ABSTRACT
This work consists definitions and basic properties of Hypergeometric series and Hypergeometric functions. Hypergeometric functions have important applications and researches in mathematics and mathematical physics. Solutions of Hypergeometric differential equation are obtained as Hypergeometric functions. Several recurrence relations and serial properties of Hypergeometric functions are obtained by different methods.
Key words: Hypergeometric series, Hypergeometric functions, differential equation, serial solutions, series manupilation, Gamma function, Pochammer symbol.
i
ÖZET
Bu çalışma Hipergeometrik seri ve Hipergeometrik fonksiyonların temel tanım ve özelliklerinden oluşmaktadır. Hipergeometrik fonksiyonlar, matematik ve matematiksel fizikte önemli uygulamalara sahiptir. Hipergeometrik diferensiyel denklemin çözümleri Hipergeometrik fonksiyonlar cinsinden elde edildi. Çeşitli indirgeme bağıntıları ve Hipergeometrik fonksiyonları serisel özellikleri de farklı metodlarla elde edildi.
Anahtar kelimeler: Hipergeometrik seriler, Hipergeometrik fonksiyonlar, diferensiyel denklemler, serisel çözümler, seri işlemleri, Gamma fonksiyonu, Pochammer sembolu.
ii
ACKNOWLEDGEMENTS
First I would like to thank my supervisor Assist. Prof. Dr. Burak Şekeroğlu who has shown plenty of encouragement, patience, and support as he guided me through this endeavor fostering my development as a graduate student.
This research was generously supported by the Department of Mathematics of the Near East University. I am grateful to all supporters.
iii
CONTENTS
ABSTRACT...i
ÖZ...ii
ACKNOWLEDGEMENTS...iii
CONTENTS...iv
CHAPTER 1, BASIC DEFINITIONS...1
1.1 Introduction...1
1.2 Series and Power Series...2
1.3 Gamma Function...7
1.4 Differential Equations...11
CHAPTER 2, HYPERGEOMETRIC SERIES...13
2.1 Definition of Hypergeometric Series...13
2.2 Sum of Hypergeometric Series...14
CHAPTER 3, THE HYPERGEOMETRIC DIFFERENTIAL EQUATION...16
CHAPTER 4, ADDITIONAL PROPERTIES OF HYPERGEOMETRIC SERIES...23
4.1 A Simple Transformation for Hypergeometric Function...23
4.2 A Quadratic Transformation for Hypergeometric Function...26
4.3 Other Quadratic Transformation for Hypergeometric Function...31
CHAPTER 5, GENERALIZED HYPERGEOMETRIC FUNCTION...34
5.1 Definition of Generalized Hypergeometric Functions...34
5.2 Convergency of Generalized Hypergeometric Functions...35
5.3 Some Special Generalized Hypergeometric Functions...36
5.4 Differential Equations of Generalized Hypergeometric Functions...37
CONCLUSION...38
REFERENCES...39
iv
