A Study of Some Special Functions and Polynomials

The subjects of special functions and orthogonal matrix polynomials of the single complex variable have a great situation in almost all disciplines Mathematics, Physics and Engineering. The present work is devoted to study these subjects and is considered to be one of several theses, which presented and still presented by the school established by Sayyed in one of the branches of complex analysis. This work investigates some subjects in complex analysis such as generalizations of Hermite matrix polynomials, the Ultraspherical matrix polynomials. Some results, which are important in this field, have been obtained in this work. In this work, Some of a new concepts of some special functions and matrix polynomials are introduced and studied in: (i) Give a review on some properties of Hermite matrix polynomials. (ii) Introduce and study of Ultraspherical matrix polynomials and their properties. We conclude our study by listing some of unsolved problems which as listed as open problems in future work:Introduce and study of Hermite-Bessel, Hermite-Laguerre, Laguerre-Laguerre, Legendre-Hermite and Hermite-Legendre matrix functions