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Generalized hypergeometric functions : transformations and group theoretical aspects

Author: K Srinivasa Rao; Vasudevan Lakshminarayanan; Institute of Physics (Great Britain),
Publisher: Bristol [England] (Temple Circus, Temple Way, Bristol BS1 6HG, UK) : IOP Publishing, [2018]
Series: IOP expanding physics.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
In 1813, Gauss first outlined his studies of the hypergeometric series which has been of great significance in the mathematical modelling of physical phenomena. This detailed monograph outlines the fundamental relationships between the hypergeometric function and special functions. In nine comprehensive chapters, Dr. Rao and Dr. Lakshminarayanan present a unified approach to the study of special functions of  Read more...
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Details

Genre/Form: Electronic books
Additional Physical Format: Print version:
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: K Srinivasa Rao; Vasudevan Lakshminarayanan; Institute of Physics (Great Britain),
ISBN: 9780750314961 0750314966 9780750314954 0750314958
OCLC Number: 1065533657
Notes: "Version: 20181001"--Title page verso.
Target Audience: Students (undergrad, grad, postgrad, and doctoral) and researchers needing special functions in their areas of study as well as practicing engineers needing a reference.
Description: 1 online resource.
Contents: 1. Hypergeometric series --
1.1. Introduction --
1.2. The Gauss differential equation --
1.3. Special functions --
1.4. Properties of the hypergeometric functions --
1.5. A conjecture 2. Group theory: basics --
2.1. Introduction --
2.2. Invariances, symmetries, and physics --
2.3. Discrete groups --
2.4. The symmetric group Sn --
2.5. An interesting property of Sn --
2.6. Representations of a group --
2.7. Lie groups and Lie algebras --
2.8. Angular momentum and the rotation group --
2.9. Compact Lie groups --
2.10. Subgroups, cosets, invariant subgroups --
2.11. Simple and semi-simple groups --
2.12. Compact groups 3. Group theory of the Kummer solutions of the Gauss differential equation --
3.1. Introduction --
3.2. The 24 solutions of the Gauss ODE --
3.3. The Riemann equation 4. Group theory of terminating and non-terminating 3F2(a, b, c; d, e; 1) transformations --
4.1. Introduction --
4.2. The Whipple notation --
4.3. Terminating 3F2(1) series --
4.4. Structure of GT and its irreps --
4.5. Scaling the WE transformation --
4.6. Non-terminating 3F2(a, b, c; d, e; 1) series --
4.7. Concluding remarks 5. Angular momentum and the rotation group --
5.1. Introduction: historical --
5.2. Angular momentum algebra --
5.3. Representations of angular momentum operators --
5.4. The rotation group --
5.5. The 3F2(1) sets --
5.6. Symmetries of the 3-j coefficient --
5.7. Inter-relationship between the sets of 3F2(1)s 6. Angular momentum recoupling and sets of 4F3(1)s --
6.1. Introduction: historical --
6.2. Addition of three angular momenta --
6.3. Symmetries of the Racah coefficient --
6.4. Two sets of 4F3(1)s --
6.5. Inter-relationship of the two sets of 4F3(1)s --
6.6. Bargmann-Shelepin arrays --
6.7. The set of Bailey transformations --
6.8. Basis states and symmetries of the 6-j coefficient 7. Double and triple hypergeometric series --
7.1. Introduction: a history --
7.2. Multiple hypergeometric series --
7.3. Definitions of the 9-j coefficient --
7.4. Symmetries of the 9-j coefficient --
7.5. The Jucys-Bandzaitis triple sum series --
7.6. Stretched 9-j coefficients --
7.7. A general transformation formula --
7.8. Transformation formulas for F⁰:3;4 ₁:1;2 --
7.9. Transformation formulas for F¹:2;2 ₀:2;2 --
7.10. Some summation formulas 8. Beta integral method and hypergeometric transformations --
8.1. Introduction --
8.2. Extensions of Euler's integral for 2F1(a, b; c; z) --
8.3. The beta integral method --
8.4. The algorithm --
8.5. New single sum hypergeometric identities from old ones 9. Gauss, hypergeometric series, and Ramanujan --
9.1. Introduction --
9.2. On some entries of Ramanujan on hypergeomtric series in his notebooks --
9.3. Entry 43, in chapter XII of Ramanujan's Notebook 1 --
9.4. The theorem of Rao, Berghe, and Krattenthaler --
9.5. Observations and concluding remarks --
9.6. Epilogue.
Series Title: IOP expanding physics.
Responsibility: K. Srinivasa Rao, Vasudevan Lakshminarayanan.

Abstract:

In 1813, Gauss first outlined his studies of the hypergeometric series which has been of great significance in the mathematical modelling of physical phenomena. This detailed monograph outlines the fundamental relationships between the hypergeometric function and special functions. In nine comprehensive chapters, Dr. Rao and Dr. Lakshminarayanan present a unified approach to the study of special functions of mathematics using Group theory. The book offers fresh insight into various aspects of special functions and their relationship, utilizing transformations and group theory and their applications. It will lay the foundation for deeper understanding by both experienced researchers and novice students.

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