content_image/412907.jpg content_image/412894.jpg content_image/412896.jpg content_image/412897.jpg content_image/412895.jpg content_image/412904.jpg content_image/412901.jpg content_image/412905.jpg content_image/412903.jpg content_image/412902.jpg content_image/412898.jpg content_image/412899.jpg content_image/412900.jpg
<img src='content_image/412907.jpg'>

## V. FRACTIONAL HYPERGEOMETRIC FUNCTION 

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 

<img src='content_image/412894.jpg'>

<img src='content_image/412896.jpg'>

<img src='content_image/412897.jpg'>

<img src='content_image/412895.jpg'>

Thus we get a solution of the above differential equation, 

Here ( a ) α k is defined as 

<img src='content_image/412904.jpg'>

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 

which reduces to the Gauss hypergeometric series when α = 1. 

The ratio of successive coefficients is 

<img src='content_image/412901.jpg'>

or 

<img src='content_image/412905.jpg'>

<img src='content_image/412903.jpg'>

<img src='content_image/412902.jpg'>

<img src='content_image/412898.jpg'>

Since 

<img src='content_image/412899.jpg'>

<img src='content_image/412900.jpg'>

