The sheffer type polynomials and their applications on (q-)umbral calculus

The main aim of this thesis is to give some identities about q-Sheffer polynomials and q-Appell polynomias on q-umbral calculus and investigate some special polynomials by using these identities. Firstly, the definition and some identities of classic umbral algebra are given. Then c_{n}-umbral calculus, which is a generalization of umbral calculus, is given and c_{n}-Bernoulli type polynomials and c_{n}-Euler type polynomials are defined. Some identities of these polynomials are also obtained. Furthermore, the methods of q-umbral calculus are investigated. By using these methods and generating functions for these numbers and polynomials, some relations of q-Bernoulli polynomials, q-Euler polynomials, q-Hermite polynomials, q-Hermite based Bernoulli polynomials ve q-Hermite based Euler polynomials are derived.