5 Thus we get a solution of the above differential equation, Here ( a ) α k is defined as This can be seen as a fractional generalization of the rising factorial And the series (40) can be seen as a fractional generaliza- tion the confluent hypergeometric function. If α = 1, it is exactly the confluent hypergeometric function. Profiles of this series (40) with different values of α are displayed in Fig. 6. For the fractional Gauss hypergeometric function, con- sider the following series There are other types of special functions in mathemat- ical physics. A most famous one is the hypergeometric function [12–14]. In this section, we will try to define a fractional generalization of the hypergeometric functions. Let’s first consider the generalization of the confluent hypergeometric differential equation: Here a and c are complex parameters. When α = 1, this is the ordinary confluent hypergeometric equation. Introducing the fractional Taylor series and substituting, we get the ratio of successive coeffi- cients which reduces to the Gauss hypergeometric series when α = 1. The ratio of successive coefficients is FIG. 6: Fractional confluent hypergeometric function. V. FRACTIONAL HYPERGEOMETRIC FUNCTION Since z $^{α}$( D $^{α}$) $^{2}$y + ( c − z $^{α}$) D $^{α}$y − ay = 0 . (36) y ( z ) = ∞ ∑ k =0 c$_{k}$z α · $^{k}$, (37) c$_{k}$$_{+1}$ · Γ( kα + α + 1) c$_{k}$ · Γ( kα + 1) = a + Γ( kα +1) Γ( kα − α +1) c + Γ( kα +1) Γ( kα − α +1) , (38) c$_{1}$ · Γ( α + 1) c$_{0}$ = a c . (39) c$_{k}$$_{+1}$ · c Γ( kα + 1) Γ( kα − α + 1) + c$_{k}$$_{+1}$ · Γ( kα + α + 1) Γ( kα − α + 1) = c$_{k}$ · ab + c$_{k}$ · ( a + b ) Γ( kα + 1) Γ( kα − α + 1) + c$_{k}$ · Γ( kα + 1) Γ( kα − α + 1) Γ( kα + 1) Γ( kα − α + 1) . (45) y ( z ) = ∞ ∑ k =0 c$_{k}$z α · $^{k}$, (46) z $^{α}$D$^{α}$y ( z ) = ∞ ∑ k =1 c$_{k}$ Γ( kα + 1) Γ( kα − α + 1) z α · $^{k}$, (47) c$_{k}$$_{+1}$ · Γ( kα + α + 1) c$_{k}$ · Γ( kα + 1) = ( a + Γ( kα +1) Γ( kα − α +1) )( b + Γ( kα +1) Γ( kα − α +1) ) ( c + Γ( kα +1) Γ( kα − α +1) ) , (44) y ( z ) = ∞ ∑ k =0 ( a ) α $_{k}$( b ) α k ( c ) α k 1 Γ( kα + 1) z α · $^{k}$, (43) ( a )$_{k}$ = a ( a + 1) ... ( a + k − 1) . (42) ( a ) α 0 = 1 , ( a ) α 1 = a, ( a ) α k = ( a ) α 1 a + Γ( α + 1) Γ(1) ) ... ( a + Γ( kα − α + 1) Γ( kα − 2 α + 1) ) , k ≥ 2 . (41) y ( z ) = ∞ ∑ k =0 ( a ) α k ( c ) α k 1 Γ( kα + 1) z α · $^{k}$. (40) or 0 0.5 1 1.5 2 2.5 3 3.5 4 0 20 40 60 80 100 120 140 160 180 =1/3 =1/2  =2/3  =1  Fractional confluent hypergeometric function a=1, c=1 