and zeta functions. Those matrix RH problems are Abelian and exactly solvable, which provides the integral repesentations for these functions. We show how to get the non-Abelian generalizations of the RH problems, leading to new examples of special functions, such as Painlev\'e transcendents.