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On Multiple Zeta Function and Associated Properties

Maged G. Bin-Saad , M. A. Pathan, Ali Z. Bin-Alhag
Turkish Journal of Analysis and Number Theory. 2018, 6(3), 84-89. DOI: 10.12691/tjant-6-3-3
Received January 01, 2018; Revised April 06, 2018; Accepted May 28, 2018

Abstract

The purpose of this paper is to introduce and investigate a new class of multiple zeta functions of variables. We study its properties, integral representations, differential relation, series expansion and discuss the link with known results.

1. Introduction

The generalized (Hurwitz’s) zeta function is defined by 1, 2

(1.1)

so that when , we have

(1.2)

where is the Riemann zeta function. The function extends (1.1) further, and this generalized Hurwitz-Lerch zeta function [ 1, p. 316], is defined by

(1.3)

A generalization of (1.3) is the Zeta function which is defined by [ 3, p.100, (1.5)]:

(1.4)

where for denotes the Pochhammer’s symbol and denotes the Gamma function. Evidently, we have

(1.5)
(1.6)
(1.7)

The zeta functions in (1.3) and (1.4) have since been extended and generalized by a number of workers (see e.g. 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14. The present sequel to these earlier papers is motivated largely by the aforementioned works of Matsumoto and Kamano 15, 16 in which the zeta function in (1.1) was generalized to the following multiple Hurwitz zeta function

(1.8)

In the present paper we introduce a new class of zeta functions which is defined by

(1.9)

Clearly, we have the following relationship

In the case when we have simply

which implies the next result.

Corollary 1.1. Let

Then

(1.10)

where is the Lauricella function of variables defined by the series (see e.g. 17 and 18),

(1.11)

By using the Hankel's contour integral for Gamma function ( for details see e.g. [ 19, Section 12.12])

(1.12)

we can derive the following interesting formula.

Corollary 1.2. Let Then

(1.13)

Proof. The result follow directly from definition (1.9) and the integral representation (1.11).

2. Integral Representations

First, by using Eulerian integral formula of the second kind (see e.g. 2):

(2.1)

and the formula(see, e.g., [ 20, p.67, Eq (22)]) :

(2.2)

we should proof the following result.

Theorem 2.1. Let and either

Then

(2.3)

Proof. Denote, for convenience, the right-hand side of equation (2.3) by I. Then in view of (2.2), it is easily seen that

Upon using (2.1) and in view of the definition (1.9), we led finally to the left-hand side of the formula (2.3). Next, by using the contour integral formula [ 2, p.14 (4)]:

(2.4)

one can derive the following contour integral representation.

Theorem 2.2. Let and

Then

(2.5)

Proof. Denote, for convenience, the right-hand side of equation (2.5) by I. Then in view of (2.2), it is easily seen that

Upon using (2.4) and in view of the definition (1.9) with the reflection formula

for Gamma function, we led finally to the left-hand side of the formula (2.5).

Further, we evaluate some definite integrals involving the function First, we recall the Eulerian integral formula of first kind (cf. e.g 12):

(2.6)

From the term-by-term integration, we can derive the following formula.

Theorem 2.3. Let

(2.7)

Proof. Denote for convenience the left-hand side of equation (2.7) by I.

Then in view of definition (1.9), it is easily seen that

Upon using (2.6) and the relation

we are finally led to the right-hand of the relation (2.7).

Finally, we prove the following result.

Theorem 2.4. Let Then

(2.8)

Proof. Setting and

in the Eulerian Beta function formula (see, e.g., [ 20, p.8, Eq. (45)]):

we find that

which, by appealing to the definition (1.9), immediately yields the assertion (2.8).

3. Operational Relations

The multiple Zeta function satisfies some operational relations. Fortunately these properties of be developed directly from the definition (1.9). First, by recalling the familiar derivative formula from calculus in terms of the gamma function 8

(3.1)

where we aim now to derive the following differential relations for

Theorem 3.1. Let Then

(3.2)

Proof. By starting from the left-hand side of (3.2) and in view of (1.9) and by using the relation (3.1), we get:

(3.3)

Now, letting in (3.3) using the formula and considering the definition (1.9), we get the right-hand side of formula (3.2).

Theorem 3.2. Let

Then

(3.4)

Proof. By starting from the left-hand side of (3.4) and in view of (1.9) and by using the relation

we get:

Now, considering the definition (1.9), we get the right-hand side of formula (3.4).

Closely associated with the derivative of the gamma function is the digamma function defined by [ 21, p.74(2.51)]:

(3.5)

Now, we wish to establish the derivative of the function with respect to the parameters

Theorem 3.3. Let Then

(3.6)

Proof. By starting from the left-hand side of formula (3.6) and by using the relation (3.5), then according to the result by using:

(3.7)

we obtain the right-hand side of formula (3.6).

Next, let us recall the definition of the Weyl fractional derivative of exponential function of order in the form (see [ 22, p.248(7.4)] ):

(3.8)

We now proceed to find the fractional derivative of the function with respect to .

Theorem 3.4. Let Then

(3.9)

Proof. Since we have

The desired result now follows by applying the formula (3.8) to the above identity.

4. Series Expansions

First we derive the following basic sums of series

Theorem 4.1. Let . Then

(4.1)

Proof. If in formula (1.9), we replace by multiply throughout by and then sums up, we get (4.1).

Further, from definition (1.9) we easily have the following interesting series relation.

Theorem 4.2. Let Then

(4.2)
(4.3)

Proof. Since

it is easily seen that :

The desired result (4.2) now follows by using definition (1.9). Also, by right-hand formula (4.3) and using (3.4) we get to left-hand side of formula (4.3).

Theorem 4.3. Let Then

(4.4)

where

(4.5)

where

(4.6)

where and

(4.7)

where are the Lauricella's hypergeometric functions of -variables (see [ 18, p.60, Eq (1), (2), (3), (4)]).

Proof. We refer to the proof of Theorem 4.2.

Theorem 4.4. Let Then

(4.8)

Proof. By starting from the right-hand side of (4.8) and in view of (1.9) and by using the relation

we get the left-hand side of formula (4.8).

Theorem 4.5. Let and Then

(4.9)

where is the Appell's function of two variables defined by the series 18

Proof. By starting from the right-hand side of (4.9) and in view of (1.9), we get

Since

Hence, the right-hand side of formula (4.9) follows.

References

[1]  Chaudhry M. A. and Zubair S. M., On a Class of Incomplete Gamma Functions with Applications, Chapman and Hall, (CRC Press Company), Boca Raton, London, New York and Washington, D. C., 2001.
In article      View Article
 
[2]  Erdélyi,A., Magnus,W., Oberhettinger, F. and Tricomi, F.G., Higher Transcendental Functions, Vol.I , McGraw – Hill book inc. New York, Toronto and London, 1953.
In article      
 
[3]  Jankov, D., Pogány T. K. and Saxena, R. K. An extended general Hurwitz-Lerch Zeta function as a Mathieu (a,λ)-series, Appl. Math. Lett., 24, 1473-1476,2011.
In article      View Article
 
[4]  Bin-Saad Maged G. , "Sums and partial sums of double power series associated with the generalized zeta function and their N-fractional calculus", Math. J. Okayama University, 49, 37-52, 2007.
In article      View Article
 
[5]  Bin-Saad Maged G.,"Hypergeometric Series Associated with the Hurwitz-Lerch Zeta Function", Acta Math. Univ. Comenianae, 2, 269-286, 2009.
In article      View Article
 
[6]  Bin-Saad Maged G. and Al Gonah, A.A., On hypergeometric type generating functions associated with generalized zeta function, Acta Math. Univ. Comenianae, 2, 253-266, 2006
In article      
 
[7]  Bin-Saad Maged G., Pathan M. A. and Hanballa Amani M.,On power series associated with generalized multiple zeta function, Math. Sci. Res. J. 17(10) 279-291, 2013.
In article      View Article
 
[8]  Choi J., Multiple gamma function and their applications, in Proc. Internat. Conf. on Analysis (editied by Y.C. Kim), Yeungnam Univeristy, Korea, 73-84,1996.
In article      
 
[9]  Choi J., Jang D. S. and Srivastava H. M., A generalization of the Hurwitz-Lerch Zeta function, Integral Transforms Spec. Funct. 19, 65-79, 2008.
In article      View Article
 
[10]  Goyal, S. and Laddha, R.K. , On the Generalized Riemann Zeta Funcion and the Generalized Lambert Transform, Ganita Sandesh, 11, 99-108, 1997.
In article      
 
[11]  Lin S. D. and Srivastava H. M., Some families of the Hurwitz-Lerch Zeta functions and associated fractional derivative and other integral representations, Appl. Math. Comput. 154, 725-733, 2004.
In article      View Article
 
[12]  Srivastava, H. M., M. J. Luo and. Raina, R. K, New results involving a class of generalized Hurwitz-Lerch Zeta functions and their applications, Turkish J. Anal. Number Theory 1(1), 26-35, 2013.
In article      View Article
 
[13]  Srivastava, H. M., R. K. Saxena, T. K. Pogány and R. Saxena, Integral and computational representations of the extended Hurwitz-Lerch Zeta function, Integral Transforms Spec. Funct. 22(7) , 487-506, 2011.
In article      View Article
 
[14]  Srivastava, H. M., D. Jankov, Pogány, D., T. K. and R. K. Saxena, Two-sided inequalities for the extended Hurwitz-Lerch Zeta function, Comput. Math. Appl. 62 (2011), 516-522
In article      View Article
 
[15]  Kamano K., The multiple Hurwitz Zeta function and a generalization of Lerch’s formula, Tokyo J. Math. 29, 61-73, 2006.
In article      View Article
 
[16]  Matsumoto, K., The analytic continuation and the asymptonic behaviour of certain multiple zeta-functions I, J. Number Theory 101, 223-243,2003.
In article      View Article
 
[17]  Srivastava, H. M. and Karlsson, P. W., Multiple Gaussian Hypergeometric Series, Halsted Press, Brisbane, London, New York, 1985.
In article      
 
[18]  Srivastava, H. M. and Manoch, H. L., A treatise on Generating Functions, Halsted Press, Brisbane, London, New York, 1984.
In article      
 
[19]  Whittaker, E. T. and Watson, G. N., A course of modern Analysis, Fourth Edition, Cambridge Uni. Press, 1952.
In article      
 
[20]  Srivastava H. M. and Choi J., Zeta and q-Zeta Functions and Associate Series and Integrals, Elsevier Science, Publishers, Amsterdam, London and New York, 2012.
In article      View Article
 
[21]  Andrews, L.C., Special Functions for Engineers and Applied Mathematician, Mac-Millan, New York, 1985.
In article      View Article
 
[22]  Miller, K.S. and Rose, B., An introduction to The Fractional Calculus and Fractional Differential Equations, New York, 1993.
In article      View Article
 
[23]  Lauricella, G., Sulle funzioni ipergeometriche a piu variabili. Rend. Circ. Mat. Palermo 7, 111-158, 1893.
In article      View Article
 

Published with license by Science and Education Publishing, Copyright © 2018 Maged G. Bin-Saad, M. A. Pathan and Ali Z. Bin-Alhag

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Maged G. Bin-Saad, M. A. Pathan, Ali Z. Bin-Alhag. On Multiple Zeta Function and Associated Properties. Turkish Journal of Analysis and Number Theory. Vol. 6, No. 3, 2018, pp 84-89. http://pubs.sciepub.com/tjant/6/3/3
MLA Style
Bin-Saad, Maged G., M. A. Pathan, and Ali Z. Bin-Alhag. "On Multiple Zeta Function and Associated Properties." Turkish Journal of Analysis and Number Theory 6.3 (2018): 84-89.
APA Style
Bin-Saad, M. G. , Pathan, M. A. , & Bin-Alhag, A. Z. (2018). On Multiple Zeta Function and Associated Properties. Turkish Journal of Analysis and Number Theory, 6(3), 84-89.
Chicago Style
Bin-Saad, Maged G., M. A. Pathan, and Ali Z. Bin-Alhag. "On Multiple Zeta Function and Associated Properties." Turkish Journal of Analysis and Number Theory 6, no. 3 (2018): 84-89.
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[1]  Chaudhry M. A. and Zubair S. M., On a Class of Incomplete Gamma Functions with Applications, Chapman and Hall, (CRC Press Company), Boca Raton, London, New York and Washington, D. C., 2001.
In article      View Article
 
[2]  Erdélyi,A., Magnus,W., Oberhettinger, F. and Tricomi, F.G., Higher Transcendental Functions, Vol.I , McGraw – Hill book inc. New York, Toronto and London, 1953.
In article      
 
[3]  Jankov, D., Pogány T. K. and Saxena, R. K. An extended general Hurwitz-Lerch Zeta function as a Mathieu (a,λ)-series, Appl. Math. Lett., 24, 1473-1476,2011.
In article      View Article
 
[4]  Bin-Saad Maged G. , "Sums and partial sums of double power series associated with the generalized zeta function and their N-fractional calculus", Math. J. Okayama University, 49, 37-52, 2007.
In article      View Article
 
[5]  Bin-Saad Maged G.,"Hypergeometric Series Associated with the Hurwitz-Lerch Zeta Function", Acta Math. Univ. Comenianae, 2, 269-286, 2009.
In article      View Article
 
[6]  Bin-Saad Maged G. and Al Gonah, A.A., On hypergeometric type generating functions associated with generalized zeta function, Acta Math. Univ. Comenianae, 2, 253-266, 2006
In article      
 
[7]  Bin-Saad Maged G., Pathan M. A. and Hanballa Amani M.,On power series associated with generalized multiple zeta function, Math. Sci. Res. J. 17(10) 279-291, 2013.
In article      View Article
 
[8]  Choi J., Multiple gamma function and their applications, in Proc. Internat. Conf. on Analysis (editied by Y.C. Kim), Yeungnam Univeristy, Korea, 73-84,1996.
In article      
 
[9]  Choi J., Jang D. S. and Srivastava H. M., A generalization of the Hurwitz-Lerch Zeta function, Integral Transforms Spec. Funct. 19, 65-79, 2008.
In article      View Article
 
[10]  Goyal, S. and Laddha, R.K. , On the Generalized Riemann Zeta Funcion and the Generalized Lambert Transform, Ganita Sandesh, 11, 99-108, 1997.
In article      
 
[11]  Lin S. D. and Srivastava H. M., Some families of the Hurwitz-Lerch Zeta functions and associated fractional derivative and other integral representations, Appl. Math. Comput. 154, 725-733, 2004.
In article      View Article
 
[12]  Srivastava, H. M., M. J. Luo and. Raina, R. K, New results involving a class of generalized Hurwitz-Lerch Zeta functions and their applications, Turkish J. Anal. Number Theory 1(1), 26-35, 2013.
In article      View Article
 
[13]  Srivastava, H. M., R. K. Saxena, T. K. Pogány and R. Saxena, Integral and computational representations of the extended Hurwitz-Lerch Zeta function, Integral Transforms Spec. Funct. 22(7) , 487-506, 2011.
In article      View Article
 
[14]  Srivastava, H. M., D. Jankov, Pogány, D., T. K. and R. K. Saxena, Two-sided inequalities for the extended Hurwitz-Lerch Zeta function, Comput. Math. Appl. 62 (2011), 516-522
In article      View Article
 
[15]  Kamano K., The multiple Hurwitz Zeta function and a generalization of Lerch’s formula, Tokyo J. Math. 29, 61-73, 2006.
In article      View Article
 
[16]  Matsumoto, K., The analytic continuation and the asymptonic behaviour of certain multiple zeta-functions I, J. Number Theory 101, 223-243,2003.
In article      View Article
 
[17]  Srivastava, H. M. and Karlsson, P. W., Multiple Gaussian Hypergeometric Series, Halsted Press, Brisbane, London, New York, 1985.
In article      
 
[18]  Srivastava, H. M. and Manoch, H. L., A treatise on Generating Functions, Halsted Press, Brisbane, London, New York, 1984.
In article      
 
[19]  Whittaker, E. T. and Watson, G. N., A course of modern Analysis, Fourth Edition, Cambridge Uni. Press, 1952.
In article      
 
[20]  Srivastava H. M. and Choi J., Zeta and q-Zeta Functions and Associate Series and Integrals, Elsevier Science, Publishers, Amsterdam, London and New York, 2012.
In article      View Article
 
[21]  Andrews, L.C., Special Functions for Engineers and Applied Mathematician, Mac-Millan, New York, 1985.
In article      View Article
 
[22]  Miller, K.S. and Rose, B., An introduction to The Fractional Calculus and Fractional Differential Equations, New York, 1993.
In article      View Article
 
[23]  Lauricella, G., Sulle funzioni ipergeometriche a piu variabili. Rend. Circ. Mat. Palermo 7, 111-158, 1893.
In article      View Article