Integral and Computational representations of Generalized Hurwitz Lerch Zeta Function φ(a 1 z,s,a)

(1)

INTEGRAL AND COMPUTATIONAL REPRESENTATIONS OF GENERALIZED HURWITZ

ZETA FUNCTION

Department of Mathematics

ARTICLE INFO ABSTRACT

This article presents a systematic investigation of various integral and computational representations for family of General

Generalized Hurwitz

representation for generalized Riemann Zeta function is obtained as special cases of main results. In this paper w

representations of generalized

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

INTRODUCTION

Definitions:

(i)The generalized Hurwitz-Lerch Zeta function is defined by [2, p. 100, eq.(1.5)]; (see, p.121] and [6, p.194]).

Φ( , , ) = ∑ ( )

( ) ………

or Φ( , , ) = ∑ ( )

( ) ……….

| | < 1, ( ) > 0, ( ) > 0

(ii)Integral representation of the Generalized (1.6)]; see also [6, p.194, eq. 2.5 (4)] is given by

Φ( , , ) =

( )∫ ………..

or Φ( , , ) =

( )∫ ………..

*Corresponding author: Harish Nagar,

Department of mathematics, Mewar University, Gangrar,

ISSN: 0975-833X

Article History:

Received 23rd_{April, 2015} Received in revised form 19th_{May, 2015} Accepted 15th_{June, 2015} Published online 31st _July,₂₀₁₅

Key words:

Generalized Riemann Zeta function, Integral representations of special functions,

Generalized Hurwitz-Lerch Zeta Function, Gamma function.

Citation:Harish Nagar and Alka Tank, 2015. “

Function φ( , , )”, International Journal of Current Research

RESEARCH ARTICLE

INTEGRAL AND COMPUTATIONAL REPRESENTATIONS OF GENERALIZED HURWITZ

ZETA FUNCTION

(

, , )

*

_{Harish Nagar and Alka Tank}

Mathematics, Mewar University, Gangrar, Chittorgarh (Raj), India

ABSTRACT

This article presents a systematic investigation of various integral and computational representations for family of General Hurwitz – Lerch Zeta function Φ( , ,

Generalized Hurwitz – Lerch Zeta function Φ( , , )are established.Certain known integral representation for generalized Riemann Zeta function is obtained as special cases of main results. In this paper we established five results, these results presents the different forms of integral representations of generalized Hurwitz-Lerch Zeta function.

This is an open access article distributed under the Creative Commons Att use, distribution, and reproduction in any medium, provided the original work is properly cited.

Lerch Zeta function is defined by [2, p. 100, eq.(1.5)]; (see, eq. [1, p.27, eq. 1.11 (1)]; see also [5,

………

……….

Generalized Hurwitz-Lerch Zeta function Φ( , , )(see, eq. [1, p. 27, eq. 1.11 (3)]; [2, p. 100, eq. (1.6)]; see also [6, p.194, eq. 2.5 (4)] is given by

………..

Department of mathematics, Mewar University, Gangrar, Chittorgarh (Raj), India.

International Journal of Current Research

Vol. 7, Issue, 07, pp.18456-18461, July, 2015

2015. “Integral and Computational representations of Generalized Hurwitz International Journal of Current Research, 7, (7), 18456-18461.

INTEGRAL AND COMPUTATIONAL REPRESENTATIONS OF GENERALIZED HURWITZ-LERCH

Chittorgarh (Raj), India

This article presents a systematic investigation of various integral and computational representations

, ). In this paper new results for are established.Certain known integral representation for generalized Riemann Zeta function is obtained as special cases of main results. In nts the different forms of integral

is an open access article distributed under the Creative Commons Attribution License, which permits

[1, p.27, eq. 1.11 (1)]; see also [5,

……… (1.1)

………. (1.2)

[1, p. 27, eq. 1.11 (3)]; [2, p. 100, eq.

……….. (1.3)

……….. (1.4) OF CURRENT RESEARCH

(2)

(iii)The generalized Hurwitz-Lerch Zeta function defined in (1.1) reduce to Riemannian Zeta function ( ) for z = 1, the Hurwitz (or generalized) Zeta function ( , ) and Lerch Zeta function ℓ ( ) defined by (see, for details, [1, Chapter I]; [5, Chapter 2]; and [1, p. 27, eq. (1)].

( ) = ∑

( )= Φ(1, , 1); ( ) > 1 ………..… (1.5)

( , ) = ∑

( )= Φ(1, , );( ( ) > 1; ∈ ℂ\ ) ……….. (1.6)

andℓ ( ) = ∑

( )= Φ , , 1 ; ( ( ) > 1; ∈ ℝ), ………. (1.7)

(iv)The integral representation of Riemannian Zeta function [1, p. 32, eq. (1)].

ζ( , ) =

( )∫ ……….... (1.8)

( ) > 1, ( ) > 0

(v)The Gamma function defined by [3, p. 884, eq. 8.312 (2)]

∫ = ( )………... (1.9)

Results Required

The following results are required here

(i)The Gamma function is defined by [3, p. 884, eq. 8.312 (2)]

Γ( ) = ∫ , ( ) > 0, > 0………. (1.10)

(ii)The sum of finite and infinite geometric progression are also required

(see, [3, p. 1, eq. (0.112); p. 8, eq. (0.231) (1)].

∑ = ( )……….. (1.11)

[ ≠ 1]

∑ = , | | < 1………. (1.12)

Main Results

We have established here the following five results, these results presents the different forms of integral representations of Generalized Hurwitz – Lerch Zeta function.

Result – 1:

Φ( , , ) =

( )∫ 1 − ( )

( ) ( ) ₊

( )∫ ( )

( ) _{……… (2.1)}

( ) > 1, | | < 1, ( ) > 0, ( ) > 0

In particular

Φ( , − , ) =

( )∫ 1 − ( )

( ) ( ) ₊

( )∫ ( )

( ) _{……… (2.2)}

Result – 2:

Φ( , , ) =

( )∫ 1 − ( )

( ) ( ) ₊

( )∫ ( )

( )_{(1 − (} ₎ ₎ ₊

( )∫ ( )

( )

(3)

Provided that, ( ) > 1, | | < 1, ( ) > 0, ∈ , ∈ In particular

Φ( , − , ) =

( )∫ 1 − ( )

( ) ( ) ₊

( )∫ ( )

( )_{(1 − (} ₎ ₎ ₊

( )∫ ( )

( ) _{………. (2.4)}

Where ( ) < −1

Result – 3:

Φ( , , ) = ∑ ( ) ( )

( ) ∫

( )

1 − ( )( ) ( )

……… (2.5) ( ) > 1, | | < 1, ( ) > 0,

In particular

Φ( , − , ) = ∑ ( ) ( )

( ) ∫

( )

1 − ( )( ) ( ) ………. (2.6)

Where ( ) < −1

Result – 4:

Φ( , , ) =

( )∫ ( ) +

( ) ( ) ∫

( )

( ) ………... (2.7)

( ) > 1, | | < 1, ( ) > 0,

In particular

Φ( , − , ) =

( )∫ ( ) +

( ) ( )∫

( )

( ) ………. (2.8)

( ) < −1, | | < 1, ( ) > 0,

Result – 5:

Φ( , , ) = ( )∑

( )₍ ₎

(1 − ( ) ) ……….. (2.9)

( ) > 1, ∈ , | | < 1, ( ) > 0,

In particular

Φ( , − , ) = ₍ ₎∑ ( )( ) (1 − ( ) ) ……… (2.10)

Where ( ) < −1

Prof of (2.1) and (2.2)

To prove the result in (2.1) we have the generalized Zeta function defined in (1.2)

Φ( , , ) = ( ) ( + )

It can be written in the following form

Φ( , , ) = ∑ ( ) ( ) + ∑

( )

( ) ………. (2.11)

On using the definition of Gamma function and changing the order of summation and integration, we have

Φ( , , ) =

(4)

Prof of (2.3) and (2.4)

From result of (2.1)

∑ ( )

( ) = ( )∫ 1 − ( )

( ) ( )

………... (2.13)

In the same way as the proof of (2.1), we have

∑ ( )

( ) = ∑

( )

( ) ∫ ……… (2.14)

Changing the order of summation and integration than using the result in (1.11) we have

∑ ( )

( ) = ( )∫

( )( ) ( )

(1 − ( ) ) ………. (2.15)

And

( ) ( + ) =

( ) Γ( )

Changing the order of summation and integration, we have

( ) ( + ) =

1

Γ( ) ( )

on using the result in (1.12)

∑ ( )

( ) = ( )∫

( )( ) ( )

……….. (2.16)

We have (2.3) after summing (2.13), (2.15) and (2.16) since

Φ( , , ) = + + . ( )

( + ) ( ∈ ), ∈

The result in (2.4) is obtained on replacing s by –s in (2.3) Prof of (2.5) and (2.6)

In the right part of (1.1) put n=0 and using of definition of Gamma function we get

= _{( )}∫ ………... (2.17)

And

( )

( + ) =( + 1) + ( ) ( + 2)

( )

( + ) =Γ( )

( ) ₊( )

Γ( )

( )

( + ) =Γ( )

( ) 1 +

1 − (1 − )

∑ ( )

( ) = ( )∫

(5)

Similarly

∑ ( )

( ) =

( ) ( ) ∫

( ) ( ) _{………... (2.19)}

∑ ( )

( ) =

( ) ( ) ∫

( ) ( ) _{……… (2.20)}

And so on for > 0 ( ) > 1, then after the summing up (2.19) and (2.20) on wards. We get (2.5) Since

Φ( , , ) = + + + + … . ( ) ( + )

The result in (2.6) is obtained on replacing s by –s in (2.5) Prof of (2.7) and (2.8)

To prove the result in (2.7), we can be written (1.2) in the following form

Φ( , , ) = ∑ ( )

( ) + ∑

( )

( ) ………. (2.21)

On using (1.10), we have

Φ( , , ) = ∑ ( ) _{( )}∫ ( ) + ∑ ( ) _{( )}∫ ( ) ………..… (2.22)

Changing the order of summation and integration

Φ( , , ) = _{( )}∫ {∑ ( ) } + _{( )}∫ {∑ ( ) } ……… (2.23)

On summing the inner series in view of (1.12), we at once arrive the desired result in (2.7) The result in (2.8) is obtained on replacing s by –s in (2.7)

Prof of (2.9) and (2.10)

To prove the result in (2.9), we can write

∑ ( ) ( ) = ∑

( ) ( ) ∫

( ) _{………(2.24)}

Changing the order of summation and integration and after using (1.11), we have

∑ ( )

( ) = ( )∫ ……….. (2.25)

Similarly

( ) ( + ) =

( ) Γ( )

( )

∑ ( )

( ) = ( )∫ ∑ ( ) ……… (2.26)

On using (1.11)

∑ ( )

( ) = ( )∫

( ) ( ) [ _{……… (2.27)}

And

∑ ( )

( ) = ( )∫

(6)

By induction, we know that the (k+1)th summation is

∑ ( )

( ) = ( )∫

( ) ( ) [

( )

……… (2.29)

For ( ) > 0, ∈ , ∈

After the summation up to (2.25)~(2.29), we get (2.9) under the conditions. The result in (2.10) is obtained on replacing s by –s in (2.9)

Special Cases

(1)If in (2.1) to (2.10), we take z=1 and =1, then these results provided the known integral representations of generalized Hurwitz Zeta function ( , ) respectively [4, pp. 9-95, eqs. (3), (4); (12), (13); (20), (21); (28), (29); (32), (33)].

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