ABSTRACT: This thesis consists of five chapters. The first Chapter gives general information about the thesis. In the second Chapter, some preliminaries and auxilary results that are used throughout the thesis are given. The original parts of the thesis are Chapters 3, 4 and 5 which are established from [35], [46] and [48]. In Chapter three, extended 2D Bernoulli and 2D Euler polynomials are introduced. Moreover, some recurrence relations are given. Differential, integrodifferential and partial differential equations of the extended 2D Bernoulli and the extended 2D Euler polynomials are obtained by using the factorization method. The special cases reduces to differential equation of the usual Bernoulli and Euler polynomials. Note that the results for the usual 2D Euler polynomials are new. In Chapter four, we consider Hermite-based Appell polynomials and give partial differential equations of them. In the special cases, we present the recurrence relation, differential, integro-differential and partial differential equations of the Hermite-based Bernoulli and Hermite-based Euler polynomials. In Chapter five, introducing k-times shift operators, factorization method is generalized. The differential equations of the Appell polynomials are obtained. For the special case k = 2, differential equation of Bernoulli and Hermite polynomials are exhibited. Keywords: 2D Bernoulli polynomial, 2D Euler polynomial, extended 2D Bernoulli polynomial, extended 2D Euler polynomial, Hermite-based Appell polynomials, factorization method. …………………………………………………………………………………………………………………………………………………………………………………………………………
