Elliptic hypergeometric series on root systems

Abstract

Let f = n≥0 c n . The series f is called hypergeometric if the ratio c n+1 /c n , viewed as a function of n, is rational. A simple example is the Taylor series exp(z) = ∞ n=0 z n /n!. Similarly, if the ratio of consecutive terms of f is a rational function of q n for some fixed q -known as the base -then f is called a basic hypergeometric series. An early example of a basic hypergeometric series is Euler's q-exponential function e q (z) = n≥0 z n / (1 − q) · · · (1 − q n ) . If we express the base as q = exp(2πi/ω) then c n+1 /c n becomes a trigonometric function in n, with period ω. This motivates the more general definition of an elliptic hypergeometric series as a series f for which c n+1 /c n is a doubly-periodic meromorphic function of n.Elliptic hypergeometric series first appeared in 1988 in the work of Date et al. on exactly solvable lattice models in statistical mechanics [12]. They were formally defined and identified as mathematical objects of interest in their own right by Frenkel and Turaev in 1997 [19]. Subsequently, Spiridonov introduced the elliptic beta integral, initiating a parallel theory of elliptic hypergeometric integrals [61]. Together with Zhedanov [62, 72] he also showed that Rahman's [43] and Wilson's [74] theory of biorthogonal rational functions -itself a generalization of the Askey scheme [33] of classical orthogonal polynomials -can be lifted to the elliptic level.All three aspects of the theory of elliptic hypergeometric functions (series, integrals and biorthogonal functions) have been generalized to higher dimensions, connecting them to root systems and Macdonald-Koornwinder theory. In [73] Warnaar introduced elliptic hypergeometric series associated to root systems, including a conjectural series evaluation of type C n . This was recognized by van Diejen and Spiridonov [14,15] as a discrete analogue of a multiple elliptic beta integral (or elliptic Selberg integral). They formulated the corresponding integral evaluation, again as a conjecture. This in turn led Rains [44,46] to develop an elliptic analogue of Macdonald-Koornwinder theory, resulting in continuous as well as discrete biorthogonal elliptic functions attached to the non-reduced root system BC n . In this theory, the elliptic multiple beta integral and its discrete analogue give the total mass of the biorthogonality measure.Although a relatively young field, the theory of elliptic hypergeometric functions has already seen some remarkable applications. Many of these involve the multivariable theory.2 Elliptic hypergeometric functions associated with root systemsIn 2009, Dolan and Osborn showed that supersymmetric indices of four-dimensional supersymmetric quantum field theories are expressible in terms of elliptic hypergeometric integrals [18]. Conjecturally, such field theories admit electric-magnetic dualities known as Seiberg dualities, such that dual theories have the same index. This leads to non-trivial identities between elliptic hypergeometric integrals (or, for so called confining theories, to integral evaluat...

Essays on the theory of elliptic hypergeometric functions

“…Its first recursive proof was obtained in [86]. In the limit p → 0 it degenerates to a multivariate 8 ϕ 7 -sum, which was found in [103].…”

“…b n . After taking out of the sum the (n + 1)-st term, this relation can be rewritten in the form [86] …”

“…These expansions into "simple" fractions for theta functions were used in the papers [33,36,86] for the proof of some exact summation and integration formulae for elliptic hypergeometric functions. Let us describe also another expansion which was used recently in [59]:…”

An elementary approach to 6j-symbols (classical, quantum, rational, trigonometric, and elliptic)

“…Plugging (3.14) into (3.9) and rewriting the result in standard notation gives This is Jackson's summation. Essentially the same method was used in [Ro2] to obtain extensions of Jackson's summation to multiple elliptic hypergeometric series related to the root systems A n and D n .…”

“…This is equivalent to the classical identity [TM,p. 34], see also [Ro2], n k=1 n j=1 θ(a k /b j ) n j=1,j =k θ(a k /a j ) = 0, a 1 · · · a n = b 1 · · · b n .…”