Biorthogonal rational functions and elliptic hypergeometric function

In the first part of the talk we shall consider general biorthogonal rational functions (DBP), which generalize orthogonal polynomials. They satisfy a three-term recurrence relation and can be designed with the help of determinants constructed from the generalized moments. For DBP with a discrete measure orthogonality relation is equivalent to orthogonality for polynomials with variable weight function proposed by A. A. Gonchar in 1978. In the second part of the talk we shall describe the system DBP, expressed in terms of elliptic hypergeometric series and generalizes all the known classical special functions. In 2000, the author introduced a new class of exactly calculated integrals, the elliptic beta integrals. The simplest of them is a measure of DBP for the above system. Moreover, it serves as a measure for more complex systems of biorthogonal functions, which are no longer rational. The talk is expository, and many technical details and proofs will be omitted.