# Series Representations at Special Values of Generalized Hurwitz-Lerch Zeta Function

Authors: S. Gaboury; A. Bayad
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

By making use of some explicit relationships between the Apostol-Bernoulli, Apostol-Euler, Apostol-Genocchi, and Apostol-Frobenius-Euler polynomials of higher order and the generalized Hurwitz-Lerch zeta function as well
as a new expansion formula for the generalized Hurwitz-Lerch zeta function
obtained recently by Gaboury and Bayad , in this paper we present some series representations for these polynomials at rational arguments. These results
provide extensions of those obtained by Apostol (1951) and by Srivastava (2000).


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## Body

## 1. Introduction

The generalized Hurwitz zeta functionζ(s,a) is defined by [1, page 88 et seq.](1)ζ(s,a):=∑n=0∞1(n+a)s(Re(s)>1;a∈ℂ∖ℤ0-;ℤ0-:={0,-1,-2,…}),
where
(2)ζ(s,1)=ζ(s)=12s-1ζ(s,12)
yields the celebrated Riemann zeta function ζ(s). The Riemann zeta function is continued meromorphically to the whole complex s-plane except for a simple pole at s=1 with residue 1.The Hurwitz-Lerch zeta functionΦ(z,s,a) is defined, as in [1, page 121 et seq.], by
(3)Φ(z,s,a):=∑n=0∞zn(n+a)s(a∈ℂ∖ℤ0-;s∈ℂwhen|z|<1;Re(s)>1when|z|=1).
Clearly, we have the following relations:
(4)Φ(1,s,a)=ζ(s,a),Φ(1,s,1)=ζ(s).The Hurwitz-Lerch zeta function has the well-known integral representation(5)Φ(z,s,a)=1Γ(s)∫0∞ts-1e-at1-ze-tdt(Re(a)>0;Re(s)>0when|z|≤1(z≠1);Re(s)>1whenz=1).Recently, Lin and Srivastava [2] investigated a more general family of Hurwitz-Lerch zeta functions. Explicitly, they introduced the function Φμ,ν(ρ,σ)(z,s,a) defined by
(6)Φμ,ν(ρ,σ)(z,s,a):=∑n=0∞(μ)ρn(ν)σnzn(a+n)s(μ∈ℂ;a,ν∈ℂ∖ℤ0-;ρ,σ∈ℝ+;ρ<σwhens,z∈ℂ;ρ=σ,s∈ℂwhen|z|<1;ρ=σ,Re(s-μ+ν)>1when|z|=1),
where (λ)κ denotes the Pochhammer symbol defined, in terms of the Gamma function, by
(7)(λ)κ:=Γ(λ+κ)Γ(λ)={λ(λ+1)⋯(λ+κ-1),(κ∈ℕ;λ∈ℂ)1,(κ=0;λ∈ℂ∖{0}).
It is easily seen that
(8)Φν,ν(σ,σ)(z,s,a)=Φμ,ν(0,0)(z,s,a)=Φ(z,s,a),(9)Φμ,1(1,1)(z,s,a)=Φμ*(z,s,a):=∑n=0∞(μ)nn!zn(n+a)s.
The function Φμ*(z,s,a) is, in fact, a generalized Hurwith-Lerch zeta function investigated by Goyal and Laddha [3, page 100, (1.5)].Let us recall some other important special cases of the Hurwitz-Lerch zeta functionΦ(z,s,a). The Lerch zeta function defined by
(10)ls(ξ):=∑n=1∞e2nπiξns(ξ∈ℝ;Re(s)>1)
is related to the Hurwitz-Lerch zeta function by the following relation:
(11)ls(ξ)=e2πiξΦ(e2πiξ,s,1).
Also, we note, as a special case of the Hurwitz-Lerch zeta functions, the Lipschitz-Lerch zeta function [1, page 122, Equation (2.5)(11)]:
(12)ϕ(ξ,a,s):=∑n=0∞e2nπiξ(n+a)s=Φ(e2πiξ,s,a)(a∈ℂ∖ℤ0-;Re(s)>0whenξ∈ℝ∖ℤ;Re(s)>1whenξ∈ℤ).
Setting z=exp(2πip/q) with p∈ℤ and q∈ℕ and using the next series identity
(13)∑n=0∞f(n)=∑j=0k-1∑n=0∞f(kn+j),
we obtain the following summation formula for the Lipschitz-Lerch zeta function ϕ(ξ,a,s):
(14)ϕ(pq,a,s)=q-s∑j=1qζ(s,a+j-1q)exp(2(j-1)pπiq)
in terms of the Hurwitz zeta function ζ(s,a) defined by (1) (see also [4, page 81, Equation (3.9)]).Finally, we recall Lerch's functional equation [5, page 29, Equation (1.11)(7)]:
(15)Φ(z,s,a)=iz-a(2π)s-1Γ(1-s)×{exp(-12sπi)Φ(e-2πia,1-s,logz2πi)c-exp[(2a+12s)πi]c×Φ(e2πia,1-s,1-logz2πi)}(Re(s)<0;|arg(-log(z)mod2πi)|≤π;a∈ℂ∖ℤ0-),
which played a central role in obtaining expansion formulas.The aim of this paper is to give some series representations of the Apostol-Bernoulli, Apostol-Euler, Apostol-Genocchi, and Apostol-Frobenius-Euler polynomials of higher order in terms of the generalized Hurwitz zeta function (1).It is worthy to mention that throughout this work, the following convention is adopted for the binomial coefficient:(16)(n-1k-1):={1,n=k=1,0,n≥1,k=0,(n-1k-1),otherwise.


## 2. Expansion Formulas forΦμ*(z,s,a) and Fourier Series

Few years ago, in order to give extensions of the works of Apostol [6] and Srivastava [4], Lin et al. [7] obtained an expansion formula for the generalized Hurwitz-Lerch zeta function Φμ*(z,s,a) defined by (9) in terms of the Hurwitz-Lerch zeta function (3) by making use of the generalized Leibniz rule for fractional derivative [8, page 95, Equation (4.3)]. They proved the following expansion formula for the general Hurwitz-Lerch zeta function Φμ*(z,s,a):
(17)Φμ*(z,s,a)=iz-aΓ(1-s)∑j=0∞(j-a+1)μ-j-1Γ(μ-j)j!×∑k=0j(j-1k-1)(1-s)kBj-k(j)·(2π)s-k-1[(e2πia,1-s+k,1-logz2πi)exp(-12(s-k)πi)[·(2π)s-k-1×Φ(e-2πia,1-s+k,logz2πi)[·(2π)s-k-1-exp[(2a+12(s-k))πi][·(2π)s-k-1×Φ(e2πia,1-s+k,1-logz2πi)],
with Re(μ)>0, s∈ℂ, |arg(-log(z)mod2πi)|≤π, a∈ℂ∖ℤ0- and where Bn(α):=Bn(α)(0) are the generalized Bernoulli numbers [9] given by the following generating function:
(18)(zez-1)αexz=∑n=0∞Bn(α)(x)znn!(|z|<2π;1α:=1).From this expansion formula, many authors found several interesting explicit formulas for the Apostol-Bernoulli, Apostol-Euler, and Apostol-Genocchi polynomials at rational arguments [10–12]. For instance, Garg et al. [10, page 814, Equation (54)] obtained the following explicit formula for the Apostol-Bernoulli polynomials ℬn(α)(x;λ) (which will be defined in Section 3):
(19)ℬn(l)(pq;e2πiξ)=i(-1)ln!(l-1)!∑j=0l-1(l-1j)(j-pq+1)l-j-1×∑k=0j(j-1k-1)(1-l+n)k×Bj-k(j)(2πq)l-n-k-1·{∑r=1qζ(1-l+n+k,ξ+r-1q)·exp[-(2(ξ+r-1)pq12(l-n-k)+2(ξ+r-1)pq)πi]-∑r=1qζ(1-l+n+k,r-ξq)·exp[(2(r-ξ)pq12(l-n-k)+2(r-ξ)pq)πi]∑r=1q‍}(n,l∈ℕ;ξ∈ℝ;0<ξ<1;p∈ℤ;q∈ℕ).
More recently, Gaboury and Bayad [13] applied a new generalized Leibniz rule for the fractional derivatives obtained by one of the authors [14] to obtain a new expansion formula involving the generalized Hurwitz-Lerch zeta function Φμ*(z,s,a). Explicitly, they proved the following theorem.Theorem 1.
The following expansion holds forΦμ*(z,s,a):
(20)Φμ*(z,s,a)=iΓ(1-s)z-asinβπsin(β+μ-ω)πΓ(μ)Γ(1-ω-a)sin(μ+β)πsin(β-ω)π·∑m=0∞∑k=0m∑j=0k(m-1k-1)(kj)×Γ(μ-a-ω-m)Γ(μ-m)m!Bm-k(m)×(2π)s-1-j(1-s)j·(ω-1+m)k-j×[exp(-(s-j)πi2)×Φ(e-2πia,1-s+j,logz2πi)-exp[(2a+s-j2)πi]×Φ(e2πia,1-s+j,1-logz2πi)(s-j)πi2](μ∈ℂ∖ℤ0-;ω∈ℂ;Re(1-β)>0;s∈ℂ;|arg(-log(z)mod2πi)|≤π;a∈ℂ∖ℤ0-).More interestingly, setting(21)z=e2πiξ,a=pq(p∈ℤ;q∈ℕ;ξ∈ℝ),
replacing s by μ-s, and then applying definition (12) and the series identity (14), we obtain the following corollary.Corollary 2.
The following expansion holds true:(22)Φμ*(e2πiξ,μ-s,pq)=iΓ(1-μ+s)sinβπsin(β+μ-ω)πΓ(μ)Γ(1-ω-(p/q))sin(μ+β)πsin(β-ω)π×∑m=0∞∑k=0m∑j=0k(m-1k-1)(kj)ccc·Γ(μ-p/q-ω-m)Γ(μ-m)m!ccc×Bm-k(m)(2πq)μ-s-1-jccc×(1-μ+s)j(ω-1+m)k-j·[∑r=1qζ(1-μ+s+j,ξ+r-1q)×exp2(r-1+ξ)pq[-iπ(2(r-1+ξ)pqμ-s-j2+2(r-1+ξ)pq)]-∑r=1qζ(1-μ+s+j,r-ξq)×exp[iπ(2(r-ξ)pqμ-s-j2+2(r-ξ)pq)]∑r=1qζ(1-μ+s+j,ξ+r-1q)](μ∈ℂ∖ℤ0-;ω∈ℂ;Re(1-β)>0;s∈ℂ;0<ξ<1;a∈ℂ∖ℤ0-).Remark 3.
It is worthy to mention that (20) can be interpreted as the Fourier series expansion of Φμ*(z,s,a) with respect to the variable a. Moreover, if we set μ=1 in (20), then, for Re(s)<0, 0<|z|<1 and a∈[0,1], we find an expansion theorem due to Erdélyi et al. and Wirtinger [5, 15], namely,
(23)Φ(z,s,a)=z-aΓ(1-s)∑n∈ℤe2πina(2πin-logz)1-s.


## 3. Generalized Hurwitz-Lerch Zeta Function at Negative Integers

In this section, we recall the definitions of the Apostol-Bernoulli, Apostol-Euler, Apostol-Genocchi, and Apostol-Frobenius-Euler polynomials. Next, we give explicit relationships between these polynomials and the generalized Hurwitz-Lerch zeta function evaluated at negative integers.Definition 4.
The generalized Apostol-Bernoulli polynomialsℬn(α)(x;λ) are defined, for λ,x∈ℂ, by the following generating function [16, 17]:
(24)(tλet-1)αext=∑n=0∞ℬn(α)(x;λ)tnn!,|t+log(λ)|<2π,|t+log(λ)|<2π,1(α∈ℂ,ifλ≠1;α∈ℕifλ=1).Definition 5.
The generalized Apostol-Euler polynomialsℰn(α)(x;λ) are defined, for λ,α and x∈ℂ, by the following generating function [18]:
(25)(2λet+1)αext=∑n=0∞ℰn(α)(x;λ)tnn!,|t+logλ|<π,∑n=0∞ℰn(α)(x;λ)tnn!,|t+logλ|<π,(α∈ℂ,λ≠-1).Definition 6.
The generalized Apostol-Genocchi polynomials𝒢n(α)(x;λ) are defined, for λ,x∈ℂ, by the following generating function [17, 19]:
(26)(2tλet+1)αext=∑n=0∞𝒢n(α)(x;λ)tnn!,|t+log(λ)|<π,|t+log(λ)|<π(α∈ℂ,ifλ≠-1;α∈ℕifλ=-1).Definition 7.
The generalized Apostol-Frobenius-Euler polynomialsℋn(α)(x;λ∣u) are defined, for λ,α,u and x∈ℂ with u≠0,1 and λ≠u, by the following generating function [20]:
(27)(1-uλet-u)αext=∑n=0∞ℋn(α)(x;λ∣u)tnn!,|t+log(λu)|<π.
Settingλ=1 and α=1 in each of these definitions gives, respectively, the Bernoulli, Euler, Genocchi and Frobenius-Euler polynomials which polynomials have been largely investigated in the literature.
Recently, Bayad and Chikhi [21] established the following relationship between the Apostol-Euler polynomials ℰn(α)(x;λ) and the generalized Hurwitz-Lerch zeta function Φα*(z,s,a).Theorem 8.
Letλ be a complex number such that |λ|≤1 and λ≠-1. Let α and x be two complex numbers such that Re(α)>0 and Re(x)>0; then for all nonnegative integer n, one has
(28)ℰn(α)(x;λ)=2αΦα*(-λ,-n,x).From this last theorem, we can find the relationships between the generalized Hurwitz-Lerch zeta functionΦα*(z,s,a) and the polynomials defined by (24), (26), and (27). By simple manipulations of their generating functions, we can rewrite each polynomials in function of the Apostol-Euler polynomials. For instance, we have
(29)(1-uλet-u)αext=(u-12u)α(2(-λ/u)et+1)αext
and thus,
(30)ℋn(α)(x;λ∣u)=(u-12u)αℰn(α)(x;-λu).In a similar way, we can deduce the two next relationships between the Apostol-Bernoulli polynomials and the Apostol-Euler polynomials and between the Apostol-Genocchi polynomials and the Apostol-Euler polynomials. Indeed, we have(31)ℬn(l)(x;λ)=l!(nl)(-12)lℰn-l(l)(x;-λ)(l∈ℕ0,λ≠1),𝒢n(l)(x;λ)=l!(nl)ℰn-l(l)(x;λ)(l∈ℕ0).
Now, with the help of Theorem 8, we obtain the following relations between each type of polynomials and the generalized Hurwitz-Lerch zeta function at negative integers:
(32)ℬn(l)(x;λ)=l!(nl)(-1)lΦl*(λ,l-n,x)(|λ|≤1;λ≠1;l∈ℕ0),(33)𝒢n(l)(x;λ)=2ll!(nl)Φl*(-λ,l-n,x)(|λ|≤1;λ≠-1;l∈ℕ0),(34)ℋn(α)(x;λ∣u)=(u-1u)αΦα*(λu,-n,x)(u≠0,1;|λu|≤1;λ≠u).


## 4. Explicit Representations for Several Polynomials at Rational Arguments

This section is devoted to the presentation of the new series representations for each type of polynomials defined in Section3. These representations are obtained by suitably combining the new expansion formula (20) and identities (28) and (32)–(34).Theorem 9.
The following explicit representation holds for the Apostol-Bernoulli polynomials at rational argument:(35)ℬn(l)(pq;e2πiξ)=i(-1)ln!Γ(l)Γ(1-ω-(p/q))×∑m=0l-1∑k=0m∑j=0k(m-1k-1)(kj)·Γ(l-(p/q)-ω-m)Γ(l-m)m!Bm-k(m)(2πq)l-n-1-j×(1-l+n)j(ω-1+m)k-j·[∑r=1qζ(1-l+n+j,ξ+r-1q)×exp[-iπ(2(r-1+ξ)pql-n-j2+2(r-1+ξ)pq)]-∑r=1qζ(1-l+n+j,r-ξq)×exp[iπ(l-n-j2+2(r-ξ)pq)]∑r=1qζ(1-l+n+j,ξ+r-1q)].(pql,n∈ℕ;ω∈ℂ;Re(1-β)>0;ξ∈ℝ,0<ξ<1;pq∈ℚ∖ℤ0-).Proof.
Settingμ=l, s=n with n,l∈ℕ in the expansion formula (22) and with the help of the representation of Apostol-Bernoulli polynomials in terms of the generalized Hurwitz-Lerch zeta function (32), the result follows.Theorem 10.
The following explicit representation holds for the Apostol-Euler polynomials at rational argument:(36)ℰn(α)(pq;e2πiξ)=iΓ(1+n)2αsinβπsin(β+α-ω)πΓ(α)Γ(1-ω-(p/q))sin(α+β)πsin(β-ω)π×∑m=0∞∑k=0m∑j=0k(kj)·(m-1k-1)Γ(α-(p/q)-ω-m)Γ(α-m)m!×Bm-k(m)(2πq)-n-1-j(1+n)j(ω-1+m)k-j·[∑r=1qζ(1+n+j,2ξ+2r-12q)ccc×exp[-iπ((2r+2ξ-1)pq-n-j2+(2r+2ξ-1)pq)]-∑r=1qζ(1+n+j,2r-2ξ-12q)×exp[iπ((2r-2ξ-1)pq-n-j2+(2r-2ξ-1)pq)]∑r=1qζ(1+n+j,2ξ+2r-12q)](12;pqα∈ℂ∖ℤ0-;ω∈ℂ;Re(1-β)>0;ξ∈ℝ,0≤ξ≤1,ξ≠12;pq∈ℚ∖ℤ0-).Proof.
Replacingz by -e2πiξ(ξ∈ℝ), a by (p/q)  (p∈ℤ;q∈ℕ), μ by α(α∈ℂ), s by -n(n∈ℕ) in (20) and using the identity (28) yield the result.Theorem 11.
The following explicit representation holds for the Apostol-Genocchi polynomials at rational argument:(37)𝒢n(l)(pq;e2πiξ)=i2ln!Γ(l)Γ(1-ω-(p/q))×∑m=0l-1∑k=0m∑j=0k(kj)(m-1k-1)·Γ(l-(p/q)-ω-m)Γ(l-m)m!Bm-k(m)(2πq)l-n-1-j×(1-l+n)j(ω-1+m)k-j·[∑r=1qζ(1-l+n+j,2ξ+2r-12q)×exp[-iπ((2r+2ξ-1)pql-n-j2+(2r+2ξ-1)pq)]-∑r=1qζ(1-l+n+j,2r-2ξ-12q)×exp[iπ((2r-2ξ-1)pql-n-j2+(2r-2ξ-1)pq)]∑r=1qζ(1-l+n+j,2ξ+2r-12q)](pql,n∈ℕ;ω∈ℂ;Re(1-β)>0;ξ∈ℝ,0≤ξ≤1,ξ≠12;pq∈ℚ∖ℤ0-).Proof.
Substitutingz by -e2πiξ(ξ∈ℝ), a by (p/q)(p∈ℤ;q∈ℕ) and s by μ-s(μ∈ℂ) in (20) gives
(38)Φμ*(-e2πiξ,μ-s,pq;)=iΓ(1-μ+s)sinβπsin(β+μ-ω)πΓ(μ)Γ(1-ω-(p/q))sin(μ+β)πsin(β-ω)π×∑m=0∞∑k=0m∑j=0k·(kj)(m-1k-1)Γ(μ-(p/q)-ω-m)Γ(μ-m)m!×Bm-k(m)(2πq)μ-s-1-j(1-μ+s)j(ω-1+m)k-j·[∑r=1qζ(1-μ+s+j,2ξ+2r-12q)×exp[-iπ(μ-s-j2+(2r+2ξ-1)pq)]-∑r=1qζ(1-μ-s+j,2r-2ξ-12q)×exp[iπ(μ-s-j2+(2r-2ξ-1)pq)]∑r=1qζ(1-μ+s+j,2ξ+2r-12q)].
Putting μ=l(l∈ℕ), s=n(n∈ℕ;n≥l) and with the help of the identity (33), the result follows.Theorem 12.
The following explicit representation holds for the Apostol-Frobenius-Euler polynomials at rational argument:(39)ℋn(α)(pq;e2πiξ∣e2πiθ)=i(1-e-2πiθ)αn!sinβπsin(β+α-ω)πΓ(α)Γ(1-ω-(p/q))sin(α+β)πsin(β-ω)π×∑m=0∞∑k=0m∑j=0k·(m-1k-1)(kj)Γ(α-(p/q)-ω-m)Γ(α-m)m!×Bm-k(m)(2πq)-n-1-j(1+n)j(ω-1+m)k-j·[∑r=1qζ(1+n+j,(ξ-θ)+r-1q)×exp[-iπ(-n-j2+2(r-1+(ξ-θ))pq)]-∑r=1qζ(1+n+j,r-(ξ-θ)q)×exp[iπ(2(r-(ξ-θ))pq-n-j2+2(r-(ξ-θ))pq)]∑r=1qζ(1+n+j,(ξ-θ)+r-1q)].(Re(1-β)ξ,θ∈ℝ,0≤ξ≤1,0<θ<1,0<|ξ-θ|<1,ξ≠θ;pq∈ℚ∖ℤ0-;α∈ℂ∖ℤ0-;ω∈ℂ;Re(1-β)>0).Proof.
The result follows by settingz=e2πi(ξ-θ)  (ξ,θ∈ℝ), a=(p/q)(p∈ℤ;q∈ℕ), μ=α(α∈ℂ), s=-n(n∈ℕ) in (20) and using the identity (34) with λ=e2πi(ξ) and u=e2πi(θ).It is worthy to note that settingl=1 in (35) gives
(40)ℬn(pq;e2πiξ)=-n!(2πq)n[∑r=1qζ(n,ξ+r-1q)cccccccc×exp[-iπ(2(r-1+ξ)pq-n2+2(r-1+ξ)pq)]cccccccc-∑r=1qζ(n,r-ξq)cccccccc×exp[iπ(-n2+2(r-ξ)pq)]∑r=1qζ(n,ξ+r-1q)](n∈ℕ;n≥1ξ∈ℝ,0<ξ<1;pq∈ℚ∖ℤ0-),
a result given by Srivastava [4, page 84, Equation (4.6)].Moreover, if we replaceα by 1 in (36), we recover a result obtained by Luo [11, page 345, Equation (4.3)], namely,
(41)ℰn(pq;e2πiξ)=2n!(2πq)n+1[∑r=1qζ(1+n,2ξ+2r-12q)×exp[-iπ((2r+2ξ-1)pq-n-12+(2r+2ξ-1)pq)]-∑r=1qζ(1+n,2r-2ξ-12q)×exp[iπ((2r-2ξ-1)pq-n+12+(2r-2ξ-1)pq)]∑r=1qζ(1+n,2ξ+2r-12q)](ξ∈ℝ,0≤ξ≤1,ξ≠12;p∈ℤ;q∈ℕ;pq∈ℚ∖ℤ0-).



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