Special Functions

Andrews, George E.; Askey, Richard; Roy, Ranjan

doi:10.1017/cbo9781107325937

Cited by 2,796 publications

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“…This leads to the m-tuple integral representation for I (m) similar to the Euler type integral representation for the general plain hypergeometric function m+1 F m ; i.e., I (m) can be interpreted as an elliptic analog of the m+1 F m -function. The recurrence (2.2) is a special realization of an integral analog of the Bailey chains discovered in [18] (the Bailey chains technique is well known as a simplest tool for proving the Rogers-Ramanujan type identities [1]). …”

Section: Continuous Biorthogonal Functionsmentioning

confidence: 99%

“…. , t 8 ) that generalizes many special functions of hypergeometric type obeying "classical" properties [1]. In particular, it was shown that this function exhibits symmetry transformations tied to the exceptional root system E 7 and satisfies the elliptic hypergeometric equation.…”

Section: §1 Introductionmentioning

confidence: 97%

“…The complexity of these systems correlates with the complexity of solutions of the Yang-Baxter equation -rational, trigonometric, or elliptic [22]. In the theory of special functions this structural hierarchy is reflected in the existence of the plain and q-hypergeometric functions [1] and their elliptic generalizations [19]. One of our aims in the present paper is to clarify certain questions about this correspondence at the elliptic level.…”

Section: §1 Introductionmentioning

confidence: 99%

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Continuous biorthogonality of an elliptic hypergeometric function

Spiridonov

2009

St. Petersburg Math. J.

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Abstract. A family of continuous biorthogonal functions related to an elliptic analog of the Gauss hypergeometric function is constructed. The key tools used for that are the elliptic beta integral and the integral Bailey chain introduced earlier by the author. The relationship with the Sklyanin algebra and elliptic analogs of the Faddeev modular double are discussed in detail. §1. Introduction Classical and quantum completely integrable systems serve as a rich source of special functions. The complexity of these systems correlates with the complexity of solutions of the Yang-Baxter equation -rational, trigonometric, or elliptic [22]. In the theory of special functions this structural hierarchy is reflected in the existence of the plain and q-hypergeometric functions [1] and their elliptic generalizations [19]. One of our aims in the present paper is to clarify certain questions about this correspondence at the elliptic level.In [17,19], the author introduced an elliptic analog of the Gauss hypergeometric function V (t 1 , . . . , t 8 ) that generalizes many special functions of hypergeometric type obeying "classical" properties [1]. In particular, it was shown that this function exhibits symmetry transformations tied to the exceptional root system E 7 and satisfies the elliptic hypergeometric equation. We start this work with showing in §2 that the V -function satisfies a simple biorthogonality relation corresponding to the continuous values of a spectral parameter. For that we use the elliptic beta integral [16] and an integral analog of the Bailey chains introduced in [18]. Our construction can be viewed as an integral generalization of Rosengren's approach to the elliptic 6j-symbols [11].In §3, we investigate the generalized eigenvalue problem for a second order finite difference operator D (a, b, c, d; p; q). The relationship of such problems with biorthogonality relations was considered in [23]. A problem similar to ours was investigated earlier in [10,11] for the discrete spectrum. Its general solution determines a basis in some space of meromorphic functions. For a simple scalar product, we derive the dual basis defined by solutions of a similar spectral problem for conjugate operators. As a result, the overlap of the basis vectors with their duals coincides with the V -function. In §4, we derive a new contiguous relation for the elliptic hypergeometric function and the biorthogonality condition associated with that.It turns out that the V -function is directly related to the Sklyanin algebra, one of the central objects in the quantum inverse scattering method [14,15]. This relationship is established in §5 on the basis of the observation, due to Rains [9,12], that the D-operator can be represented as a linear combination of all four generators of the Sklyanin algebra.2000 Mathematics Subject Classification. Primary 33C75, 81R12.

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Section: Continuous Biorthogonal Functionsmentioning

confidence: 99%

Section: §1 Introductionmentioning

confidence: 97%

Section: §1 Introductionmentioning

confidence: 99%

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Continuous biorthogonality of an elliptic hypergeometric function

Spiridonov

2009

St. Petersburg Math. J.

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“…To define the Hecke operator T p on S k (X * 6 ) for a prime p relatively prime to 6, we pick an element α in O of norm p and consider the double coset (2), and…”

Section: 4mentioning

confidence: 99%

Ramanujan-type identities for Shimura curves

Yang

2016

Isr. J. Math.

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ABSTRACT. In 1914, Ramanujan gave a list of 17 identities expressing 1/π as linear combinations of values of hypergeometric functions at certain rational numbers. Since then, identities of similar nature have been discovered by many authors. Nowadays, one of the standard approaches to this kind of identities uses the theory of modular curves. In this paper, we will consider the case of Shimura curves and obtain Ramanujan-type formulas involving special values of hypergeometric functions and products of Gamma values. These products of Gamma values are related to periods of elliptic curves with complex multiplication by Q( √ −3) and Q( √ −4).

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“…See [5] for an introduction to basic notions and terminology or, for a more detailed account, the monograph [1].…”

Section: Using Classical Hypergeometric Machinerymentioning

confidence: 99%