### Abstract

The aim of this research note is to prove the following new transformation formula (1 - x)^{-2a} _{3}F_{2} [ _{c + 3/2, d} ^{a, a + 1/2, d + 1}; x^{2}/(1 - x)_{2} ] = _{4}F_{3} [ _{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 = √16d^{2} - 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 |

### Fingerprint

### Keywords

- Gauss summation theorem
- Kummer transformation
- Quadratic transformation formula

### ASJC Scopus subject areas

- Algebra and Number Theory
- Analysis
- Applied Mathematics
- Geometry and Topology

### Cite this

*Mathematical Communications*,

*14*(2), 207-209.

**On an extension of a quadratic transformation formula due to Kummer.** / Rakha, Medhat A.; Rathie, Navratna; Chopra, Purnima.

Research output: Contribution to journal › Article

*Mathematical Communications*, vol. 14, no. 2, pp. 207-209.

}

TY - JOUR

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

UR - http://www.scopus.com/inward/record.url?scp=74549156389&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=74549156389&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:74549156389

VL - 14

SP - 207

EP - 209

JO - Mathematical Communications

JF - Mathematical Communications

SN - 1331-0623

IS - 2

ER -