Euler and q-Euler polynomials and their approximation properties

This study consists of four parts. The first section is introduction and detailed description of important advance information. The other sections include the basic parts of the study.The first chapter is examined under five main headings. Very important information is given in this chapter to analysis each main topic. Especially, important theorems and results are examined by giving the -adic analysis of the basic building blocks of work; Quantum calculus with the rapid development in many areas recently; -Volkenborn integral is used to build many numbers and polynomials; Bernoulli numbers and polynomials, some functions such as the Zeta function, some numbers and polyonomials, which has got relations with Euler numbers and polynomials,The second part is defined and the generalizations of Euler numbers and polynomials. In addition, other polynomials, the numbers and functions are related to each other.In the third section -Euler numbers and polynomials are constructed and their important results are given.In the fourth and final section, the weighted generalized -Euler numbers and polynomials are introduced and also important theorems, proofs and result are given. Similarly, approaches to -extension Zeta type functions of -weighted twisted -Euler polynomials are examined. In addition, the results and findings gained in the work are given.Key words: Bernoulli numbers, Bernoulli polynomials, q-Bernoulli numbers, q-Bernoulli polynomials, Dirichlet character, p-adic q-integral, Second kind Stirling numbers, Mellin transform, Riemann Zeta function, Euler numbers, Euler polynomials, -Euler numbers, -Euler polynomials.