New transformations for elliptic hypergeometric series on the root system A n

Rosengren, Hjalmar

doi:10.1007/s11139-006-0070-6

Cited by 15 publications

(12 citation statements)

References 23 publications

(38 reference statements)

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“…The transformations (1.3.9) and (1.3.10) were obtained by Rosengren [53], together with two more A n transformations that are not surveyed here. The transformation (1.3.11) was obtained independently by Kajihara and Noumi [31] and Rosengren [54]. Both these papers contain further transformations that can be obtained by iterating (1.3.11).…”

Section: Notesmentioning

confidence: 99%

Elliptic hypergeometric series on root systems

Rosengren

2004

Advances in Mathematics

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113

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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...

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Section: Notesmentioning

confidence: 99%

Elliptic hypergeometric series on root systems

Rosengren

2004

Advances in Mathematics

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113

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“…We remark that an elliptic analogue of the Milne-Gustafson summation is due to Kajihara-Noumi [32], and Rosengren [41,42], independently. In these references a specialization of (1.3) is generalized to involve elliptic analogues of the q-products, and the resulting summation…”

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confidence: 86%

“…, x n ; q, q s ) in the case κ = ∅. On the other hand one viewpoint of (1.3) is as a multi-dimensional 1 ψ 1 summation associated to the root system A n−1 (see e.g.[37] and references therein), and such generalized hypergeometric function identities allow natural generalizations to include Macdonald polynomials [33,12].We remark that an elliptic analogue of the Milne-Gustafson summation is due to Kajihara-Noumi [32], and Rosengren [41,42], independently. In these references a specialization of (1.3) is generalized to involve elliptic analogues of the q-products, and the resulting summation…”

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The q-Dixon–Anderson integral and multi-dimensionalψ11summations

Ito

Forrester

2015

Journal of Mathematical Analysis and Applications

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“…A generalization of the elliptic Bailey transformation to multiple elliptic hypergeometric series, called the duality transformation formula of type A n , was obtained independently by Kajihara -Noumi [7] and by Rosengren [14]. A characteristic feature of this formula is that the summations of the two sides are taken over multi-indices of possibly different dimensions, relevant to different sets of variables.…”

Section: Introductionmentioning

confidence: 99%