  This is author's Habilitation Thesis (Dr. Sci. dissertation) submitted at the
beginning of September 2004. It is written in Russian and is posted due to the
continuing requests for the manuscript.
  The content: 1. Introduction, 2. Nonlinear chains with the discrete time and
their self-similar solutions, 3. General theory of theta hypergeometric series,
4. Theta hypergeometric integrals, 5. Biorthogonal functions, 6. Elliptic
hypergeometric functions with |q|=1, 7. Conclusion, 8. References.
  It contains an outline of a general heuristic scheme for building univariate
special functions through self-similar reductions of spectral transformation
chains, which allowed construction of the differential-difference q-Painleve
equations, as well as of the most general known set of elliptic biorthogonal
functions comprising all classical orthogonal polynomials and biorthogonal
rational functions. One of the key results of the thesis consists in the
discovery of genuinely transcendental elliptic hypergeometric functions
determined by the elliptic hypergeometric integrals. The whole theory of such
integrals can be built from the univariate elliptic beta integral -- the most
complicated known definite integral with exact evaluation, which generalizes
the ordinary binomial theorem and its q-extension, Euler's beta integral, the
measure for Askey-Wilson polynomials, and many other previously established
results on ordinary and q-hypergeometric functions.
