Generalizations of Dixon's Theorem on the Sum of A 3 F 2

Lavoie, J. L.; Grondin, Frédéric; Rathie, Arjun K.; Arora, Kasturi L

doi:10.2307/2153407

Cited by 65 publications

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“…The aim of this paper is to obtain three interesting results of reducibility of Kampé de Fériet function by following the same technique developed by Bailey. The results are obtained with the help of contiguous Gauss's second summation formula obtained earlier by Lavoie et al [6]. The results obtained by Bailey [2] and Rathie and Nagar [9] follow special cases of our main findings.…”

Section: Introductionsupporting

confidence: 73%

“…The following known theorems due to Watson and Lavoie et al [6] will be required in our present investigations. Gauss's second summation theorem [3]:…”

Section: Results Requiredmentioning

confidence: 99%

“…by utilizing well-known Gauss's second summation formulas obtained earlier by Lavoie et al [6]. On the other hand, just as the Gauss function 2 F 1 was extended to p F q by increasing the number of the numerator and denominator parameters, the four Appell functions were unified and generalized by Apell and Kampé de Fériet [1] who defined a general hypergeometric function of two variables.…”

Section: Introductionmentioning

confidence: 99%

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On Certain Reducibility of Kampe De Feriet Function

Kim¹

2009

Honam Mathematical Journal

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Section: Introductionsupporting

confidence: 73%

“…The following known theorems due to Watson and Lavoie et al [6] will be required in our present investigations. Gauss's second summation theorem [3]:…”

Section: Results Requiredmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On Certain Reducibility of Kampe De Feriet Function

Kim¹

2009

Honam Mathematical Journal

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“…By straightforward computations and by using summation formulae for hypergeometric series (i.e. Dixon's identities and contiguous identities [35]). …”

Section: Prop 53 With the Values (64) Yields Immediately Thatmentioning

confidence: 99%

Correlation functions of the KdV hierarchy and applications to intersection numbers over M¯g,n

Bertola

Dubrovin

Yang

2016

Physica D: Nonlinear Phenomena

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We derive an explicit generating function of correlations functions of an arbitrary tau-function of the KdV hierarchy. In particular applications, our formulation gives closed formulae of a new type for the generating series of intersection numbers of ψ-classes as well as of mixed ψ-and κ-classes in full genera.Keywords. KdV hierarchy; tau-function; wave function; correlation function; intersection number. Contents Introduction and resultsThe famed Korteweg-de Vries (KdV) equationhas long been known to be integrable. It belongs to an infinite family of pairwise commuting nonlinear evolutionary PDEs called the KdV hierarchy. The hierarchy can be described in terms of isospectral deformations of the Lax operator( 1.2) Namely, the k-th equation of the KdV hierarchy, in the normalization of the present paper, reads( 1.4) Here, the independent variables t 0 , t 1 , t 2 , ... are called times. The symbol L 2k+1 2 + stands for the differential part of the pseudo-differential operator L 2k+1 2 , see e.g. the book [16] for details. The k = 1 equation of (1.3) coincides with (1.1). As customary in the literature, we shall identify t 0 with the spatial variable x.The notion of tau-function for the KdV hierarchy was introduced by the Kyoto school [50,27,15] during the 1970s-1980s. In 1991, E. Witten, in his study of two-dimensional quantum gravity [52], conjectured that the generating function of the intersection numbers of ψ-classes on the Deligne-Mumford moduli spaces M g,n of stable algebraic curves is a tau-function of the KdV hierarchy. Witten's conjecture was later proved by M. Kontsevich [34]; see [33,49,44] for several alternative proofs. Moreover the so-called "tau structures" of KdV-like hierarchies became one of the central subjects in the study of the deep relation between integrable hierarchies and Gromov-Witten invariants [22,20,21].

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“…It should also be remarked here that whenever hypergeometric and generalized hypergeometric functions reduce to express in terms of Gamma functions, the results are very important from the applicative point of view. Therefore, the classical summation theorems such as those of Gauss, Gauss's second, Bailey and Kummer for the series 2 F 1 and Dixon, Watson, Whipple and Saalschütz for the series 3 F 2 and their rather recent extensions and generalizations (see [7], [8], [9], [10] and [11]) play an important role in the theory of hypergeometric and generalized hypergeometric series. For applications of the above-mentioned classical summation theorems, we refer to [2], [5], [6], [10], [11], [12] and [13].…”

Section: Introductionmentioning

confidence: 99%

APPELL'S FUNCTION F₁AND EXTON'S TRIPLE HYPERGEOMETRIC FUNCTION X₉

Choi¹,

Rathie²

2013

The Pure and Applied Mathematics

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Abstract. In the theory of hypergeometric functions of one or several variables, a remarkable amount of mathematicians's concern has been given to develop their transformation formulas and summation identities. Here we aim at presenting explicit expressions (in a single form) of the following weighted Appell's function F1:in terms of Exton's triple hypergeometric X9. The results are derived with the help of generalizations of Kummer's second theorem very recently provided by Kim et al. A large number of very interesting special cases including Exton's result are also given.

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Generalizations of Dixon's Theorem on the Sum of A 3 F 2

Cited by 65 publications

References 2 publications

On Certain Reducibility of Kampe De Feriet Function

On Certain Reducibility of Kampe De Feriet Function

Correlation functions of the KdV hierarchy and applications to intersection numbers over M¯g,n

APPELL'S FUNCTION F₁AND EXTON'S TRIPLE HYPERGEOMETRIC FUNCTION X₉

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Generalizations of Dixon's Theorem on the Sum of A 3 F 2

Cited by 65 publications

References 2 publications

On Certain Reducibility of Kampe De Feriet Function

On Certain Reducibility of Kampe De Feriet Function

Correlation functions of the KdV hierarchy and applications to intersection numbers over M¯g,n

APPELL'S FUNCTION F1AND EXTON'S TRIPLE HYPERGEOMETRIC FUNCTION X9

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APPELL'S FUNCTION F₁AND EXTON'S TRIPLE HYPERGEOMETRIC FUNCTION X₉