A Study on A New Class of q-Bernoulli, q-Euler and q-Genocchi Polynomials

This thesis is aimed to study a new class of Bernoulli, Euler and Genocchi polynomials in the means of quantum forms. To achieve this aim, we introduce a new class of q-exponential function and using properties of this function; we reach to the interesting formulae. The q-analogue of some familiar relations as an addition theorem for these polynomials is found. Explicit relations between these classes of polynomials are given. In addition, the new differential equations related to these polynomials are studied. Moreover, improved q-exponential function creates a new class of q-Bernoulli numbers and like the ordinary case, all the odd coefficient becomes zero and leads us to the relation of these numbers and q-trigonometric functions. At the end we introduce a unification form of q-exponential function. In this way all the properties of these kinds of polynomials investigated in a general case. We also focus on two important properties of q-exponential function that lead us to the symmetric form of q- Euler, q-Bernoulli and q-Genocchi numbers. These properties and the conditions of them are studied. Keywords: q-Exponential Function, q-Calculus, q-Polynomials, q-Bernoulli, q- Euler, q-Genocchi, q-Trigonometric Functions.