q-Polynomials and Location of Their Zeros

ABSTRACT: In this thesis, we define the q-Bernoulli numbers and polynomials, q-Euler numbers and polynomials, q-Frobenius-Euler numbers and polynomials and q-Genocchi numbers and polynomials of higher order in two variables x and y, by using two q-exponential functions. We also prove some properties and relationships of these polynomials and q-analogue of the Srivastava and Pinter addition theorem. Furthermore, we represent the figures of the q-Bernoulli, q-Euler and q-Genocchi numbers and polynomials. We find the solutions of these q-polynomials, for n 2 N, x and q 2 C by using a computer package MathematicaR ⃝ software. Finally, we discuss the reflection symmetries of these q-polynomials. Keywords: q-analogues of Bernoulli - Euler - Genocchi - Frobenius-Euler numbers and Polynomials, Srivastava Pinter addition Theorems, shapes and roots of q-polynomials …………………………………………………………………………………………………………………………