Theta hypergeometric integrals

By V. Spiridonov. 2003

162 citations recorded by the source corpus.

We define a general class of (multiple) integrals of hypergeometric type associated with the Jacobi theta functions. These integrals are related to theta hypergeometric series through the residue calculus. In the one variable case, we get theta function extensions of the Meijer function. A number of multiple generalizations of the elliptic beta integral [S2] associated with the root systems $A_n$ and $C_n$ is described. Some of the $C_n$-examples were proposed earlier by van Diejen and the author, but other integrals are new. An example of the biorthogonality relations associated with the elliptic beta integrals is considered in detail.

Subfields: Algebra and Number Theory, Applied Mathematics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/f0332b7cf062dfb215e94e3c60057f009bd9ca1f

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In Algebra and Number Theory

In Applied Mathematics

In Statistical and Nonlinear Physics

Citations in the corpus

Transformations of elliptic hypergometric integrals (2003, methodology): “The following identity was originally conjectured by van Diejen and Spiridonov [6] (their “Type II” integral): Theorem 6.1.”
Extensions of the well-poised and elliptic well-poised Bailey lemma (2003, unknown): “series we refer the reader to [6,7,8,9,10,12,14,15, 16 ,18,19,20,21,23].”
$BC_n$-symmetric abelian functions (2004, background): “Another indication that an elliptic extension should exist was the existence of elliptic analogues [27, 26], both discrete and continuous, of the Askey-Wilson orthogonal polynomials [2].”
A Bailey Tree for Integrals (2003, methodology): “Actually, all the changes of the orders of integrations in multiple integrals used in [10], [11], [12], [13], [14], [5] may be interpreted as integral generalizations of the Bailey’s change of the order of summations for series [1], [2].”
Elliptic Hypergeometry of Supersymmetric Dualities (2009, methodology): “10) served as an entry ticket to the large class of new exactly computable integrals discussed in [19, 20, 21, 65, 83, 93], which is essentially extended by the conjectures presented in this paper.”
Some systems of multivariable orthogonal q-Racah polynomials (2004, unknown): “For some related results see, e.g., [6], [7], [8], [9], [12], [16] and [ 19 ].”
FROM 4d SUPERCONFORMAL INDICES TO 3d PARTITION FUNCTIONS (2011, background): “Selberg integral of [34] and a multidimensional extension of the elliptic analogue of the Gauss hypergeometric function of [2, 35].”
Inversions of integral operators and elliptic beta integrals on root systems (2004, methodology): “As already mentioned in the introduction this integral was conjectured by the first author [24] and subsequently proven by Rains [17, Corollary 4.”
Advanced Determinant Calculus: A Complement (2005, background): “For example, determinant evaluations involving elliptic functions have come into the picture in the theory of multiple elliptic hypergeometric series, see [88, 134, 138, 139, 141, 163, 174].”
Essays on the theory of elliptic hypergeometric functions (2008, background): “In this case it is necessary to take l = s and choose the parameters tj , t̃j , and wj , w̃j in such a way that all Γ-functions are combined to the modified elliptic gamma functions G(u;ω) (it is in this way that this function was built in [11]).”