Asymptotic expansion and taylor coefficients of Hurwitz Zeta and Lerch Zeta functions

The aim of this thesis is to study polyexponantial functions with the help of geometric and exponantial polynomials and to investigate expansions of ApostolBernoulli functions and Frobenius Eulerian polynomials in a di§erent way. In the Örst section of the thesis we introduce Bernoulli polynomials, Riemann zeta, Hurwitz zeta, polyexponantial functions, Dirichlet series H(s) and H(s,z). In the second section, main theorems and some relations which holds by Apostol-Bernoulli, Riemann zeta, Hurwitz zeta, polyexponantial functions and Dirichlet series H(s,z) are given. In the third section, some properties of the (xD) operator are given. Afterwards series with Hurwitz and Lerch zeta function coe¢ cients are studied. In the Önal section, series with polyexponantial coe¢ cients are studied and Frobenius Eulerian polynomials and numbers are obtained in terms of geometric polynomials. KEY WORDS: Frobenius Eulerian Polynomials, Geometric Polynomials, Exponantial Polynomials, Riemann zeta Function, Hurwitz zeta Function, Lerch zeta Function, Polyexponantials, Dirichlet Series.