Incomplete Riemann-Liouville fractional derivative operators and incomplete hypergeometric functions

TL;DR

This paper introduces incomplete hypergeometric functions based on incomplete Pochhammer ratios and explores their properties, including integral representations and derivative formulas, along with defining an incomplete Riemann-Liouville fractional derivative operator.

Contribution

It presents new incomplete hypergeometric functions and an associated fractional derivative operator, expanding the theoretical framework of special functions.

Findings

Derived integral and transformation formulas for incomplete hypergeometric functions

Established generating relations using the incomplete Riemann-Liouville fractional derivative

Introduced new incomplete Pochhammer ratios linked to incomplete beta functions

Abstract

In this paper, the incomplete Pochhammer ratios are defined in terms of the incomplete beta function $B_{y}(x,z)$. With the help of these incomplete Pochhammer ratios, we introduce new incomplete Gauss, confluent hypergeometric and Appell's functions and investigate several properties of them such as integral representations, derivative formulas, transformation formulas, and recurrence relation. Furthermore, an incomplete Riemann-Liouville fractional derivative operators are introduced. This definition helps us to obtain linear and bilinear generating relations for the new incomplete Gauss hypergeometric functions.

Equations

$$\gamma (s,x):={\int }_0^x{t}^{s-1}{e}^{-t}dt\left(\mathrm{Re}(s)>0;x\geqq 0\right)$$

$$\Gamma (s,x):={\int }_x^{\infty }{t}^{s-1}{e}^{-t}dt(x\geqq 0;\mathrm{Re}(s)>0\text{ when }x=0),$$

$$\gamma (s,x)+\Gamma (s,x)=\Gamma (s)\left(\mathrm{Re}(s)>0\right).$$

$$\left(\lambda\right)_{\nu}:=\frac{\Gamma\left(\lambda+\nu\right)}{\Gamma\left(\lambda\right)}=\left\{\begin{array}[]{ll}1&\left(\nu=0;\text{ }\lambda\in\mathbb{C}\backslash\left\{0\right\}\right)\\ \lambda\left(\lambda+1\right)...\left(\lambda+\nu-1\right)\text{ \ \ \ }&\left(\nu\in\mathbb{N};\text{ }\lambda\in\mathbb{C}\right)\end{array}\right\}$$

$${\left(\lambda ;x\right)}_{\nu }:=\frac{\gamma (\lambda +\nu ,x)}{\Gamma \left(\lambda \right)}\left(\lambda ,\nu \in \mathbb{C};x\geqq 0\right)$$

$${\left[\lambda ;x\right]}_{\nu }:=\frac{\Gamma \left(\lambda +\nu ,x\right)}{\Gamma \left(\lambda \right)}\left(\lambda ,\nu \in \mathbb{C};x\geqq 0\right).$$

$${\left(\lambda ;x\right)}_{\nu }+{\left[\lambda ;x\right]}_{\nu }={\left(\lambda \right)}_{\nu }\left(\lambda ,\nu \in \mathbb{C};x\geqq 0\right),$$

$${}_{2}\gamma_{1}\left[\begin{array}[]{cccc}\left(a,x\right)&.&b&;\\ &&c&;\end{array}\begin{array}[]{c}z\end{array}\right]:=\mathop{\displaystyle\sum}\limits_{n=0}^{\infty}\frac{(a;x)_{n}(b)_{n}}{(c)_{n}}\frac{z^{n}}{n!}$$

$${}_{2}\Gamma_{1}\left[\begin{array}[]{cccc}(a,x)&.&b&;\\ &&c&;\end{array}\begin{array}[]{c}z\end{array}\right]:=\mathop{\displaystyle\sum}\limits_{n=0}^{\infty}\frac{[a;x]_{n}(b)_{n}}{(c)_{n}}\frac{z^{n}}{n!}.$$

$${}_{2}\gamma_{1}\left[\begin{array}[]{cccc}\left(a,x\right)&.&b&;\\ &&c&;\end{array}\begin{array}[]{c}z\end{array}\right]+~{}_{2}\Gamma_{1}\left[\begin{array}[]{cccc}(a,x)&.&b&;\\ &&c&;\end{array}\begin{array}[]{c}z\end{array}\right]=~{}_{2}F_{1}\left[\begin{array}[]{cccc}a&.&b&;\\ &&c&;\end{array}\begin{array}[]{c}z\end{array}\right].$$

$$B_y(x,z):={\int }_0^y{t}^{x-1}(1-t{)}^{z-1}dt,\mathrm{Re}(x)>\mathrm{Re}(z)>0,0\le y<1$$

$$B_y(x,z):=\frac{y^x}{x}(1-y{)}^z{}_2{F}_1(1,x+z;1+x;y).$$

$$B_y(b+n,c-b)+B_{1-y}(c-b,b+n)=B(b+n,c-b),n\in {\mathbb{N}}_0:=\mathbb{N}\cup \left\{0\right\}.$$

$${\left[b,c;y\right]}_n:=\frac{B_y\left(b+n,c-b\right)}{B\left(b,c-b\right)}$$

$${\left\{b,c;y\right\}}_n:=\frac{B_{1-y}(c-b,b+n)}{B(b,c-b)}$$

$${\left[b,c;y\right]}_n+{\left\{b,c;y\right\}}_n=\frac{{\left(b\right)}_n}{{\left(c\right)}_n}.$$

$${\left[b,c;y\right]}_n:=\frac{1}{B(b,c-b)}\frac{y^{b+n}}{b+n}(1-y{)}^{c-b}{}_2{F}_1(1,c+n;b+n+1;y)$$

$${\left\{b,c;y\right\}}_n:=\frac{1}{B(b,c-b)}\frac{(1-y{)}^{c-b}}{c-b}{y}^{b+n}{}_2{F}_1(1,c+n;1+c-b;1-y).$$

$$[b,c;y{]}_n=\frac{{\left(-1\right)}^n\Gamma \left(c\right)}{\Gamma \left(c-b+n\right)\Gamma \left(b\right)}{y}^{b+n}\frac{d^n}{d{y}^n}\left[y^{-b}{B}_y(b,c-b+n)\right],$$

$$\{b,c;y{\}}_n=\frac{\Gamma (b+n)}{\Gamma (b+2n)}\frac{1}{B(b,c-b)}(1-y{)}^{c-b}\frac{d^n}{d{y}^n}((1-y{)}^{-c+b+n}{B}_{1-y}(c-b-n,b+2n)).$$

$$[b,c;y{]}_n=\frac{y^{b+n}}{B(b,c-b)}{\int }_0^1{u}^{b+n-1}(1-uy{)}^{c-b-1}du.$$

$$y^{-b}{B}_y(b,c-b+n)={\int }_0^1{u}^{b-1}(1-uy{)}^{c-b+n-1}du.$$

$${}_2{F}_1(a,[b,c;y];x):=\sum _{n=0}^{\infty }(a{)}_n[b,c;y{]}_n\frac{x^n}{n!},$$

$${}_2{F}_1(a,\left\{b,c;y\right\};x):=\sum _{n=0}^{\infty }(a{)}_n{\left\{b,c;y\right\}}_n\frac{x^n}{n!},$$

$${}_1{F}_1([a,b;y];x):=\sum _{n=0}^{\infty }[a,b;y{]}_n\frac{x^n}{n!},$$

$${}_1{F}_1(\left\{a,b;y\right\};x):=\sum _{n=0}^{\infty }{\left\{a,b;y\right\}}_n\frac{x^n}{n!}$$

$${}_2{F}_1(a,[b,c;y];x)+{}_2{F}_1(a,\left\{b,c;y\right\};x)={}_2{F}_1(a,b;c;x)$$

$${}_1{F}_1([a,b;y];x)+{}_1{F}_1(\left\{a,b;y\right\};x)={}_1{F}_1(a;b;x).$$

$${}_2{F}_1(a,[b,c;y],x)$$

$$\mathrm{Re}(c)$$

$${}_2{F}_1(a,[b,c;y],x)=\frac{1}{B(b,c-b)}{\int }_0^y{t}^{b-1}(1-t{)}^{c-b-1}(1-xt{)}^{-a}dt,$$

Full text

Key words : incomplete gamma functions, Pochhammer symbols, incomplete Pochhammer ratios, incomplete beta functions, incomplete hypergeometric functions, incomplete Appell’s functions, generating relations.

1

Introduction

In recent years , some extensions of the well known special functions have been considered by several authors (see, for example, [3], [4], [6], [15], [16], [17], [18], [19], [29] ). The familiar incomplete gamma fuctions $\gamma(s,x)$ and $\Gamma(s,x)$ are defined by

[TABLE]

and

[TABLE]

respectively. They satisfy the following decomposition formula:

[TABLE]

The function $\Gamma(s)$ and its incomplete versions $\gamma(s,x)$ and $\Gamma(s,x),$ play important roles in the study of analytical solutions of a variety of problems in diverse areas of science and engineering [14].

The widely used Pochhammer symbol $\left(\lambda\right)_{\nu}$ $\left(\lambda,\nu\in\mathbb{C}\right)$ is defined, in general, by

[TABLE]

In terms of the incomplete gamma functions $\gamma(s,x)$ and $\Gamma(s,x)$, the incomplete Pochhammer symbols $\left(\lambda;x\right)_{\nu}$ and $\left[\lambda;x\right]_{\nu}$ $\left(\lambda;\nu\in\mathbb{C};\text{ }x\geqq 0\right)$ were defined as follows [5]:

[TABLE]

and

[TABLE]

In view of (1), these incomplete Pochhammer symbols $\left(\lambda;x\right)_{\nu}$ and $\left[\lambda;x\right]_{\nu}$ satisfy the following decomposition relation:

[TABLE]

where $\left(\lambda\right)_{\nu}$ is the Pochhammer symbol given by (2).

The incomplete Gauss hypergeometric functions were defined by means of the incomplete gamma functions as follows [5]:

[TABLE]

and

[TABLE]

In view of (1), these incomplete Gauss hypergeometric functions are satisfy the following decomposition relation:

[TABLE]

It should be also mentioned that, for all that, H. M. Srivastava, M. Aslam Chaudhry and Ravi P. Agarwal discussed some properties and some interesting applications of these families of incomplete hypergeometric functions [24]. In recent years, the gamma function $\Gamma\left(z\right)$ and the Pochhammer symbol $\left(\lambda\right)_{v}$ were used to extend the generalized hypergeometric functions and their multivariate versions. After this works, incomplete hypergeometric functions have become one of the hot topics of recent years [1], [6], [7], [8], [9], [15], [20], [21], [22], [23], [26], [27], [28], [29].

On the other hand, fractional derivative operators found applications in many diverse areas of mathematical, physical and engineering problems. Because of this reason these operators have been an active research in recent years [2], [10], [11], [12], [13], [14], [30]. The use of fractional derivative operators in obtaining generating relations for some special functionscan be found in [16], [25]. In the present paper, we are aimed to introduce new incomplete hypergeometric functions with the aid of incomplete Pochhammer ratios and investigate their certain properties. Moreover, we introduce incomplete Riemann-Liouville fractional derivative operators and we obtain some generating relations for these new incomplete hypergeometric function with the aid of these new defined operators. The organization of the paper as follows:

In Section 2, the incomplete Pochhammer ratios are introduced by using the incomplete beta function and some derivative formulas to involving these new incomplete Pochhammer ratios are investigated. In Section 3, new incomplete Gauss hypergeometric functions and confluent hypergeometric functions are introduced with the help of these incomplete Pochhammer ratios and integral representations, derivative formulas, transformation formulas and recurrence relation are obtained for them. In Section 4, we define new incomplete Appell’s functions $F_{1}[a,b,c;d;x,z;y]$, $F_{1}\{a,b,c;d;x,z;y\}$, $F_{2}[a,b,c;d,e;x,z;y]$ and $F_{2}\{a,b,c;d,e;x,z;y\}$ and obtain their integral representations. In Section 5, we introduce incomplete Riemann- Liouville fractional derivative operator and show that the incomplete Riemann-Liouville fractional derivative of some elementary fuctions give the new incomplete fuctions defined in Sections 3 and 4. Finally, in the last section, we obtain linear and bilinear generating relations for incomplete hypergeometric functions.

2 The incomplete Pochhammer Ratio

The incomplete beta function is defined by

[TABLE]

and can be expressed in terms of the Gauss hypergeometric function

[TABLE]

The incomplete beta function satisfy the following relation:

[TABLE]

In terms of the incomplete beta function $B_{y}(x,z),$ the incomplete Pochhammer ratios $\left[b,c;y\right]_{n}$ and $\left\{b,c;y\right\}_{n}$ are introduced as follows:

[TABLE]

and

[TABLE]

where $0\leq y<1.$ It is clear from (11) that

[TABLE]

In view of (10) , we have the following relations

[TABLE]

and

[TABLE]

In the following theorem, we investigate the $n-th~{}$derivatives of the incomplete beta fuction by means of incomplete Pochhammer ratios.

Theorem 1

The following derivative formulas hold true:

[TABLE]

and

[TABLE]

Proof. Using (9) and (12), we immediately obtain the following equation:

[TABLE]

On the other hand, we have

[TABLE]

Taking derivatives $n$ times on both sides of (19) with respect to $y$, we can obtain a derivative formula for the incomplete beta function $[b,c;y]_{n}$ asserted by (17). Formula (18) can be proved in a similar way.

3 The new incomplete Gauss and confluent hypergeometric functions

In this section, we introduce new incomplete Gauss and confluent hypergeometric functions by

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

where $0\leq y<1.$

An immediate consequence of (14) and the definitions (20), (21), (22) and (23) are the following decomposition formulas

[TABLE]

and

[TABLE]

Theorem 2

The following integral representation holds true:

[TABLE]

Proof. Replacing the incomplete Pochhammer ratio $[b,c;y]$ in the definition (20) by its integral representation given by (9) and interchanging the order of summation and integral which is permissible under the conditions given in the hypothesis of the Theorem, we find

[TABLE]

which can be written as follows:

[TABLE]

In a similar way, we have the following theorem:

Theorem 3

The following integral representation holds true:

[TABLE]

Theorem 4

The following result holds true:

[TABLE]

Proof. Putting $x=1$ in (24), we obtain

[TABLE]

Using the Euler’s integral representation for (3), we have

[TABLE]

Using transformation formula

[TABLE]

in (32), we obtain

[TABLE]

Considering (34) in (32), we get

[TABLE]

Theorem 5

The following result holds true:

[TABLE]

Theorem 6

The following integral representations hold true:

[TABLE]

and

[TABLE]

Proof. Replacing the incomplete Pochhammer ratio $[a,b;y]$ in the definition (22) by its integral representation given by (9), we are led to the desired result (37). Formula (38) can be proved in a similar way.

Theorem 7

The following integral representation holds true:

[TABLE]

Proof. It is known that from the Euler’s formula

[TABLE]

Taking$~{}u=y^{k}~{}$and the remaining part as $dv$ and applying the integration by parts, we get

[TABLE]

By rearranging the terms we get the result.

Corollary 8

Taking $k=1$ in Theorem 7, we get the following result:

[TABLE]

Theorem 9

The following integral representation holds true:

[TABLE]

Proof. It is known that

[TABLE]

Taking $u=(1-y)^{k}$ and the rest as $dv$ and using integration by parts, we get the result.

Corollary 10

Taking $k=1$ in Theorem 9, we get the following result:

[TABLE]

Theorem 11

The following derivative formula holds true:

[TABLE]

Proof. Using (27), differentiating on both sides with respect to $x,$ we obtain

[TABLE]

which is (43) for $n=1$. The general result follows by the principle of mathematical induction on $n.$

Theorem 12

The following derivative formula holds true:

[TABLE]

Theorem 13

We have the following difference formula for ${}_{2}F_{1}(a,[b,b+h;y];x)$ :

[TABLE]

Proof. Recalling that the Mellin transform operator is defined by

[TABLE]

we observe that ${}_{2}F_{1}(a,[b,b+h;y];x)$ is the Mellin transform of the function

[TABLE]

where

[TABLE]

is the Heaviside unit function. Observing the fact that

[TABLE]

we can write that

[TABLE]

where $\frac{\partial}{\partial t}(H(t))=\delta(t-t_{0}),$

[TABLE]

is the Dirac delta function. Applying Mellin transform on both sides (3) and using (46) and the fact that

[TABLE]

we have

[TABLE]

This completes the proof.

In the following theorems, we give transformation formulas:

Theorem 14

The following transformation formula holds true:

[TABLE]

Proof. Using (27), we obtain

[TABLE]

The substitution $s=1-t$ in (49) leads to

[TABLE]

Theorem 15

The following transformation formula holds true:

[TABLE]

Theorem 16

The following transformation formulas hold true:

[TABLE]

and

[TABLE]

Proof. The proofs of (51) and (52) are direct consequences of Theorem 6.

4 The incomplete Appell’s functions

In this section, we introduce the incomplete Appell’s functions $F_{1}[a,b,c;d;x,z;y]$, $F_{1}\{a,b,c;d;x,z;y\}$, $F_{2}[a,b,c;d,e;x,z;y]$ and $F_{2}\{a,b,c;d,e;x,z;y\}$ by

[TABLE]

and

[TABLE]

and

[TABLE]

and

[TABLE]

We proceed by obtaining the integral representations of the functions $F_{1}[a,b,c;d;x,z;y]$, $F_{1}\{a,b,c;d;x,z;y\}$, $F_{2}[a,b,c;d,e;x,z;y]$ and $F_{2}\{a,b,c;d,e;x,z;y\}.$

Theorem 17

For the incomplete Appell’s functions $F_{1}[a,b,c;d;x,z;y]$ and $F_{1}\{a,b,c;d;x,z;y\},$ we have the following integral representation:

[TABLE]

and

[TABLE]

Proof. Replacing the integral representation for incomplete beta function which is given by (9), we find that

[TABLE]

which can be written as

[TABLE]

Whence the result. Formula (58) can be proved in a similar way.

Theorem 18

For the incomplete Appell’s functions $F_{2}[a,b,c;d,e;x,z;y]$ and $F_{2}\{a,b,c;d,e;x,z;y\},$ we have the following integral representation:

[TABLE]

and

[TABLE]

Proof. Replacing the integral representation for incomplete beta function which is given by (9), we get

[TABLE]

Considering the fact that the series involved are uniformly convergent and we have a right to interchange the order of summation and integration, we get

[TABLE]

Formula (60) can be proved in a similar way.

5 Incomplete Riemann-Liouville fractional derivative operator

In this section, we introduce and investigate the incomplete Riemann-Liouville fractional derivative operators. The Riemann-Liouville fractional derivative of order $\mu$ is defined by

[TABLE]

Now, we define the incomplete Riemann-Liouville fractional derivative operators $D_{z}^{\mu}[f(z);y]$ and $D_{z}^{\mu}\{f(z);y\}$ by

[TABLE]

and its counterpart is by

[TABLE]

We start our investigation by calculating the incomplete fractional derivatives of some elementary functions.

Theorem 19

Let $\mathop{\mathrm{R}e}(\lambda)>-1,~{}\mathop{\mathrm{R}e}(\mu)<0.$ Then

[TABLE]

Proof. Using (5) and (9), we get

[TABLE]

Whence the result.

Theorem 20

Let $\mathop{\mathrm{R}e}(\lambda)>-1,~{}\mathop{\mathrm{R}e}(\mu)<0.$ Then

[TABLE]

Theorem 21

Let $\mathop{\mathrm{R}e}(\lambda)>0,~{}\mathop{\mathrm{R}e}(\alpha)>0,~{}\mathop{\mathrm{R}e}(\mu)<0$ and$~{}\left|z\right|<1.~{}$Then

[TABLE]

and

[TABLE]

Proof. Direct calculations yield

[TABLE]

By (26), we can write

[TABLE]

Hence the proof is completed. Formula (67) can be proved in a similar way.

Theorem 22

Let $\mathop{\mathrm{R}e}(\lambda)>\mathop{\mathrm{R}e}(\mu)>0,~{}\mathop{\mathrm{R}e}(\alpha)>0,~{}\mathop{\mathrm{R}e}(\beta)>0$ ; $\left|az\right|<1~{}$and $\left|bz\right|<1.$ Then

[TABLE]

and

[TABLE]

Proof. We have

[TABLE]

By (57), we can write

[TABLE]

Whence the result. Formula (69), can be proved in a similar way.

Theorem 23

Let $\mathop{\mathrm{R}e}(\lambda)>\mathop{\mathrm{R}e}(\mu)>0,\mathop{\mathrm{R}e}(\alpha)>0,\mathop{\mathrm{R}e}(\beta)>0,\mathop{\mathrm{R}e}(\gamma)>0$ ; $\left|\frac{t}{1-z}\right|<1$and $\left|t\right|+\left|z\right|<1$ we have

[TABLE]

and

[TABLE]

Proof. Using Theorem 19 and (55), we get

[TABLE]

Hence proof is completed. Formula (71), can be proved in a similar way.

6

Generating Functions

Now, we obtain linear and bilinear generating relations for the incomplete hypergeometric functions ${}_{2}F_{1}(a,\left[b,c;y\right];x)$ by following the methods described in [4]. We start with the following theorem:

Theorem 24

For the incomplete hypergeometric functions we have

[TABLE]

and

[TABLE]

where $\left|z\right|<\min\{1,\left|1-t\right|\}$ and  $\mathop{\mathrm{R}e}(\lambda)>0$,$~{}\mathop{\mathrm{R}e}(\beta)>\mathop{\mathrm{R}e}(\alpha)>0.$

Proof. Considering the elementary identity

[TABLE]

and expanding the left hand side, we have for $\left|t\right|<\left|1-z\right|$ that

[TABLE]

Now, multiplying both sides of the above equality by $z^{\alpha-1}$ and applying the incomplete fractional derivative operator $D_{z}^{\alpha-\beta}[f(z);y]$ on both sides , we can write

[TABLE]

Interchanging the order, which is valid for $\mathop{\mathrm{R}e}(\alpha)>0$ and $\left|t\right|<\left|1-z\right|,$ we get

[TABLE]

Using Theorem 21, we get the desired result. Formula (73), can be proved in a similar way.

The following theorem gives another linear generating relation for the incomplete hypergeometric functions.

Theorem 25

For the incomplete hypergeometric functions we have

[TABLE]

and

[TABLE]

where $\mathop{\mathrm{R}e}(\lambda)>0,~{}\mathop{\mathrm{R}e}(\rho)>0$,$~{}\mathop{\mathrm{R}e}(\beta)>\mathop{\mathrm{R}e}(\alpha)>0;$ $\left|t\right|<\frac{1}{1+\left|z\right|}.$

Proof. Considering

[TABLE]

and expanding the left hand side, we have for $\left|t\right|<\left|1-z\right|$ that

[TABLE]

Now, multiplying both sides of the above equality by $z^{\alpha-1}(1-z)^{-\rho}$ and applying the fractional derivative operator $D_{z}^{\alpha-\beta}[f(z);y]$ on both sides, we get

[TABLE]

Interchanging the order, which is valid for $\mathop{\mathrm{R}e}(\alpha)>0$ and $\left|zt\right|<\left|1-t\right|,$ we get

[TABLE]

Using Theorem 21 and 22, we get the desired result. Generating relation (75), can be proved in a similar way.

Finally we have the following bilinear generating relation for the incomplete hypergeometric functions.

Theorem 26

For the incomplete hypergeometric functions we have

[TABLE]

and

[TABLE]

where $\mathop{\mathrm{R}e}(\lambda)>0,~{}\mathop{\mathrm{R}e}(\gamma)>0$,$~{}\mathop{\mathrm{R}e}(\beta)>0,~{}\mathop{\mathrm{R}e}(\delta)>0,~{}\mathop{\mathrm{R}e}(\alpha)>0;$ $\left|t\right|<\frac{1-\left|z\right|}{1+\left|x\right|}$ and $\left|z\right|<1.$

Proof.* Replacing $t~{}$by$~{}(1-x)t$ in (72), multiplying the resulting equality by $x^{\gamma-1}$ and then applying the incomplete fractional derivative operator $D_{x}^{\gamma-\delta}[f(x);y],$ we get*

[TABLE]

Interchanging the order, which is valid for $\left|z\right|<1,\left|\frac{1-x}{1-z}t\right|<1$ and $\left|\frac{z}{1-t}\right|+\left|\frac{xt}{1-t}\right|<1,$ we can write that

[TABLE]

*Using Theorems 21 and 23, we get (76). Generating relation (77), can be proved in a similar way.   * **

7

Conclusion

Incomplete Pochhammer ratios are defined in (12) and (13) by using the incomplete beta functions. Several properties of these functions are obtained. Incomplete hypergeometric functions are introduced with the help of these incomplete Pochhammer ratios and certain properties such as integral representations, derivative formulas, transformation formulas and recurrence relation are investigated. Furthermore, incomplete Riemann-Liouville fractional derivative operators are defined. The incomplete Riemann-Liouville fractional derivatives for the some elementary functions are given. Linear and bilinear generating relations for incomplete hypergeometric functions are obtained.