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Sfb 288 Differential Geometry and Quantum Physics |
Abstract for Sfb Preprint No. 272
Special functions of the isomonodromy type A. V. Kitaev
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 "non-linear" special functions, i.e.,the 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 from the isomonodromy point of view, but the practical one too: (1) this approach is already known as the extremely useful one in the theory of the Painlevé functions, (2) many properties (some of them can be new ones) of the multi-variable special functions can be obtained on the regular basis, and (3) it 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 the q-special functions.
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