Applications of q-calculus to special function

The objective of this thesis is to extend $q$-analogues of special functions at classical analysis and generalize some properties on the domains of these functions for all real numbers. For this purpose, the concepts of neutrix and neutrix limit, which was developed by Van der Corput, have been used. This study consists of eight sections. In the second section after introduction , some definitions and properties which will be used frequently in the other sections about quantum calculus and then in the third section, definitions of neutrix and neutrix limit with their examples are given. From the forth section, the results obtained in this study is presented. In the forth section, the properties on $q$-gamma function $\Gamma_q(x)$ which is the $q$-analogue of classical gamma function are given. The fifth section consists of some results on incomplete $q$-gamma function and its first derivative obtained by using neutrix and neutrix limit. In the sixth section, $q$-analogue of polygamma function $\psi_{q,n}(x)$ is extended for all real numbers by the aid of neutrix and neutrix limit. In the seventh section, some results which are obtained by using neutrix limit of $q$-beta function at zero and negative integers for which this function is not definite actually are presented. In the eight and last section, some inequalities for $q$-gamma $\Gamma_q(x)$, beta $B(x,y)$, $q$-beta $B_q(x,y)$ functions are given respectively.