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
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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
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(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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