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Équations différentielles motiviques et au–delà

Higher Bessel Functions: product formulas, Landau-Ginzburg type models and corresponding integer sequences

11 rue Pierre et Marie Curie
75231 Paris Cedex 05

Product formulas for the standard Bessel functions as well as special functions of the same type have been known since the works of N.Sonine and L.Gegenbauer from 1880. Such formulas may be found in various contexts and recently appeared in the works of Golyshev-Mellit-Rubtsov-van Straten and Kontsevich-Odesskii. Also such types of formulas are deeply connected with formal group laws by Buchstaber. In my talk I will report on the progress with V. Rubtsov and D. van Straten on the multiplication kernels for higher rank versions of the Bessel equation. I will demonstrate how the periods for the certain Landau-Ginzburg type models appear in the special multiplication formulas. Such LG-type models enjoy nice integral properties and allow to define an analogue of mirror map and instanton numbers. Moreover, I will explain how to express a general multiplication kernel as a generating function for the integer sequences which generalize the Pascal triangle. During the talk I will give an exposition of already known and obtained results, discuss the connections and show some experimental observations.