on hypergeometric series associated with the generalized zeta functions

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Oct 19, 2016 - Here, we aim to derive basic properties of generalized zeta function for ... general Hurwitz – Lerch zeta functions and fractional derivative.
SCIREA Journal of Mathematics http://www.scirea.org/journal/Mathematics October 19, 2016 Volume 1, Issue1, October 2016

ON HYPERGEOMETRIC SERIES ASSOCIATED WITH THE GENERALIZED ZETA FUNCTIONS

Maged G. Bin-Saad1,* and Amani M. Hanballa1 1

Department of Mathematics, Aden University, Aden, Kohrmaksar , P.O.Box 6014,Yemen

E-mail:[email protected] *Corresponding

author

Abstract In this work we will introduce and study a generalized zeta function of the general Hurwitz – Lerch zeta functions. Here, we aim to derive basic properties of generalized zeta function for the general Hurwitz – Lerch zeta functions include some integral representations for several general Hurwitz – Lerch zeta functions and fractional derivative. A number of known and new results which introduced are generalization of a known results introduced by Shy-Der and Srivastava [6].

Key words:Hurwitz-Lerch zeta functions; Mellin-Barnes integrals; hypergeometric functions; hypergeomteric-type generating functions.

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1. Introduction For Re(z )  1 , the Riemann zeta function  (z ) is defined as follows [1,p.88] 

1 , z n 1 n

 (z )  

(1.1)

which can, except for a simple pole at z  1 with its residue 1. be continued moromorphically to the whole complex z -plane. The Hurwitz (or generalized) zeta function denoted usually by  (z , a) and so for Re(z )  1 , 

1 , z n  0 (a  n )

 (z , a )  

(1.2)

can just as  (z ) . be continued moromorphically to the whole complex z -plane except for a simple pole at z  1 with its residue 1.

When a  1, (1.2) reduces to (1.1). In terms of

Hurwitz-Lerch zeta function (x , z , a) defined by [5,p.121] and [1]: 

xn , x  1, a  0, 1, 2,.... z n  0 (a  n )

(x , z , a )  

(1.3)

Equivalently, it has the integral expression 

(x , z , a ) 

1 t z 1e at (1  xe t )1dt ,  ( z ) 0

(1.4)

provided Re(a)  0 and either x  1 and Re(z )  0 or x  1 and Re(z )  1 . So that obviously

(1, z , a)   (z , a) .

(1.5)

The zeta function in (1.3) has since been extended by Goyal and Laddha [3,p.100(1.5)], in the form :

(  )n x n , z n! n  0 (a  n ) 

* (x , z , a )   where x  1, Re(a)  0,   1 .

Equivalently , it has the integral expression : 54

(1.6)



1   (x , z , a )  t z 1e at (1  xe t )   dt ,  ( z ) 0 *

(1.7)

provided that   1, Re(a)  0, x  1 and either Re(z )  0 or Re(z )  Re( ) according to

x  1 or x  1 . In [6] Shy-Der and Srivastava introduced the general family of the Hurwitz-Lerch zeta function in the form: 

( ) n

n 0

( ) n

 (,, ) (x , z , a )   where   , a,  \

 0

,  , 



xn , (a  n )z

,    when x , z  ,    , z 

(1.8)

x  1,   

and Re(z    )  1 when x  1 . Equivalently , it has the following integral expressions :



(  , )  ,

(  ;  ), (1;1);   1  z 1  at k t  (x , z , a )  t e 2 1  x e  dt . (z ) 0  ( ;  ); 

(1.9)

In this paper we will introduce and study a generalized zeta function of the general Hurwitz – Lerch zeta function

( , ) ,

(x, z, a ) . This work aims to derive basic properties of generalized

zeta function for the general Hurwitz – Lerch zeta functions

include some integral

representations for several general Hurwitz – Lerch zeta functions and fractional derivative. A number of known and new results which introduced are generalization of a known results introduced in [6]. In section 2 we will introduce analogous definition of the general family of Hurwitz-Lerch zeta function denoted by representation for

( , ) , ,

( , ) , ,

(x, z, a ) with a known special cases. Some integral

(x, z, a ) are discussed in section 3 and fractional derivative in section

4.

55

( , ) , ,

2. The Function

(x, z, a )

In this section we will give analogous definition of the function

( , ) ,

(x, z, a )

which is defined by (1.3) in its form : ( , ) , ,

(x, z, a )

( )

z

( , ) ,

x , z,

( )n

a n 0

  ,  , a,  \ when x

1,

 0

, , 



( ) n (a

,    when x, z

and Re(z

)

xn

,

1 when x

n)z

,

(2.1)

and 1.

Clearly, we have ( , ) , ,1

(1,1) ,1,1

(x, z, a)

(0,0) , ,1

(x, z, a)

*

Obviously, when

x

a

(x, z, a)

(x, z, a),

(2.2)

(x, z, a).

(2.3)

1 , (2.1) reduce to (1.8). Further, on putting

1 in (2.2), it reduces to (1.1). Next, on putting x

1 and

1 in (2.3), it reduces

to (1.2).

3. Integral representations By using the Fox-Wright extension of the generalized hypergeometric

p

Fq function with p

numerator and q denominator parameters, defined by:

(1 ; A1 ), ( 2 ; A 2 ),..., ( p , A p );  p     j 1 ( j )A jn z n z q , p q  n  0  j 1 (  j ) B jn n !  ( 1 ; B1 ), (  2 , B 2 )..., ( q , B q );   

where the argument z , the complex parameters  j ( j  1,..., p ) and  j ( j  1,..., q ) , and the positive real parameters A j ( j  1,..., p ) and B j ( j  1,..., q ) are so constrained

56

that the series in (3.1) converges absolutely [1,p.183] and Eulerian integral formula of second kind [1] :

(n  a )  z 



1 t z 1e  ( n a )t dt ,(min Re (z ), R e(n  a )  ( z ) 0

  0) .

it is easy to derive the following integral representation. Theorem 1: Let Re(a)

x

( , ) , ,

0 when x

0, Re(z )

1(x

1) Re(z )

1 when

1 .Then (  ;  ), (1;1);   ( ) z z 1  a t  t  t e 2 1  xe  dt . (z ) 0   ( ;  );

(x, z, a)

(3.1)

Proof: By starting from the left-hand side of formula (3.1), then in view of definition (2.1), it is easily seen that ( , ) , ,

(x, z, a )

( )

z

( , ) ,

x , z,

a

.

Upon using the integral formula (1.9), we are led finally to the right-hand side of formula (3.1). In terms of the Gauss hypergeometric function 2 F 1 ( see [7] ):

 a  n b  n 2 F1  a ,b ;c ; x    c n n 0 

xn , c  0,  1,  2, n!

,

1 as follows :

a special cases of (3.1), occur when



(1,1) , ,1

(x, z, a)

1 t z 1e at 2 F1   ,1; ; xe t  dt . (z ) 0

By using the familiar Euler transformation formula [1,p.64] 2

F1( , ; ; x ) ( arg(1

x)

(1

x)

2

,(0

F1(

; ; x ),

, ), 57

0

),

(3.2)

we get equivalently integral formula for (3.2) in form : 

(1,1) , ,1

1 t z 1e at (1  xe t )   1 2 F1    ,  1; ; xe t  dt .  ( z ) 0

(x, z, a )

(3.3)

For   1 , (3.3) reduces to the well- known result of Goyal and Laddha [3] 

(1,1) ,1,1

1 t z 1e at (1  xe t )  dt  * (x , z , a ) .  ( z ) 0

(x, z, a )

Theorem 2: Let Re( )

0,

(3.4)

. Then



( , ) , ,

( ) z  1 t (0, )   a t e   ,v  xt , z ,  dt , (x, z, a ) =  (  ) 0  

(3.5)

and

( )  z 1 1 a )  (Log )  1 (0, dt .  ,v  x (Log ) , z ,  (  ) 0 t t    1

( , ) , ,

(x, z, a ) =

(3.6)

Proof: Denote, for convenience, the right-hand side of formula (3.5) by I. Then it is easily seen that

I  ( )

z



1 a )   t  1e t  (0, dt .  ,v  xt , z ,  (  ) 0    

(  )0 n x n t    n 1e t dt . z  n  0 (  )( ) n (a   n ) 0 



Upon using the Hankel's formula [6, p.13(2)]

1 (z )

1 2 i

(0 )

et t

z

dt ,

arg(t )

,z

.

(3.7)

and definition (2.1) , we are led finally to the left-hand side of formula (3.5). In the same manner one can derive the formula (3.6).

1 ,the integral representations

In their special cases, when (3.5) and (3.6) reduce to the following forms: 

1 t  1e t  (0,1)   (x , z , a )   ,1 (xt , z , a )dt ,  (  ) 0 *

58

(3.8)

and

1 1 1   (Log )  1  (0,1)  ,1  x (Log ), z , a  dt ,  (  ) 0 t t   1

* (x , z , a ) 

(3.9)

respectively. Theorem 3: Let 0 ( , ) , ,

(x, z, a)

Re( )

Re( ) ,

. Then

  xt  ( ) z ( ) t  1 a  , z ,  dt .     (   )(  ) 0 (1  t )   (1  t )

(3.10)

Proof: Denote, for convenience, the right-hand side of formula (3.10) by I. Then it is easily seen that

I  ( )

z

  xt  ( ) t  1 a  , z ,  dt .     (   )(  ) 0 (1  t )   (1  t )

Now, in view of integral formula  (,, ) (x , z , a ) 



( ) t  1 xt   ( , z , a )dt , (   )(  ) 0 (1  t ) (1  t )

(0  Re( )  Re( ),  



),

and the definition (2.1), we obtain the left-hand side of formula (3.10).

1 , (3.10) reduces to the following result :

Obviously, for

  (x , z , a )  *

sin( )





 xt  t  1 0 (1  t )   (1  t ) , z , a  dt .

(3.11)

Furthermore we derive the following result . Theorem 4: Let ( , ) , ,

/

0

, arg(w )   .Then

,

( ) z ( ) (x, z, a ) = 2 i

(0  )

w



a  e  (,,0)  xw  , z ,  dw .  

 w

Proof: Denote, for convenience, the right-hand side of formula (3.12) by I . Then in 59

(3.12)

view of (1.8),it is easily seen that

( ) nxn

I n 0

( ) z n) 2 i

(a

(0  )

w

 (  n ) w

e dw .



Upon using the formula (3.7) and definition (2.1) , we are led finally to left-hand side of formula (3.12).

1 , (3.12) reduces to the following result :

Obviously, for

  (x , z , a ) 

(0  )

1

*

2 i

w

1 w

1 e  (1,0) , z , a )dw ,  ,1 (xw

(3.13)



( arg(w )   ) . Furthermore , in its limit case when ρ→0, (3.12) would simplify immediately to the following formula : (0, ) , ,

(x, z, a )

( ) z ( ) 2 i

(  /

 0

(0  )

w



, 



a  e   xw  , z ,  dw ,  

 w

(3.14)

, arg(w )   ) .

If we replace the Hankel’s loop in (3.12) to (3.14) by a suitable Mellin – Barnes contour integral we get : Theorem 5 : Let Re( )

0, ,

.Then  i 

( , ) , ,

(x, z, a)

* (x , z , a ) 

( ) z ( ) a  w  e w (,,0)  xw  , z ,  dw ,  2 i    i  1

 i 

2 i  

1 w 1e w  (1,0) , z , a )dw ,  ,1 ( xw

(3.15)

(3.16)

i 

and  i 

(0, ) , ,

(x, z, a)

( ) z ( ) a  w  e w   xw  , z ,  dw .  2 i    i 

Proof : Denote, for convenience, the right-hand side of formula (3.15) by I . Then it is easily seen that

60

(3.17)

 i 

( ) z ( ) a  I  w  e w  (,,0)  xw  , z ,  dw .  2 i    i  Now, in view of integral formula[23] 

(  , )  ,

 i 

( ) (x , z , a )  w  e w (,,0) (xw  , z , a )dw ,  2 i  i 

(Re( )  0,  , 



),

and definition (2.1), we are led finally to the left-hand side of formula (3.15). In the same manner one can derive formulas (3.16) and (3.17).

4. Fractional Derivative We recall the Riemann - Liouville fractional derivative operator Dx [2,p.181] and [4]

x

1 Dx f (x )

(

(x

)

1

t)

f (t )dt,(Re( )

0)

0

m

d Dx dx m

(4. 1) m

f (x ) , m

1

Re( )

m(m

N)

Clearly , we have : (

Dx {x }

1)

(

1)

x

,

(Re( )

1,

0) .

(4.2)

It is easy to derive the following results. Theorem 5: Let Re( )

Dx

x

1

( )

z

.Then

0,

x , z,

a

( ) x ( )

1

( , ) , ,

(x , z, a ) .

Proof: Denote, for convenience, the left-hand side of formula (4.3) by F l . Then in view of (1.3),it is easily seen that

61

(4.3)

Fl

Dx

( )

n 1

x 0 (a

z n

n )z

.

Upon using the relation (4.2), we are led finally to right -hand side of formula (4.3).

1 , (4.3) would reduce to the following result :

In particular , when

Dx

x

1

( ) x ( )

(x , z, a )

1

( , ) ,

(x , z, a ) .

(4.4)

1 in (4.3) , formula (4.3) reduces to the

On putting interesting result

* (x , z , a)

1 Dx ( )

1

x

1

(x, z, a ) .

(4.5)

1 in (4.3) , we arrive to the following interesting result

When ( , ) ,1,1

(x , z, a )

1 Dx ( )

1

x

1

(x , z, a ) .

(4.6)

REFERENCES [1]

Erdélyi, A., Magnw,W., Oberhettinger, F. and Tricomi, F.G.,Higher Transcendental Functions, Vol.I , McGraw – Hill book inc. New York, Toronto and London, 1953.

[2]

Erdélyi, A., Magnw,W., Oberhettinger, F. and Tricomi, F.G., Tables of Integral Transforms, Vol.2 , McGraw – Hill book inc. New York, Toronto and London, 1954.

[3]

Goyal, S, P. and Laddha, R. K., On the generalized Riemann zeta functions and the generalized Lambert transform, Gahnite Sandesh, Vol.2 Dec.(1997), 99-108.

[4]

Miller, K.S.and Rose, B., An introduction to The Fractional Calculus and Fractional Differential Equations, New York, 1993.

[5]

Srivastava, H. M., and Choi, J., Series associated with the zeta and Related function, Kluwee Academic Publishers, Dordrecht,2001

[6]

Shy-Der Lin, Srivastava, H. M., Some families of the Hurwitz-Lerch zeta functions and associated fractional derivative and other integral representations, Applied Mathematics and Computation 154(2004) 725-733.

[7]

Srivastava, H. M. and Karlsson, P. W., Multiple Gaussian Hypergeometric Series, Halsted Press, Brisbane, New York and Toronto 1985.

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