This book has been written to present a simple and up-to-date introduction to the hypergeometric functions and its series representation. It is designed to convey in an effective manner the fundamental properties of such functions. In order to make the book accessible to mathematicians, physicists, and engineers, the author has the work self-contained as possible. Hence, this book starts with an introductory chapter showing all the needed tools for the derivation that are carried out in the later chapters. In the second chapter we derive all the Kummer''s solutions and the logarithmic solutions of the hypergeometric differential equation. Showing the representations of simple functions, like the trigonometric and exponential functions in terms of the hypergeometric series. Due to the importance of the special functions in mathematical analysis, physics, engineering applications. In this book, we present the Legendre functions of the first and the second kind in terms of the hypergeometric functions. This representation has the advantage of obtaining the analytic continuation in the complex-plane of the special functions leading to variety of applications.