On matrix expressions of some families of Appell polynomials

In this thesis, we study on matrix extension of some classical special functions. We give addition and multiplication formulas for Hermite matrix polynomials. We obtain other generating functions for Hermite matrix polynomials and write Hermite matrix polynomials as a hypergeometric matrix functions. We introduce generalized Hermite matrix polynomials. We obtain Burchnall operational formula and Nielsen identity for Hermite matrix polynomials by choosing appropriate parameters from the properties of generalized Hermite matrix polynomials. We define modified Laguerre matrix polynomials by modifiying the generating function of Laguerre matrix polynomials. We obtain three term matrix recurrence relation, Rodrigues formula, second-order matrix differential equation and several families of bilinear and bilateral generating matrix functions for modified Laguerre matrix polynomials. Moreover a new generalization of the Laguerre-type matrix polynomials is introduced. We generalize the second kind Chebyshev matrix polynomials and focus on their properties. Using their integral representation we investigate operational rules associated with operators corresponding to these matrix polynomials. Furthermore we obtain a matrix differential equation of second kind Chebyshev matrix polynomials. We obtain some properties of generalized Humbert matrix polynomials such as matrix recurrence relations, matrix differential equation and an integral representation. Moreover, we obtain a series transformation formula involving Gegenbauer matrix polynomials. Then we provide a number of applications. We get a functional equation of the gamma matrix function. Moreover we give the infinite product expansion of sin matrix function. We define Riemann zeta matrix function and evaluate some matrix integrals. Finally we prove a functional equation of Riemann zeta matrix function.