Integral expressions for Hilbert-type infinite multilinear form and related multiple Hurwitz-Lerch Zeta functions

Available online at www.ispacs.com/jiasc Volume 2012, Year 2012 Article ID jiasc-00006, 12 pages

doi:10.5899/2012/jiasc-00006 Research Article

Integral expressions for Hilbert–type infinite

multilinear form and related multiple

Hurwitz–Lerch Zeta functions

Ram K. Saxena1_{, Tibor K. Pog´any}2∗

(1)Department of Mathematics and Statistics, Jai Narain Vyas University, Jodhpur – 342004, India (2)Faculty of Maritime Studies, University of Rijeka, 51000 Rijeka, Croatia

Copyright 2012 c⃝Ram K. Saxena and Tibor Pog´any. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The article deals with different kinds integral expressions concerning multiple Hurwitz– Lerch Zeta function (introduced originally by Barnes [4]), Hilbert–type infinite multilinear form and its power series extension. Here Laplace integral forms and multiple Mellin– Barnes type integral representation are derived for these special functions. As a special cases of our investigations we deduce the integral expressions for the Matsumoto’s multiple Mordell–Tornheim Zeta function, that is, for Tornheim’s double sum i.e. Mordell–Witten Zeta, for the multiple Hurwitz Zeta and for the multiple Hurwitz–Euler Eta function, re-cently studied by Choi and Srivastava [7].

Keywords : Multiple Hurwitz–Lerch Zeta function; Hilbert–type infinite multilinear form; Mul-tiple Hurwitz–Lerch Zeta power series; Tornheim’s double sum; Mordell–Witten Zeta function; Matsumoto’s multiple Mordell–Tornheim Zeta function; Dirichlet–series; Cahen’s Laplace integral formula; Mellin–Barnes type integral.

1

Introduction

The general Hurwitz–Lerch Zeta function (HLZ), denoted by Φ(z, s, a), is defined as [10], also see [22]

Φ(z, s, a) =∑ n≥0

zn

(n+a)s, (1.1)

where a ∈ C_\ Z−

0;ℜ(s) > 1 when |z| = 1 and s ∈ C when |z| < 1 and continues

meromorphically to the complex s–plane, except for the simple pole at s = 1, with its residue equal to 1.

The function Φ(z, s, a) has many special cases such as Riemann Zeta [10], Hurwitz Zeta [22] and Lerch Zeta function [28]. Some other special cases involve the polylogarithm (or Jonqi`ere’s function) Lis(z) = zΦ(z, s,1) [28], and the Lipschitz–Lerch Zeta function Φ(e2πiξ, s,1) [22]. A multiple HLZ is studied by Choi et al. [8]. Other generalizations of HLZ are discussed by Lin–Srivastava [15], Garg–Jain–Kalla [12, 13], Lin–Srivastava– Wang [16], Goyal–Laddha [14], Srivastava–Saxena–Pog´any–Saxena [23] and others. The generalizedd–ple Hurwitz Zeta functionζd(s;a|z1,· · ·, zd) has been introduced and studied by Barnes [4] (see also the earlier papers [1, 2, 3]), for ℜ(s) > d, d∈ N_{, by the multiple} series

ζd(s;a|ν) :=

∑

n∈Nd

0

1 (a+n◦ν)s ;

here and in what follows x = (x1,· · · , xd), N0 = N∪ {0} = {0,1,2,· · · }, and x◦y =

∑d

j=1xjyj stands for the inner product ofd–dimensional vectorsx,y. Choi and Srivastava devoted their article [7] to different type integral representations either for ζd(s;a|1) or to the so–called alternating multiple generalized Hurwitz Zeta (or, equivalently Hurwitz– Euler Eta function)

ηd(s, a) :=

∑

n∈Nd

0

(−1)n◦1

(a+n◦1)s

where ℜ(s)>0;a >0;d∈N _{[7]. Obviously η1(s,1)≡η(s) is the Dirichlet Eta function.} Inspiring by ηd(s, a) we will consider the extended variant of Hurwitz–Euler Eta (at the same time the alternating variant ofζd), defined by

ηd(s;a|ν) :=

∑

n∈Nd

0

(−1)n◦₁

(a+n◦ν)s, ηd(s;a|1)≡ηd(s, a).

Another kind of special function closely connected to HLZ and/or their expansions is the Hilbert–type infinite bilinear form [19]

Ha_λ,_,b_ρ(s) := ∑ m,n≥1

ambn (λn+ρn)s

,

to derive deeper general Hilbert’s double series theorems with respect to the non–homoge-neous kernel (λm+ρm)−s. Here λ

_N = λ = (λn)n≥1, ρ

_N =ρ = (ρn)n≥1 are restrictions

coming from monotone increasing functions λ, ρ:R_{+ 7→}R_{+ which behave}

lim x→∞

{ _λ

ρ

}

Also, Hilbert–type infinite d–linear form H_d(a, s;α,λ) was introduced by Draˇsˇci´c Ban– Peˇcari´c–Peri´c–Pog´any [9], by the followingd–ple series:

H_d(a, s;α,λ) := ∑ n∈Nd

d

∏

j=1 αj(nj)

(a+λ1(n1) +· · ·+λd(nd))s

, (1.3)

for all a, s >0, d∈N_{2 ={2,3,· · · }. Here αj = (αj(nj)), nj ≥1, j= 1, dare nonnegative} sequences andλ1,· · ·, λd:R+7→R+are monotone increasing positive functions possesing

property (1.2). Let us remark that the main authors’ goal in [9] was to derive a sharp discrete multiple Hilbert–type inequalities for (1.3), by finding sharp upper bounds for the related Laplace–integral expressions obtained for H_d(a, s;α,λ).

It is straightforward that H_d(a, s;α,λ) significantly generalizes ζd(s;a|ν), as

H_d(a, s;1,I ◦ν) +a−s=ζd(s;a|ν) ; (1.4)

here I denotes the identity transformation applied to summation indices d–ple sequence n. On the other hand integral representations achieved for H_d(a, s;1,I ◦ν) in [19] and [9] can be easily extended to less restrictive real sequences αj, j = 1, d covered by this Hurwitz–Euler Eta function as well:

H_d(a, s; (−1),I ◦ν) +a−s =ηd(s;a|ν), (1.5)

writing (−1) := (−1)n◦1_.

In 1950 Tornheim [25] introduced the double series

T(p, q, r) = ∑ n∈N2

1

np₁nq₂(n1+n2)r p, q, r≥0, r+ min{p, q}>1,

In honour to the author it is called Tornheim double series (also known in literature as Witten Zeta function or Mordell series). Later, various continuations of the domain for the functionT(p, q, r) are given [5, 11, 24, 26, 27]. We point out that

T(p, q, r) =H₂(0, r;N−pN−q_{,I ◦1),}

where N−p_N−q _{denotes the coefficients sequence (n}−p

1 n

−q

2 ), n1, n2 ≥ 1. Matsumoto [17]

introduced the related, so–called Mordell-Tornheimd–ple Zeta function in the form

ζM T,d(r;s) =

∑

n∈Nd

1

nr1

1 · · ·n

rd

d (n1+· · ·+nd)s

,

where r := (r1, . . . , rd);rj, s ∈ C. The series (1.5) is absolutely convergent for ℜ(rj) > 1, j = 1, d and ℜ(s) > 0. It is obvious that by taking d = 2 and real parameters (1.5) it reduces to (1.4). Convergence conditions for ζM T,d(r;s) are prescribed in [17]. It is interesting to mention that

ζM T,d(r;s) =Hd(0, s;N−r,I ◦1), (1.6)

whereN−r _:=(_n−r1

1 · · ·n

−rd d

)

Obviously, the most general special function here considered is the Hilbert–type infinite

d–linear formH_d(a, s;α,λ) defined by (1.3). Finally, it is known that [10]

Φ(z, s, a) = 1 Γ(s)

∫ ∞

0

ts−1_e−(a−1)t et_−z dt .

Another integral representation results are derived by Draˇsˇci´c Ban–Peˇcari´c–Peri´c–Pog´any [9]

ζM T,d(r;s) =

1

2d_Γ(s) ∏d j=1

Γ(rj)

∫

Rd₊+1

xs−1

edx+t1 +2···+td

d

∏

j=1

trj_j −1

sinhx+₂tj dxdt,

where r1 ≥0,· · ·, rd≥0,min1≤j≤d(rj) +d(s−1)>0 and dt=∏d_j=1dtj. In the case of Mordell–Tornheim Zeta function this result reduces to [9]

T(p, q, r) = 1 4Γ(p)Γ(q)Γ(r)

∫ ∞

0

∫ ∞

0

∫ ∞

0

xr−1_tp−1 1 t

q−1

2 e

−x−t1 +t2 2

sinhx+t1

2 sinh

x+t2

2

dxdt1dt2.

Laplace integral representation formulae, established by the associated Dirichlet series’ have been presented also in [9] (see also [19] for the case d= 2, a= 0).

At this point let us introduce thed–variate Hilbert type infinited–linear form function

e

H_d(a, s;α,λ|z) := ∑ n∈Nd

d

∏

j=1

αj(nj)zjλj(nj)

(a+λ1(n1) +· · ·+λd(nd))s

. (1.7)

Clearly He_d(a, s;α,λ|1) ≡ H_d(a, s;α,λ). So, the associated Zeta and Eta functions be-come

e

ζd(s;a|ν,z) :=

∑

n∈Nd

0

d

∏

j=1 z_jνjnj

(a+n◦ν)s, ηed(s;a|ν,z) :=

∑

n∈Nd

0

d

∏

j=1

(−zνj_j )nj

(a+n◦ν)s. (1.8)

In this research article we establish various integral representation formulae for this type of multiple series and some of its special cases as ζd andηd: 1. multiple Laplace–integral expression and 2. Mellin–Barnes type of integral formulae. Finally, multidimensional Mellin transforms of ζd andηdare also discussed.

2

Laplace–integral representation formula for

H

e

_{(s, a;}

α

,

λ

|

z

)

In the beginning, we recall the widely known Gamma function property

∫

R+

xs−1_e−Ax_dx= Γ(s)

Theorem 2.1. Let monotoneλ= (λ1,· · ·, λd) :Rd+7→Rd+have property (1.2)and possess

restrictions λj

_N= (λj(nj))nj≥1, j = 1, d. Parameters a >0, s > 0, ν = (ν1,· · ·, νd) >0

and variable z= (z1,· · · , zd)∈Cd satisfy convergence condition

lim sup m→∞

|αj(m)|1/m|zj|λj(m)/m

λj(m)s/(d·m)

<1, j= 1, d . (2.10)

Then we have the Laplace–integral representation formula

e

H_d(a, s;α,λ|z) =

∫

Rd

+ { _d

∏

j=1 [λ−1

j_∑(tj)]

nj=1

αj(nj)zjλj(nj)

}

· dt

(a+t◦1)s+d, (2.11)

where [x] denotes the integer part of somex and λ−1 _{stands for the inverse of λ.}

Proof. First, we establish the convergence of the Hilbert’sd–linear form (1.3). By virtue of the arithmetic mean–geometric mean (A–G) inequality applied to the positive denominator of He_d, we obtain

eH_d(a, s;α,λ|z)≤

d

∏

j=1

∑

nj≥1

|αj(nj)||zj|λj(nj) (a+λj(nj))s/d

∼

d

∏

j=1

∑

nj≥1

|αj(nj)||zj|λj(nj)

λj(nj)s/d ;

by virtue of the Cauchy’s convergence testHe_d absolutely converges.

On the other hand let us express the denominator using the Gamma function formula (2.9):

e

H_d(a, s;α,λ|z) = 1 Γ(s)

∫

R+

xs−1e−ax d

∏

j=1

Dj(zj, x) dx , (2.12)

where the Dirichlet series

Dj(zj, x) :=

∑

nj≥1

αj(nj)z_jλj(nj)e−λj(nj)x

has the Cahen’s Laplace integral representation formula [6], which has been exactly proved by Perron [18] (also see e.g. [21]):

Dj(zj, x) =x

∫

R+

e−xtj

{ ∑

n:λj(n)≤tj

αj(n)zjλj(n)

}

dtj j= 1, d .

As λj is monotone increasing, it possesses an unique inverse λ−_j1, so 1 ≤n ≤[λ−_j1(tj)]. Substituting the previous Cahen’s integrals into (2.12):

e

H_d(s, a;α,λ|z) =

∫

Rd₊+1

xs+d−1_e−(a+t◦1)x

{ _d ∏

j=1 [λ−1

j_∑(tj)]

nj=1

αj(nj)zλjj (nj)

}

dxdt.

Calculating the innerx–integral by (2.9) withA=a+t◦1, we complete the proof.

Theorem 2.2. Assume thata >0,ν >0;d∈N_{2 andℜ(s)> dwhen|zj|= 1, whiles∈}C

when |zj|νj <1, j= 1, d. Then for all |zj|<1, j = 1, d we have

e

ζd(s;a|ν,z) =a−s+ d

∏

j=1 z_jνj zνj_j −1 ·

∫ Rd + d ∏ j=1

(z_jνj[tj/νj]−1)

(a+t◦1)s+d dt (2.13)

e

ηd(s;a|ν,z) =a−s+ d

∏

j=1 z_jνj zνj_j + 1·

∫ Rd + d ∏ j=1 (

(−z_jνj)[tj/νj]−1)

(a+t◦1)s+d dt. (2.14)

Proof. By applying the A–G inequality to ζed we deduce the inequality

eζd(s;a|ν,z)

≤d−s

d

∏

j=1

ν_j−s/d ∑

n∈Nd

0

d

∏

j=1

|zj|νjnj

(

a dνj +nj

)s/d

={d

d

∏

j=1

ν_j1/d}−s

d

∏

j=1

Φ(|zj|νj,

s d,

a d·νj

)

.

The resulting HLZ functions converge for a/(d·νj) ̸∈ Z−

0;ℜ(s) > d when |zj| = 1 and s∈C_{when |zj|}νj _{<1, j= 1, d, (see(1.1)). However, these constraints are guaranteed by} the assumptions of the Theorem.

Next, having in mind (1.4), (1.8) and (2.11), and because λ−_j1(x) =x/νj, j= 2, d, we get

e

ζd(s;a|ν,z) =a−s+Hed(a, s;1,I ◦ν|z) =a−s+

∫ Rd + d ∏ j=1 [tj/νj_∑ ]

nj=1 zνjnj_j

(a+t◦1)s+d dt, (2.15)

which is equivalent to the first assertion (2.13) of the Theorem 2.2. Relation (2.14) we prove in completely similar manner.

By taking z_jνj = 1, j= 1, din Theorem 2.2, we deduce from (2.15) the following.

Corollary 2.1. Let a >0, s∈C_{,ℜ(s)> d and ν >0. Then it holds true}

ζd(s;a|ν) =a−s+

∫ Rd + d ∏ j=1 [ tj νj ]

· dt

(a+t◦1)s+d (2.16)

ηd(s;a|ν) =a−s+

∫ Rd + d ∏ j=1 sin2 ( π 2 [ tj νj ])

· dt

(a+t◦1)s+d. (2.17)

Proof. Relation (2.16) is straightforward. Since

[tj/νj]

∑

nj=1

(−1)nj = 1 2

(

1−(−1)[tj/νj])= sin2

( π 2 [ tj νj ]) ,

The sequel part of this section we devote to new integral representations for Mat-sumoto’s multiple Mordell–Tornheim Zeta function ζM T,d(r;s), that is for Tornheim’s double sum (or Mordell–Witten Zeta function) T(p, q, r). We recall that relation (1.6) connects Hilbert’s d–linear formHe_d(a, s;α,λ) with ζM T,d(r;s) and directly generates the associated d–variate power series

e

ζM T,d(r;s|z) :=Hed(0, s;N−r,I ◦1|z) =

∑

n∈Nd d

∏

j=1 z_jnj

nr_(n◦1)s ,

wherenr:=nr1

1 · · ·nrdd . Obviously

e

T(p, q, r|z) :=ζeM T,2

(

(p, q);r|(z1, z2)

)

.

Now, employing Theorem 2.1 we deduce the following result.

Theorem 2.3. Let d ∈ N_{2, rj > 0, s ∈} C _{when |zj| < 1, j = 1, d and rj +ℜ(s)/d > 1}

when |zj|= 1, j = 1, d. Then we have

e

ζM T,d(r;s|z) = d

∏

j=1 zj Γ(rj) ·

∫

R2d

+

d

∏

j=1

xrj_j −1{1−(e−xj_z j)[tj]

}

exj _−z j

dtdx

(t◦1)s+d. (2.18)

Proof. First of all, let us establish the convergence domain of ζeM T,d(r;s|z). By the A–G inequality we conclude:

eζM T,d(r;s|z)

≤d−s

d

∏

j=1

∑

nj≥1

|zj|nj

nrj_j +s/d

=d−s d

∏

j=1

Li_rj+s/d(|zj|).

The resulting polylogarithmic/de Jonqi`ere’s functions converge according to the theorem’s assumed parameter space, that is for d∈ N_{2, rj > 0, s ∈} C _{when |zj| < 1, j = 1, d and}

rj +ℜ(s)/d >1 when|zj|= 1, j = 1, d. Further, specfying αj(xj) = x

−rj

j , λj(xj) = I(x), j = 1, d and ν ≡ 1, Theorem 2.1 gives (2.18). Indeed, by quoted specifications we get

e

ζM T,d(r;s|z) =

∫ Rd + d ∏ j=1 [tj]

∑

nj=1 z_jnj nrj_j

dt (t◦1)s+d

= ∫ Rd + d ∏ j=1 1 Γ(rj)

∫

R+

xrj_j −1 [tj]

∑

j=1

(

e−xjzj)njdxj ·

dt (t◦1)s+d.

Now, obvious steps lead to the asserted integral expression.

Putting zj = 1, j= 1, din (2.18), we get the following

Corollary 2.2. Let d∈N_{2, rj >0 ands∈}C_{, rj+ℜ(s)/d >1. Then we have}

ζM T,d(r;s) = d

∏

j=1

1 Γ(rj)

·

∫

R2d

+

d

∏

j=1

xrj_j −1(1−e−xj[tj])

exj ₋₁

Finally, the related integral expressions for the Tornheim double sum become:

e

T(p, q, r|z) = z1z2 Γ(p)Γ(q)·

∫

R4 +

xp₁−1x₂q−1(1−(e−x1_z

1)[t1]

)(

1−(e−x2_z

2)[t2]

)

(ex1−z

1)(ex1−z1)(t1+t2)s+2

dtdx,

T(p, q, r) = 1 Γ(p)Γ(q)·

∫

R4 +

xp₁−1xq₂−1(1−e−x1[t1])(_1−e−x2[t2])

(ex1 −1)(ex1 −1)(t

1+t2)s+2

dtdx.

3

Mellin–Barnes integral formulae

This section deals with the derivation of Mellin–Barnes integral representation for the functionHe_d(a, s;α,λ|z) defined in (1.7).

Theorem 3.1. Let all paramaters and variables of He_d(a, s;α,λ|z) satisfy (2.10). Then there holds true the formula

e

H_d(a, s;α,λ|z) = 1 (2πi)d

∫

iRd d

∏

j=1

{

Γ(−ξj)Γ(1 +ξj)(−zj)ξj

}

(

a+λ1(ξ1) +· · ·+λd(ξd)

)s dξ, (3.19)

where |arg(−zj)|< π, j = 1, d. Here

∫

iRd stands for d–ple line integral on the imaginary

axis, that is ∫₋i∞_i∞ and dξ := dξ1· · ·dξd.

Proof. Assuming that non of the poles of the Gamma functions Γ(−ξj) atξj =nj,nj ∈N0, j = 1, d coincide with any poles of Γ(1 +ξj) which ones are situated at ξj = −1−nj,

nj ∈N0 and for zeros of the denominator functiona+λ1(ξj) +· · ·+λd(ξd), which cannot be real. Here the integration paths in the complex ξj–planes, which start at the point

−i∞ and terminate at the point i∞ separate the poles of Γ(−ξj), j = 1, d from the poles

ξj =−1−nj, nj ∈N0. Evaluating the residues the functions Γ(−ξj), j= 1, d atξj =nj, applying the calculus of residues, we obtain (3.19).

Corollary 3.1. Let a∈C_\Z−

0;s∈C,ℜ(s)> d and ν >0. Then we have

e

ζd(s;a|ν,z) = 1 (2πi)d

∫

iRd d

∏

j=1

{

Γ(−ξj)Γ(1 +ξj)(−zj)ξj

}

(

a+ν ◦ξ)s dξ, (3.20)

where |arg(−zj)|< π, j = 1, d.

Corollary 3.2. Let a∈C_\Z−

0;s∈C,ℜ(s)> d, ν >0. Then

ζd(s;a|ν) = 1 (2πi)d

∫

iRd d

∏

j=1

{

Γ(−ξj)Γ(1 +ξj)(−1)ξj}

(

a+ν◦ξ)s dξ.

Remark 3.1. For d= 1, ν = 1 Corollary 3.1 reduces to

Φ(z, s, a) = 1 2πi

∫

iR

Γ(−ξ)Γ(1 +ξ)(−z)ξ (a+ξ)s dξ ,

If, however, we calculate the residues of Γ(1+ξj) at the simple polesξj =−1−nj, nj ∈ N_{0, j= 1, d, we arrive at the following analytic continuation formula.}

Theorem 3.2. Let all paramaters and variables of He_d(a, s;α,λ|z) satisfy (2.10). Then there holds true the formula

e

H_d(a, s;α,λ|z) = ∑ n∈Nd

0

d

∏

j=1

(−zj)−nj−1

(

a+λ1(1 +nj) +· · ·+λd(1 +nd)

)s,

where a∈C_\Z−

0;s∈C,ℜ(s)> d.

Corollary 3.3. Let a∈C_\Z−

0;s∈C,ℜ(s)> dandν >0. Then, for|arg(−zj)|< π, j = 1, d:

e

ζd(s;a|ν,z) =

∑

n∈Nd

0

d

∏

j=1

(−zj)−nj−1

(a+1◦ν+n◦ν)s.

Corollary 3.4. Suppose thata∈C_\Z−

0;s∈C,ℜ(s)> d,ν >0. Then

ζd(s;a|ν) = (−1)d

∑

n∈Nd

0

d

∏

j=1

(−1)−nj

(a+1◦ν+n◦ν)s.

Remark 3.2. For d= 1, ν = 1 Corollary 3.3 reduces to

Φ(z, s, a) =−1

z

∑

n∈N0

(−z)−n (a+ 1 +n)s,

where |arg(−z)|< π,ℜ(s)>1.

Relations (1.5), (1.8) and (2.11) connect Hilbert’s d–linear d–variate sum He_d and the related Hurwitz–Euler Eta functions ηd,eηd. By similar considerations as for ζd,ζed, we deduce the following results (avoiding the proofs which only modestly differ of here exposed ones).

Corollary 3.5. Let a∈C_\Z−

0;s∈C,ℜ(s)> d. Also assume ν >0. Then we have

e

ηd(s;a|ν,z) = 1 (2πi)d

∫

iRd d

∏

j=1

{

Γ(−ξj)Γ(1 +ξj)zjξj

}

(

a+ν◦ξ)s dξ, (3.21)

where |arg(−zj)|< π, j = 1, d.

Corollary 3.6. Lettinga∈C_\Z−

0;s∈C,ℜ(s)> d and ν >0, it follows

ηd(s;a|ν) = 1 (2πi)d

∫

iRd d

∏

j=1

{

Γ(−ξj)Γ(1 +ξj)}

(

4

Mellin transforms

By the application of the multidimensional Mellin inversion formula, the following results readily follow from (3.20) and (3.21). Because it simplicity, we omit their proofs.

Theorem 4.1. Let a∈C_\Z−

0;s∈C,ℜ(s)> d and ν >0. Then we have

∫

Rd

+

zξ1−1

1 · · ·zξd

−1

d ζed(s;a|ν,z)dz= d

∏

j=1

Γ(ξj)Γ(1−ξj)

(

a−ν◦ξ)s ,

where |arg(−zj)|< π, j = 1, d.

Corollary 4.1. Let a∈C_\Z−

0;s∈C,ℜ(s)> d and ν >0. Then we have

∫

Rd

+

zξ1−1

1 · · ·z

ξd−1

d ηed(s;a|ν,z)dz= (−1)ξ◦1−d d

∏

j=1

Γ(ξj)Γ(1−ξj)

(

a−ν◦ξ)s ,

where |arg(−zj)|< π, j = 1, d.

Acknowledgements

The authors would like to express their sincere thanks to an unknown Reviewer who provided constructive helpful advices and suggestions to improve the article.

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