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 RiemannHilbert 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 LiovilleArnold, 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 wellknown properties of the singlevariable special functions can be
rederived from the isomonodromy point of view. The examples of relevant
RiemannHilbert (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 nonAbelian generalizations of
the RH problems, leading to new examples of special functions,
such as Painlev\'e transcendents.
