Contiguous relations and their computations for 2F1 hypergeometric series

Adel K. Ibrahim; Medhat A. Rakha

doi:10.1016/j.camwa.2008.04.018

Contiguous relations and their computations for ₂F₁ hypergeometric series

Adel K. Ibrahim, Medhat A. Rakha^*

^*Corresponding author for this work

Mathematics & Statistics

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

12 Citations (Scopus)

Abstract

The hypergeometric function ₂F₁ [a₁, a₂ ; a₃ ; z] plays an important role in mathematical analysis and its application. Gauss defined two hypergeometric functions to be contiguous if they have the same power-series variable, if two of the parameters are pairwise equal, and if the third pair differs by ±1. He showed that a hypergeometric function and any two other contiguous to it are linearly related. In this paper, we present an interesting formula as a linear relation of three shifted Gauss polynomials in the three parameters a₁, a₂ and a₃. More precisely, we obtained a recurrence relation including ₂F₁ [a₁ + α₁, a₂ ; a₃ ; z], ₂F₁ [a₁, a₂ + α₂ ; a₃ ; z] and ₂F₁ [a₁, a₂ ; a₃ + α₃ ; z] for any arbitrary integers α₁, α₂ and α₃.

Original language	English
Pages (from-to)	1918-1926
Number of pages	9
Journal	Computers and Mathematics with Applications
Volume	56
Issue number	8
DOIs	https://doi.org/10.1016/j.camwa.2008.04.018
Publication status	Published - Oct 2008

Keywords

Computer algebra
Contiguous function relation
F hypergeometric function
Gauss hypergeometric function
Linear recurrence relation

ASJC Scopus subject areas

Modelling and Simulation
Computational Theory and Mathematics
Computational Mathematics

Access to Document

10.1016/j.camwa.2008.04.018

Cite this

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title = "Contiguous relations and their computations for 2F1 hypergeometric series",

abstract = "The hypergeometric function 2F1 [a1, a2 ; a3 ; z] plays an important role in mathematical analysis and its application. Gauss defined two hypergeometric functions to be contiguous if they have the same power-series variable, if two of the parameters are pairwise equal, and if the third pair differs by ±1. He showed that a hypergeometric function and any two other contiguous to it are linearly related. In this paper, we present an interesting formula as a linear relation of three shifted Gauss polynomials in the three parameters a1, a2 and a3. More precisely, we obtained a recurrence relation including 2F1 [a1 + α1, a2 ; a3 ; z], 2F1 [a1, a2 + α2 ; a3 ; z] and 2F1 [a1, a2 ; a3 + α3 ; z] for any arbitrary integers α1, α2 and α3.",

keywords = "Computer algebra, Contiguous function relation, F hypergeometric function, Gauss hypergeometric function, Linear recurrence relation",

author = "Ibrahim, {Adel K.} and Rakha, {Medhat A.}",

note = "Funding Information: The authors wish to thank the referees for their helpful comments. The second author{\textquoteright}s work is funded by a grant from Sultan Qaboos University (IG/SCI/DOMS/06/01). ",

year = "2008",

month = oct,

doi = "10.1016/j.camwa.2008.04.018",

language = "English",

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pages = "1918--1926",

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AU - Ibrahim, Adel K.

AU - Rakha, Medhat A.

N1 - Funding Information: The authors wish to thank the referees for their helpful comments. The second author’s work is funded by a grant from Sultan Qaboos University (IG/SCI/DOMS/06/01).

PY - 2008/10

Y1 - 2008/10

N2 - The hypergeometric function 2F1 [a1, a2 ; a3 ; z] plays an important role in mathematical analysis and its application. Gauss defined two hypergeometric functions to be contiguous if they have the same power-series variable, if two of the parameters are pairwise equal, and if the third pair differs by ±1. He showed that a hypergeometric function and any two other contiguous to it are linearly related. In this paper, we present an interesting formula as a linear relation of three shifted Gauss polynomials in the three parameters a1, a2 and a3. More precisely, we obtained a recurrence relation including 2F1 [a1 + α1, a2 ; a3 ; z], 2F1 [a1, a2 + α2 ; a3 ; z] and 2F1 [a1, a2 ; a3 + α3 ; z] for any arbitrary integers α1, α2 and α3.

AB - The hypergeometric function 2F1 [a1, a2 ; a3 ; z] plays an important role in mathematical analysis and its application. Gauss defined two hypergeometric functions to be contiguous if they have the same power-series variable, if two of the parameters are pairwise equal, and if the third pair differs by ±1. He showed that a hypergeometric function and any two other contiguous to it are linearly related. In this paper, we present an interesting formula as a linear relation of three shifted Gauss polynomials in the three parameters a1, a2 and a3. More precisely, we obtained a recurrence relation including 2F1 [a1 + α1, a2 ; a3 ; z], 2F1 [a1, a2 + α2 ; a3 ; z] and 2F1 [a1, a2 ; a3 + α3 ; z] for any arbitrary integers α1, α2 and α3.

KW - Computer algebra

KW - Contiguous function relation

KW - F hypergeometric function

KW - Gauss hypergeometric function

KW - Linear recurrence relation

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