# New theory of hypergeometric functions

# New theory of hypergeometric functions

The lecture will be devoted to the new vintage in the theory of special functions, a unification of the Bessel, hypergeometric, spherical and Whittaker functions, their p-adic and difference counterparts, and of course the theta-functions (associated with root systems) in one definition. The latter was suggested;13 years ago, but a reasonably complete analytic theory of such global spherical functions was created only recently, including the Harish-Chandra asymptotic formula and many more. These global functions generalize the classical q-hypergeometric(basic) series introduce by Heine in 1846, but the new approach is very different even for one variable. Algebraically, the global functions are actually similar to the Bessel functions. For instance, the corresponding Fourier transform is essentially involutive (like for the classical Hankeltransform), which is broken in the Harish-Chandra theory and its p-adic counterpart. The construction is based on DAHA, q-deformations of the classical Heisenberg-Weyl algebras. The Lie theory was expected to provide such unification (the Gelfand program) but it did not materialize. The new theory actually begins with nonsymmetric global spherical functions (eigenfunctions of first order opertators) and their counterparts everywhere (including nonsymmetric Schur functions), which is generally beyond the Lie theory. It clarifies why we need to go back to the Heisenberg algebras and their deformations. The case of rank one will be mainly considered; no special knowledge of representation theory is assumed.