The subjects of special functions of single complex variable and several complex variables have a great situation in almost all disciplines Mathematics, Physics and Engineering. The present work is devoted to study these subjects and is considered one of several theses, which were presented and are still presenting by the school established by Sayyed in one of the branches of complex analysis. This thesis is divided into six main chapters. The main aim of Chapter 1 is to define a new family of special functions, say, $p$-Bessel function of one complex variable. The aim of Chapter 2 is to define and study of new matrix polynomial, say, the Hermite-Hermite matrix polynomials of one complex variable. Chapter 3 deals with the study of the Chebyshev matrix polynomials to be studied of one complex variable. Chapter 4, we introduce a generalization of the Hermite matrix polynomials of two complex variables. Chapter 5,6, we introduced and studied a matrix function, say, the Kummer matrix function and Horn matrix function of two complex variables. The results of this chapters are variant and significant and so it are interesting and capable to develop its study in the future.

Autorentext

A. Shehata has got his B.Sc.(Applied Mathematics-Elasticity)2006.He got his greduated Ph.D.degree in(Pure Mathematics-Mathematical Analysis-Complex Analysis+Special Functions)Department of Mathematics-Faculty of Sciences-Assiut Unviersity,Egypt.2009. The filed of interests are Mathematical, Complex Analysis, Special Functions and their applications