INTEGRAL SOLUTIONS OF HYPERGEOMETRIC q-DIFFERENCE SYSTEMS WITH |q| = 1

Nishizawa, Michitomo; Ueno, Koichi

doi:10.1142/9789812810199_0009

Cited by 5 publications

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“…For a description of the properties of the double sine function and some of its applications, see [JM,KLS,NU,Ru2]. Indeed, for Im(σ) > 0 we have Im(σ −1 ) < 0, and (2.8) is well defined.…”

Section: Thenmentioning

confidence: 99%

Theta hypergeometric integrals

Spiridonov

2004

St. Petersburg Math. J.

123

277

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A general class of (multiple) hypergeometric type integrals associated with the Jacobi theta functions is defined. These integrals are related to theta hypergeometric series via the residue calculus. In the one variable case, theta function extensions of the Meijer function are obtained. A number of multiple generalizations of the elliptic beta integral associated with the root systems An and Cn is described. Some of the Cn-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.2000 Mathematics Subject Classification. Primary 33C67, 33D70.

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Section: Thenmentioning

confidence: 99%

Theta hypergeometric integrals

Spiridonov

2004

St. Petersburg Math. J.

123

277

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“…Systematic studies of hyperbolic and elliptic special functions have commenced only recently, see e.g. [17], [20], [26] for the hyperbolic case and [12], [23], [24] for the elliptic case. A basic step in the development of special functions of a given type is the derivation of the associated beta integrals.…”

Section: Introductionmentioning

confidence: 99%

Hyperbolic beta integrals

Stokman

2005

Advances in Mathematics

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Hyperbolic beta integrals are analogues of Euler's beta integral in which the role of Euler's gamma function is taken over by Ruijsenaars' hyperbolic gamma function. They may be viewed as (q, q )-bibasic analogues of the beta integral in which the two bases q and q are interrelated by modular inversion, and they entail q-analogues of the beta integral for |q| = 1. The integrals under consideration are the hyperbolic analogues of the Ramanujan integral, the Askey-Wilson integral and the Nassrallah-Rahman integral. We show that the hyperbolic Nassrallah-Rahman integral is a formal limit case of Spiridonov's elliptic Nassrallah-Rahman integral.

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“…Note that although (48) is a formal identity if both sides converge as in (49) it holds as a function identity.…”

Section: Topological Vertex and Partition Function Of The Resolved Co...mentioning

confidence: 99%

“…The appearence of poles prevents infinite products (81) from converging. When |q| = 1 but q is not a root of unity G (1) q = Γ q is constructed explicitly in [48] via Shintani's double sine function (see [60]). This case is complementary to the classical one q = 1 and the poles are located at the points −n − m/τ with n, m ∈ Z and q = e 2πiτ .…”

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confidence: 99%

Quantum Barnes Function as the Partition Functionof the Resolved Conifold

Koshkin

2008

International Journal of Mathematics and Mathematical Sciences

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We suggest a new strategy for proving large N duality by interpreting Gromov-Witten, Donaldson-Thomas and Chern-Simons invariants of a Calabi-Yau threefold as different characterizations of the same holomorphic function. For the resolved conifold this function turns out to be the quantum Barnes function, a natural qdeformation of the classical one that in its turn generalizes Euler's gamma function. Our reasoning is based on a new formula for this function that expresses it as a graded product of q-shifted multifactorials.

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INTEGRAL SOLUTIONS OF HYPERGEOMETRIC q-DIFFERENCE SYSTEMS WITH |q| = 1

Cited by 5 publications

References 0 publications

Theta hypergeometric integrals

Theta hypergeometric integrals

Hyperbolic beta integrals

Quantum Barnes Function as the Partition Functionof the Resolved Conifold

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