Genelleştirilmiş hipergeometrik fonksyionlar cinsinden cebirsel fonksiyonlar

The aim of this thesis is to describe the solutions of a given general trinomialalgebraic equation satisfying some special conditions by means of generalized hypergeometricfunctions.Including the introduction, this dissertation consists of five chapters. Inthe first chapter, the gamma function and the beta function are introduced. Some basicproperties of the gamma function and related theorems are reviewed.In the second chapter, Gauss hypergeometric function(as a model case) is studied in detail.Besides, its differential equation, two different integral representations and monodromystudy of Gauss hypergeometric differential equation are given.In the third chapter, generalized hypergeometric(or Pochhammer) function, which is usedas our tool to obtain our results in the fourth chapter and its differential equation areintroduced. Moreover, monodromy study of generalized hypergeometric function is brieflystated.In the fourth chapter, the Newton-Puiseux algorithm which is used to find solutions ofan algebraic equation in fractional power series around the origin is explained. However,even though this algorithm enables us to obtain all solutions of an algebraic equation,when it comes to deal with algebraic equations of higher degree, application of the Newton'salgorithm becomes a hard task due to higher order terms. In the second section of chapterfour, the Mellin transformation of a solution of a general trinomial algebraic equation isdescribed. Further, the relation between such a transformation and gamma function isstudied. By the aid of this relation solutions of a general trinomial algebraic equationthat arise from the Mellin inversion formula are reviewed. In the last section of chapterfour, we applied this argument and the Newton-Puiseux algorithm to a given generaltriniomial algebraic equation satisfying some special conditions and compared the results.Moreover, we obtained hypergeometric type differential equation which is satisfied by thesolutions of a given trinomial algebraic equation mentioned above. These applications are supplemented with well-studied examples. In the final chapter, the conclusion of this dissertation is given.