The Erwin Schroedinger Institute for Mathematical Physics

Elliptic Hypergeometric Functions in Combinatorics, Integrable Systems and Physics

Dates:  March 20-24, 2017.

Organizers:  Christian Krattenthaler (U Vienna), Masatoshi Noumi (Kobe U), Simon Ruijsenaars (U Leeds), Michael J. Schlosser (U Vienna), Vyacheslav P. Spiridonov (JINR, Dubna and NRU HSE, Moscow), and S. Ole Warnaar (U Queensland).

Aims:  Special functions of the hypergeometric type were investigated over centuries in two instances, as the ordinary and q-hypergeometric functions. It was a complete surprise when on the verge of the millennium the elliptic hypergeometric functions forming the third type of hypergeometric functions have been discovered. First they appeared implicitly as elliptic solutions of the Yang-Baxter equation (Date, Jimbo, Kuniba, Miwa, Okado, 1988). As shown by Frenkel and Turaev (1997), these solutions have a form of a terminating series of hypergeometric type satisfying remarkable elliptic function identities. The genuinely transcendental elliptic hypergeometric functions are defined by the integrals discovered by Spiridonov (2000) using the elliptic gamma function of Ruijsenaars (1997). Shortly thereafter, most of the old achievements of the theory of hypergeometric functions were lifted to the elliptic level, including generalizations of the Askey-Wilson and Macdonald polynomials. Nowadays, with the efforts of many researchers (van Diejen, Felder, Noumi, Rains, Rosengren, Sakai, Schlosser, Varchenko, Warnaar, Zhedanov, and others) elliptic hypergeometric functions have been related to various areas of mathematics, including integrable systems and generalizations of the Painlevé equations, combinatorics and mathematical physics. The maybe most remarkable application is found in the four-dimensional quantum field theory, where elliptic hypergeometric integrals emerge as superconformal indices and their symmetries describe the Seiberg dualities (1995). This workshop brings together leading experts on elliptic hypergeometric functions from different areas which are still developing with great speed. The main aim of the workshop consists in giving an opportunity to specialists to exchange new results and ideas, and to young scientists to get an introduction to key subjects.


  • Elliptic integrable systems and elliptic Painlevé equations
  • Univariate and multivariate elliptic hypergeometric series and biorthogonal functions
  • Elliptic determinants and theta functions on root systems
  • Combinatorics of elliptic hypergeometric functions
  • Elliptic hypergeometric integrals in quantum field theory

The following people will give introductory lectures (titles are tentative!):

  • Fokko van de Bult:  Hypergeometric functions and integrals
  • Masatoshi Noumi:  Discrete Painlevé equations and special functions
  • Simon Ruijsenaars:  Quantum integrable systems of elliptic Calogero-Moser type
  • Michael J. Schlosser:  Elliptic hypergeometric combinatorics
  • Vyacheslav P. Spiridonov:  Applications of the elliptic hypergeometric integrals

Further, there will be talks by participants.

Attendance (restricted to 60 participants) is by invitation only. The Workshop is already full!

Proceedings: While there will be no formal proceedings, participants have the opportunity to submit articles related to the topics of the workshop to a special issue of SIGMA dedicated to Elliptic hypergeometric functions and their applications. Prospective authors are kindly invited to submit their work according to the submission rules for special issues of SIGMA. All articles will go through the standard peer reviewing procedure of SIGMA. Manuscripts can be submitted at any time, the reviewing process starts immediately and the papers will be published as soon as a positive decision is made. The deadline for submissions is January 31, 2018.