## P-adic q-measure and its applications

Thesis Type: Doctorate

Institution Of The Thesis: Uludağ Üniversitesi, Turkey

Approval Date: 2009

Thesis Language: Turkish

Student: HACER ÖZDEN

Supervisor: İSMAİL NACİ CANGÜL

Abstract:

The results obtained in this thesis are generating functions of q-Euler numbers and polynomials, higher order twisted q-Euler numbers and polynomials, twisted Hurwitz type q-Euler zeta function, twisted Euler l -function, p-adic q-Euler measure and its applications, and also applications of integral.This study consists of nine chapters of which the first one is the Introduction.In the second chapter, some basic definitions and notions which will be used in other chapters are given.In the third chapter, some notions, definitions and theorems from p-adic analysis are given.In the fourth chapter, a brief history of q-analysis is given. Afterwards, basic notions and results of q-analysis are recalled.In the fifth chapter, new generating functions of q-Euler numbers and polynomials are given. Also distribution relation (Raabe rule) of q-Euler polynomials is obtained.In the sixth chapter, the generating functions of twisted (h,q) -Euler numbers and polynomials and higher order twisted (h,q)-Euler numbers and polynomials are defined by means of p-adic q-Volkenborn integral and multiple p-adic q-Volkenborn integral. Also distribution relations for q-Euler polynomials are obtained.In the seventh chapter, interpolation functions of twisted (h,q)-Euler numbers and polynomials are defined.In the eighth chapter, p-adic q-Euler measure is built. Some relations between this measure and p-adic integral are given. Using these relations, some applications in p-adic analysis are given. Also twisted Daehee numbers and polynomials are defined. Distribution relation and properties of these polinomials are obtained.In the nineth chapter, applications of p-adic q-measure are discussed.