Elliott's identity and hypergeometric functions

Balasubramanian, R.; Naik, S.; Ponnusamy, Saminathan; Вуоринен, Матти

doi:10.1016/s0022-247x(02)00126-9

Cited by 11 publications

(4 citation statements)

References 4 publications

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“…Identity (2.3) is related to several well-known formulas for hypergeometric functions such as Legendre's identity, Elliott's identity and Anderson-Vamanamurthy-Vuorinen's identity. See [4] for details.…”

Section: Resultsmentioning

confidence: 99%

Convergence analysis of a Padé family of iterations for the matrix sector function

Karp¹,

Lin²

2011

Preprint

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The main purpose of this paper is to give a solution to a conjecture concerning a Padé family of iterations for the matrix sector function that was recently raised by B. Laszkiewicz et al in [A Padé family of iterations for the matrix sector function and the matrix pth root, Numer. Linear Algebra Appl. 2009; 16:951-970]. Using a sharpened version Schwarz's lemma, we also demonstrate a strengthening of the conjecture.

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Section: Resultsmentioning

confidence: 99%

Convergence analysis of a Padé family of iterations for the matrix sector function

Karp¹,

Lin²

2011

Preprint

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“…In [26] Balasubramanian, Naik, Ponnusamy, and Vuorinen obtained a differentiation formula for an expression involving hypergeometric series that implies Elliott's identity. This paper contains a number of other significant results, including a proof that Elliott's identity is equivalent to a formula of Ramanujan [55,p.…”

Section: Legendre's Relation and Generalizationsmentioning

confidence: 99%

Topics in Special Functions III

Anderson

Вуоринен

Zhang

2014

Analytic Number Theory, Approximation Theory, and Special Functions

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The study of quasiconformal maps led the first two authors in their joint work with M. K. Vamanamurthy to formulate open problems or questions involving special functions [14,16]. During the past two decades, many authors have contributed to the solution of these problems. However, most of the problems posed in [14] are still open.The present paper is the third in a series of surveys by the first two authors, the previous papers [19,21] being written jointly with the late M. K. Vamanamurthy. The aim of this series of surveys is to review the results motivated by the problems in [14,16] and related developments during the past two decades. In the first of these we studied classical special functions, and in the next we focused on special functions occurring in the distortion theory of quasiconformal maps. Regretfully, Vamanamurthy passed away in 2009, and the remaining authors dedicate the present work as a tribute to his memory. For an update to the bibliographies of [19] and [21] the reader is referred to [23].In 1993 the following monotone rule was derived [17, Lemma 2.2]. Though simple to state and easy to prove by means of the Cauchy Mean Value Theorem, this l'Hôpital Monotone Rule (LMR) has had wide application to special functions by many authors. Vamanamurthy was especially skillful in the application of this rule. We here quote the rule as it was restated in [20, Theorem 2].

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“…Legendre's formula was generalized by Elliott in 1904 to a linear combination of three products of the Gauss hypergeometric functions containing three independent parameters [3,Theorem 3.2.8]. Another two-parameter generalization of Legendre's identity was found in [2] which was soon thereafter generalized in [5] to an identity with four independent parameters covering both Elliott's formula and the formula from [2]. Another identity for a linear combination of three products of the Gauss functions with coefficients depending quadratically on the argument and containing three independent parameters was given in [11,Theorem 3.2].…”

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confidence: 99%

A new identity for the sum of products of the generalized hypergeometric functions

Karp

Kuznetsov

2021

Proc. Amer. Math. Soc.

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Reduction formulas for sums of products of hypergeometric functions can be traced back to Euler. This topic has an intimate connection to summation and transformation formulas, contiguous relations and algebraic properties of the (generalized) hypergeometric differential equation. Over recent several years, important discoveries have been made in this subject by Gorelov, Ebisu, Beukers and Jouhet and Feng, Kuznetsov and Yang. In this paper, we give a generalization of Feng, Kuznetsov and Yang identity covering also Ebisu's and Gorelov's formulas as particular cases.

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Elliott's identity and hypergeometric functions

Cited by 11 publications

References 4 publications

Convergence analysis of a Padé family of iterations for the matrix sector function

Convergence analysis of a Padé family of iterations for the matrix sector function

Topics in Special Functions III

A new identity for the sum of products of the generalized hypergeometric functions

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