Special Functions of the Isomonodromy Type

We introduce a new notion, a special function of the isomonodromy type, and show that most of the functions known in applied mathematics and mathematical physics as special functions belong to this type. This definition provides a unified approach to the theories of ‘linear’ special functions, i.e., classical higher transcendental functions, and ‘nonlinear’ special functions, i.e., functions of the Painlevé type. We also show that our definition has not only a conceptual (methodological) value; many well-known properties of the single-variable special functions can be re-derived not only from the isomonodromy point of view, but also the practical one too: (1) the isomonodromy approach is already known as an extremely useful one in the theory of Painlevé functions, (2) many properties (some of them can be new ones) of the multi-variable special functions can be obtained on a regular basis, and (3) the definition gives rise to several interesting mathematical questions which are discussed in the paper. We also make some remarks concerning the analogous description (via q-deformations) of q-special functions.