Journal of Advances in Mathematics and Computer Science

This original research article fractional operators are redefined, Fractional operators have been deduced from nature of the operator by Riemann, Wilhelm Leibnitz, Neils Henrik Abel, Liouville etc. these fractional operator have some deviation from reality. This article is an attempt to find expression of fractional operator by inductive method which exactly reflects the reality, expression for explicit fractional derivative for 0 < α < 1, given by 

D^αx^m = exp \(\left(\lim\limits_{r\to1}{^m_{m-\alpha}} \frac{i}{e^{-2 \pi i t}-1}\left({ }_0^{2 \pi} \frac{e^{i t \theta}}{1-r e^{-i \theta}} d \theta\right) d t\right) x^{m-\alpha}\) 

and corresponding derivatives and integrals of other special functions, and it’s representation in terms of classical classes of elementary functions, is possible iff non elementary integral, \(^{2 \pi}_0\) \(\frac{e^{i t \theta}}{1-r e^{-i \theta}} d \theta\) could be evaluated. An odd function changes to even function by operating integer order derivative operator, what are intermediary states? here intermediary state means \(\frac{d^\alpha \sin x}{d x^\alpha}=a_0 x^{1-\alpha}-a_1 x^{3-\alpha}+a_2 x^{5-\alpha}-\ldots ; 0<\alpha<1,\) where a_n is function of α and n only. A periodic odd or even functions could be represented in a countable orthogonal
basis, do the intermediary functions have countable orthogonal basis for their representation? only if 0 < α < 1, is a
rational number.