Right Type Departmental Bulletin Paper

Discrete and $q$-discrete analogues ofMittag-Leffler function are pre sented. Their relations to fractional difference are also investigated. Applications of these functions to numerical analysis and integrable systems are also made. 1Fractional derivative Fractional derivative goes back to the Leipniz’s note in his list to $\mathrm{L} $ ’Hospital in 1695 and we now have many definitions of fractional derivatives $[11, 13] $. In the last few decades, many authors pointed out that derivatives and integrals of fractional order, especially 1/2-derivative, are very suitable for the description of physical phenomena (See ref. [14] for example.). We first define affactional integral operator $I^{a} $ as follows. Definition 1Let $a $ be a nonnegative real number and $u(t)(0<t) $ be piece-wise continuous on $(0, \infty) $ and integrable on any subinterval $[0, \infty) $. Then for $t>0 $ , we call $I^{a}u(t) = \int_{0}^{t}K(a;t-s)u(s)\mathrm{d}s $ (1) $I^{0}u(t)=u(t) $ (2) the fmctional integral of $u $ of order $a $. $K(a;t) $ is a monomial given by $K(a;t) \equiv\frac{t^{a-1}}{\Gamma(a)} $ $(t>0, a>0) $. (3) Fractional derivatives of order $a>0 $ are defined by acombination of normal derivative and ffactional integral in the following two manners