Applications of some special polynomials on q-umbral calculus

In this thesis, q-umbral calculus is studied. By giving some definitions, theorems and properties belonging to q-umbral calculus, q-Sheffer sequences with their q-Appell and q-associated sequences that are subclass of these sequences are investigated. In addition, by making use of the methods of q-umbral calculus, some fundamental properties of q-Euler and q-Bernoulli polynomials that are families of special polynomials are researched on applications of q-umbral calculus. As the application of q-umbral calculus, modified Apostol type q-Bernoulli polynomials, a new family of polynomial, are defined and new identities are obtained. Furthermore, these polynomials are related to q-Euler polynomials by means of q-umbral calculus methods.