Some Results on Laguerre Type and Mittag-Leffler Type Functions

This thesis includes four chapters. In the first chapter, we give general information and some preliminaries that is used throughout the thesis. In Chapter 2, by defining a new class of 2D-Mittag-Leffler functions E(a;b;h;x ;l) g;k (x;y) and 2D-Laguerre polynomials L(a;b;g;h;x ) n;m (x;y), the two-dimensional fractional integral and two-dimensional fractional derivative properties are derived for them. Moreover, linear generating function for L(a;b;g;h;x ) n;m (x;y) in terms of E(a;b;h;x ;l) g;k (x;y) is obtained. Also, the double Laplace transform of these classes are investigated. A general singular integral equation containing L(a;b;g;h;x ) n;m (x;y) in the kernel is considered and the solution is obtained in terms of E(a;b;h;x ;l) g;k (x;y). Lastly, we obtain the image of E(a;b;h;x ;l) g;k (x;y) under the action of Marichev-Saigo-Maeda integral operators and some consequences are also exhibited. In Chapter 3, linear and mixed multilateral generating functions for the general class of 2D-Laguerre polynomials L(a;b;g;h;x ) n;m (x;y) are derived. Furthermore, a finite summation formula for L(a;b;g;h;x ) n;m (x;y) is obtained. Moreover, series relation between L(a;b;g;h;x ) n;m (x;y) and product of confluent hypergeometric functions is derived with the help of two-dimensional fractional derivative operator. In Chapter 4, new classes of bivariate Mittag-Leffler functions E(g) a;b;k (x;y) and 2DKonhauser- Laguerre polynomials kL(a;b) n (x;y) are introduced. Some of them associated with fractional calculus are given. Also, a convolution type integral equation with the polynomials kL(a;b) n (x;y) in the kernel is considered and the solution is obtained by means of E(g) a;b;k (x;y). Furthermore, a double linear generating function is obtained for the polynomials kL(a;b) n (x;y) in terms of E(g) a;b;k (x;y). Finally, some miscellaneous properties of E(g) a;b;k (x;y) and kL(a;b) n (x;y) are exhibited. Keywords: Mittag-Leffler functions, Laguerre and Konhauser polynomials, Laplace transform, fractional integrals and derivatives, generating functions, convolution integral equation, singular integral equation