On an extension of a quadratic transformation formula due to Kummer

Medhat A. Rakha; Navratna Rathie; Purnima Chopra

On an extension of a quadratic transformation formula due to Kummer

Medhat A. Rakha, Navratna Rathie, Purnima Chopra

Research output: Contribution to journal › Article › peer-review

Abstract

The aim of this research note is to prove the following new transformation formula (1 - x)^-2a ₃F₂ [ _{c + 3/2, d} ^{a, a + 1/2, d + 1}; x²/(1 - x)₂ ] = ₄F₃ [ _{2c + 2, 2d + 1/2A - 1/2, 2d - 1/2A - 1/2} ^{2a, c, 2d + 1/2A +1/2, 2d - 1/2A + 1/2}; 2x ] valid for |x| < 1/2 and if |x| = 1/2, then Re(c - 2a) > 0, where A = √16d² - 16cd - 8d + 1. For d = c + 1/2, we get quadratic transformations due to Kummer. The result is derived with the help of the generalized Gauss's summation theorem available in the literature.

Original language	English
Pages (from-to)	207-209
Number of pages	3
Journal	Mathematical Communications
Volume	14
Issue number	2
Publication status	Published - Dec 2009
Externally published	Yes

Keywords

Gauss summation theorem
Kummer transformation
Quadratic transformation formula

ASJC Scopus subject areas

Analysis
Algebra and Number Theory
Geometry and Topology
Applied Mathematics

Cite this

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title = "On an extension of a quadratic transformation formula due to Kummer",

abstract = "The aim of this research note is to prove the following new transformation formula (1 - x)-2a 3F2 [ c + 3/2, d a, a + 1/2, d + 1; x2/(1 - x)2 ] = 4F3 [ 2c + 2, 2d + 1/2A - 1/2, 2d - 1/2A - 1/2 2a, c, 2d + 1/2A +1/2, 2d - 1/2A + 1/2; 2x ] valid for |x| < 1/2 and if |x| = 1/2, then Re(c - 2a) > 0, where A = √16d2 - 16cd - 8d + 1. For d = c + 1/2, we get quadratic transformations due to Kummer. The result is derived with the help of the generalized Gauss's summation theorem available in the literature.",

keywords = "Gauss summation theorem, Kummer transformation, Quadratic transformation formula",

author = "Rakha, {Medhat A.} and Navratna Rathie and Purnima Chopra",

year = "2009",

month = dec,

language = "English",

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pages = "207--209",

journal = "Mathematical Communications",

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publisher = "Udruga Matematicara Osijek",

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T1 - On an extension of a quadratic transformation formula due to Kummer

AU - Rakha, Medhat A.

AU - Rathie, Navratna

AU - Chopra, Purnima

PY - 2009/12

Y1 - 2009/12

N2 - The aim of this research note is to prove the following new transformation formula (1 - x)-2a 3F2 [ c + 3/2, d a, a + 1/2, d + 1; x2/(1 - x)2 ] = 4F3 [ 2c + 2, 2d + 1/2A - 1/2, 2d - 1/2A - 1/2 2a, c, 2d + 1/2A +1/2, 2d - 1/2A + 1/2; 2x ] valid for |x| < 1/2 and if |x| = 1/2, then Re(c - 2a) > 0, where A = √16d2 - 16cd - 8d + 1. For d = c + 1/2, we get quadratic transformations due to Kummer. The result is derived with the help of the generalized Gauss's summation theorem available in the literature.

AB - The aim of this research note is to prove the following new transformation formula (1 - x)-2a 3F2 [ c + 3/2, d a, a + 1/2, d + 1; x2/(1 - x)2 ] = 4F3 [ 2c + 2, 2d + 1/2A - 1/2, 2d - 1/2A - 1/2 2a, c, 2d + 1/2A +1/2, 2d - 1/2A + 1/2; 2x ] valid for |x| < 1/2 and if |x| = 1/2, then Re(c - 2a) > 0, where A = √16d2 - 16cd - 8d + 1. For d = c + 1/2, we get quadratic transformations due to Kummer. The result is derived with the help of the generalized Gauss's summation theorem available in the literature.

KW - Gauss summation theorem

KW - Kummer transformation

KW - Quadratic transformation formula

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On an extension of a quadratic transformation formula due to Kummer

Abstract

Keywords

ASJC Scopus subject areas

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