Applications of some special functions on q-umbral analysis, Fourier series and Nevanlinna theory

In this thesis, q-Apostol polynomials that are a new family of polynomials are defined, and these polynomials, by q-umbral analysis method, are shown some relations between other family of polynomials. Following methods of Bayad and Luo, Fourier series expansion of Apostol Frobenius-Euler polynomials are obtained, and by using this expansion, new summation formulae are shown. By making use of Fourier series expansion of Genocchi polynomials, equalities between function and function with Genocchi polynomials are found. Furthermore, formulae with Riemann zeta function including periodic Genocchi function, Genocchi polynomials and Digamma functions are constructed. Nevanlinna theory that is used intensively in complex analysis are applied to generating functions of Bernoulli and Euler numbers, and Nevanlinna characteristic functions are computed.