Etingof Trace, Path Hypergeometric Functions and Integrable Systems

Under a certain condition, we find the explicit formulas for the trace functions of certain intertwining operators among gl(n)-modules, introduced by Etingof in connection with the solutions of the Calogero-Sutherland model. If n = 2, the master function of the trace function is exactly the classical Gauss hypergeometric function. When n > 2, the master functions of the trace functions are a new family of multiple hypergeometric functions, whose differential property and integral representation are dominated by certain polynomials of integral paths connecting pairs of positive integers. Moreover, we define and explicitly find similar trace functions for sp(2n), which give rise to solutions of the Olshanesky-Perelomov model of type C. The master functions of the trace functions for sp(2n) are similar new multiple path hypergeometric functions. Analogous multiple path hypergeometric functions for orthogonal Lie algebras are defined and studied.