# New Relations Involving an Extended Multiparameter Hurwitz-Lerch Zeta Function with Applications

Authors: H. M. Srivastava; Sébastien Gaboury; Richard Tremblay
Journal: International Journal of Analysis
 (2014)

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## Abstract

We derive several new expansion formulas involving an extended multiparameter Hurwitz-Lerch zeta function introduced and studied recently by Srivastava et al. (2011).  These expansions are obtained by using some fractional calculus methods such as the generalized Leibniz rules, the Taylor-like expansions in terms of different functions, and the generalized chain rule. Several (known or new) special cases are also given.


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## Body

## 1. Introduction

The Hurwitz-Lerch zeta functionΦ(z,s,a) which is one of the fundamentally important higher transcendental functions is defined by (see, e.g., [1, page 121et seq.]; see also [2] and [3, page 194et seq.])
(1)Φ(z,s,a)≔∑n=0∞‍zn(n+a)s,(a∈ℂ∖ℤ0-;s∈ℂwhen|z|<1;ℜ(s)>1when|z|=1).
The Hurwitz-Lerch zeta function contains, as its special cases, the Riemann zeta function ζ(s), the Hurwitz zeta function ζ(s,a), and the Lerch zeta function ℓs(ξ) defined by
(2)ζ(s):=∑n=1∞‍1ns=Φ(1,s,1)=ζ(s,1)(ℜ(s)>1),(3)ζ(s,a)≔∑n=0∞‍1(n+a)s=Φ(1,s,a)(ℜ(s)>1;a∈ℂ∖ℤ0-),(4)ℓs(ξ):=∑n=1∞‍e2nπiξ(n+1)s=Φ(e2πiξ,s,1)(ℜ(s)>1;ξ∈ℝ),
respectively.The Hurwitz-Lerch zeta function is connected with other special functions of analytic number theory such as the polylogarithmic function (orde Jonquière’s function) Lis(z):
(5)Lis(z)≔∑n=1∞‍znns=zΦ(z,s,1)(s∈ℂwhen|z|<1;ℜ(s)>1when|z|=1)
and the Lipschitz-Lerch zeta function ϕ(ξ,a,s) (see [1, page 122, Equation 2.5  (11)])
(6)ϕ(ξ,a,s):=∑n=0∞‍e2nπiξ(n+a)s=Φ(e2πiξ,s,a)(a∈ℂ∖ℤ0-;ℜ(s)>0whenξ∈ℝ∖ℤ;ℜ(s)>1whenξ∈ℤ).
The Hurwitz-Lerch zeta function Φ(z,s,a) defined in (7) can be continuedmeromorphically to the whole complex s-plane, except for a simple pole at s=1 with its residue 1. It is well known that
(7)Φ(z,s,a)=1Γ(s)∫0∞‍ts-1e-at1-ze-tdt(ℜ(a)>0;ℜ(s)>0when|z|≦1(z≠1);ℜ(s)>1whenz=1).Motivated by the works of Goyal and Laddha [4], Lin and Srivastava [5], Garg et al. [6], and other authors, Srivastava et al. [7] (see also [8]) investigated various properties of a natural multiparameter extension and generalization of the Hurwitz-Lerch zeta function Φ(z,s,a) defined by (7) (see also [9]). In particular, they considered the following functions:
(8)Φλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(z,s,a):=∑n=0∞‍∏j=1p‍(λj)nρjn!∏j=1q‍(μj)nσjzn(n+a)s(12p,q∈ℕ0;λj∈ℂ(j=1,…,p);a,μj∈ℂ∖ℤ0-(j=1,…,q);ρj,σk∈ℝ+(j=1,…,p;k=1,…,q);Δ>-1whens,z∈ℂ;Δ=-1,s∈ℂwhen|z|<∇*;Δ=-1,ℜ(Ξ)>12when|z|=∇*)
with
(9)∇*:=(∏j=1p‍ρj-ρj)·(∏j=1q‍σjσj),Δ:=∑j=1q‍σj-∑j=1p‍ρj,Ξ:=s+∑j=1q‍μj-∑j=1p‍λj+p-q2.
Here, and for the remainder of this paper, (λ)κ denotes the Pochhammer symbol defined, in terms of the gamma function, by
(10)(λ)κ≔Γ(λ+κ)Γ(λ)={λ(λ+1)⋯(λ+n-1)(κ=n∈ℕ;λ∈ℂ)1(κ=0;λ∈ℂ∖{0});
it is being understoodconventionally that (0)0:=1 and assumedtacitly that the Γ-quotient exists (see, for details, [10, page 21et seq.]).In their work, Srivastava et al. [7, page 504, Theorem 8] also proved the following relation for the function Φλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(z,s,a):(11)Φλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(z,s,a)=∏j=1q‍Γ(μj)∏j=1p‍Γ(λj)·H¯p+1,q+21,p+1[-z∣(1-λ1,ρ1;1),…,(1-λp,ρp;1),(1-a,1;s)(0,1),(1-μ1,σ1;1),…,(1-μp,σp;1),(-a,1;s)]provided that both sides of (11) exist.Definition 1.
TheH¯(z) involved in the right-hand side of (11) is the generalized Fox’s H-function introduced by Inayat-Hussain [11, page 4126]
(12)H¯(z)=H¯(z)𝔭,𝔮m,n[z∣(aj,Aj;αj)j=1n,(aj,Aj)j=n+1𝔭(bj,Bj)j=1m,(bj,Bj;βj)j=m+1𝔮]:=12πi∫𝔏‍χ(s)zsds(∏j=m+1𝔮‍{Γ(1-bj+Bjs)}βj)-1z≠0;i=-1;χ(s)≔(∏j=1m‍Γ(bj-Bjs)·∏j=1n‍{Γ(1-aj+Ajs)}αj)×(∏j=n+1𝔭‍Γ(aj-Ajs)·∏j=m+1𝔮‍{Γ(1-bj+Bjs)}βj)-1).
Here the parameters
(13)Aj>0(j=1,…,𝔭),Bj>0(j=1,…,𝔭),
and the exponents
(14)αj(j=1,…,𝔭),βj(j=1,…,𝔮)
can take noninteger values and 𝔏=𝔏iτ;∞ is a Mellin-Barnes type contour starting at the point τ-i∞ and terminating at the point τ+i∞(τ∈ℝ) with the usual indentations to separate one set of poles from the other set of poles.Buschman and Srivastava [12, page 4708] established that the sufficient conditions for the absolute convergence of the contour integral in (12) are given by
(15)Λ:=∑j=1m‍Bj+∑j=1n‍|αj|Aj-∑j=m+1𝔮‍|βj|Bj-∑j=n+1𝔭‍Aj>0
and the region of absolute convergence is
(16)|arg(z)|<12πΛ.
Note that when
(17)α1=⋯=αn=1,βm+1=⋯=β𝔮=1,
the H¯-function reduces to the well-known Fox’s H-function (see [13]).This paper is devoted to extending several interesting results obtained recently by Srivastava et al. [14] (see also [15, 16]) to the extended multiparameter Hurwitz-Lerch zeta function Φλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(z,s,a) introduced and studied by Srivastava et al. [7]. In Section 2, we give the representation of the fractional derivatives based on the Pochhammer’s contour of integration. Section 3 aims at recalling some major fractional calculus theorems, that is, two generalized Leibniz rules and three Taylor-like expansions as well as a fundamental relation linked to the generalized chain rule for the fractional derivatives. In the two remaining sections, we, respectively, present and prove the main results of this paper and we give some special cases.


## 2. Pochhammer Contour Integral Representation for Fractional Derivative

The most familiar representation for the fractional derivative of orderα of zpf(z) is the Riemann-Liouville integral [17] (see also [18–20]); that is,
(18)𝒟zα{zpf(z)}=1Γ(-α)∫0z‍f(ξ)ξp(ξ-z)-α-1dξ(ℜ(α)<0;ℜ(p)>1),
where the integration is carried out along a straight line from 0 to z in the complex ξ-plane. By integrating by part m times, we obtain
(19)𝒟zα{zpf(z)}=dmdzm{𝒟zα-m{zpf(z)}}.
This allows us to modify the restriction ℜ(α)<0 to ℜ(α)<m (see [20]).Another representation for the fractional derivative is based on the Cauchy integral formula. This representation, too, has been widely used in many interesting papers (see, for example, the works of Osler [21–24]).The relatively less restrictive representation of the fractional derivative according to parameters appears to be the one based on the Pochhammer’s contour integral introduced by Lavoie et al. [25] and Tremblay [26].Definition 2.
Letf(z) be analytic in a simply connected region ℛ of the complex z-plane. Let g(z) be regular and univalent on ℛ and let g-1(0) be an interior point of ℛ. Then, if α is not a negative integer, p is not an integer, and z is in ℛ∖{g-1(0)}, we define the fractional derivative of order α of g(z)pf(z) with respect to g(z) by
(20)Dg(z)α{[g(z)]pf(z)}=e-iπpΓ(1+α)4πsin⁡(πp)×∫C(z+,g-1(0)+,z-,g-1(0)-;F(a),F(a))‍·f(ξ)[g(ξ)]pg′(ξ)[g(ξ)-g(z)]α+1dξ.
For nonintegers α and p, the functions g(ξ)p and [g(ξ)-g(z)]-α-1 in the integrand have two branch lines which begin, respectively, at ξ=z and ξ=g-1(0), and both branches pass through the point ξ=a without crossing the Pochhammer contour P(a)={C1∪C2∪C3∪C4} at any other point as shown in Figure 1. Here F(a) denotes the principal value of the integrand in (20) at the beginning and the ending point of the Pochhammer contour P(a) which is closed on the Riemann surface of the multiple-valued function F(ξ) (see Figure 2).Figure 1
Pochhammer’s contour.Figure 2
Multiloops contour.Remark 3.
In Definition2, the function f(z) must be analytic at ξ=g-1(0). However, it is interesting to note here that if we could also allow f(z) to have an essential singularity at ξ=g-1(0), then (20) would still be valid.Remark 4.
In case the Pochhammer contour never crosses the singularities atξ=g-1(0) andξ=zin (20), then we know that the integral is analytic for allpand for allαand forzin ℛ∖{g-1(0)}. Indeed, in this case, the only possible singularities ofDg(z)α{g(z)pf(z)}areα=-1,-2,…andp=0,±1,±2,…, which can directly be identified from the coefficient of the integral (20). However, by integrating by partsNtimes the integral in (20) by two different ways, we can show thatα=-1,-2,…andp=0,1,2,…are removable singularities (see, for details, [25]).In their work, Srivastava et al. [7] made use of the following fractional calculus result obtained by Srivastava et al. [27, page 97, Equation (2.4)]:(21)Dzν{zλ-1H¯p,qm,n(ωzκ)}=zλ-ν-1H¯(z)p+1,q+1m,n+1×[ωzκ∣(1-λ,κ;1),(aj,Aj;αj)j=1n,(aj,Aj)j=n+1p(bj,Bj)j=1m,(aj,Aj;βj)j=m+1q,(1-λ+ν,κ;1)](ℜ(λ)>0;κ>0),in order to derive the following important fractional derivative formula for this work:(22)Dzν-τ{zν-1Φλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(zκ,s,a)}=∏j=1q‍Γ(μj)∏j=1p‍Γ(λj)zτ-1·H¯p+2,q+31,p+2[-zκ∣(1-λ1,ρ1;1),…,(1-λp,ρp;1),(1-ν,κ;1),(1-a,1;s)(0,1),(1-μ1,σ1;1),…,(1-μp,σp;1),(1-τ,κ;1),(-a,1;s)]=Γ(ν)Γ(τ)zτ-1Φλ1,…,λp,ν;μ1,…,μq,τ(ρ1,…,ρp,κ,σ1,…,σq,κ)(zκ,s,a)(ℜ(ν)>0;κ>0).This fractional calculus formula was obtained by using the Riemann-Liouville representation for the fractional derivative. Adopting the Pochhammer based representation for the fractional derivative, these last restrictions becomeκ+ν-1,not a negative integer, and κ>0.The parameters involved in the fractional derivative formula (22) can be specialized to deduce other results. For example, settingp-1=q=1in (22) and making the following substitutionsρ1↦ρ,ρ2↦σ,σ1↦κ,λ1↦λ,λ2↦μ, andμ1↦νlead to
(23)Dzν-τ{zν-1Φλ,μ;ν(ρ,σ,κ)(zκ,s,a)}=Γ(ν)Γ(λ)Γ(μ)zτ-1H¯4,41,4×[-zκ∣(1-λ,ρ;1),(1-μ,σ;1),(1-ν,κ;1),(1-a,1;s)(0,1),(1-ν,κ;1),(1-τ,κ;1),(-a,1;s)]=Γ(ν)Γ(λ)Γ(μ)zτ-1H¯3,31,3×[-zκ∣(1-λ,ρ;1),(1-μ,σ;1),(1-a,1;s)(0,1),(1-τ,κ;1),(-a,1;s)]=Γ(ν)Γ(τ)zτ-1Φλ,μ;τ(ρ,σ,κ)(zκ,s,a)(νnot a negative integer;κ>0).
Furthermore, if we putρ=σ=κ=1in (23), then we obtain
(24)Dzν-τ{zν-1Φλ,μ;ν(1,1,1)(z,s,a)}=Γ(ν)Γ(τ)zτ-1Φλ,μ;τ(z,s,a)=Γ(ν)Γ(λ)Γ(μ)zτ-1H¯3,31,3×[-z∣(1-λ,1;1),(1-μ,1;1),(1-a,1;s)(0,1),(1-τ,1;1),(-a,1;s)](νnot a negative integer;κ>0).
Finally, lettingλ=νin (24), this yields after elementary calculations
(25)Φμ*(z,s,a)=Γ(τ)Γ(ν)Γ(μ)H¯3,31,3×[-z∣(1-ν,1;1),(1-μ,1;1),(1-a,1;s)(0,1),(1-τ,1;1),(-a,1;s)].Another fractional derivative formula that will be very useful in this work is given by the following formula:(26)Dzα{zβΦλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(zκ,s,a)}=Γ(1+β)Γ(1+β-α)zβ-α·Φλ1,…,λp,1+β;μ1,…,μq,1+β-α(ρ1,…,ρp,κ,σ1,…,σq,κ)(zκ,s,a)(βnot a negative integer,κ>0).
This last result can be established with the help of the following well known formula [28, page 83, Equation (2.4)]:
(27)Dzα{zp}=Γ(1+p)Γ(1+p-α)zp-α(ℜ(p)>-1).
Adopting the Pochhammer based representation for the fractional derivative modifies the restriction to the case whenpis not a negative integer.


## 3. Some Fundamental Theorems Involving Fractional Calculus

In this section, we recall six fundamental theorems related to fractional calculus that will play central roles in our work. Each of these theorems is the generalized Leibniz rules for fractional derivatives, the Taylor-like expansions in terms of different types of functions, and a fundamental formula related to the generalized chain rule for fractional derivatives.First of all, we give two generalized Leibniz rules for fractional derivatives. Theorem5 is a slightly modified theorem obtained in 1970 by Osler [22]. Theorem 6 was given, some years ago, by Tremblay et al. [29] with the help of the properties of Pochhammer’s contour representation for fractional derivatives.Theorem 5.
(i) Letℛ be a simply connected region containing the origin. (ii) Let u(z) andv(z)satisfy the conditions of Definition 2 for the existence of the fractional derivative. Then, for ℜ(p+q)>-1 andγ∈ℂ, the following Leibniz rule holds true:
(28)Dzα{zp+qu(z)v(z)}=∑n=-∞∞‍(αγ+n)Dzα-γ-n×{zpu(z)}Dzγ+n{zqv(z)}.Theorem 6.
(i) Letℛ be a simply connected region containing the origin. (ii) Let u(z) andv(z)satisfy the conditions of Definition 2 for the existence of the fractional derivative. (iii) Let 𝒰⊂ℛ be the region of analyticity of the functionu(z)and let 𝒱⊂ℛ be the region of analyticity of the functionv(z). Then, for
(29)z≠0,z∈𝒰∩𝒱,ℜ(1-β)>0,
the following product rule holds true:
(30)Dzα{zα+β-1u(z)v(z)}=zΓ(1+α)sin⁡(βπ)sin⁡(μπ)sin⁡[(α+β-μ)π]sin[(α+β)π]sin⁡[(β-μ-ν)π]sin⁡[(μ+ν)π]·∑n=-∞∞‍Dzα+ν+1-n{zα+β-μ-1-nu(z)}Dz-1-ν+n{zμ-1+nv(z)}Γ(2+α+ν-n)Γ(-ν+n).Next, in 1971, Osler [30] established the following generalized Taylor-like series expansion involving fractional derivatives.Theorem 7.
Letf(z)be an analytic function in a simply connected region ℛ. Letαandγbe arbitrary complex numbers and
(31)θ(z)=(z-z0)q(z)
withq(z)a regular and univalent function without any zero in ℛ. Letabe a positive real number and
(32)K={([c]the largest integer not greater thanc)0,1,…,[c]([c]thelargestintegernotgreaterthanc)}.
Letb and z0 be two points in ℛ such thatb≠z0and let
(33)ω=exp⁡(2πia).
Then the following relationship holds true:
(34)∑k∈K‍c-1ω-γkf(θ-1(θ(z)ωk))=∑n=-∞∞‍[θ(z)]cn+γΓ(cn+γ+1)·Dz-bcn+γ{f(z)θ′(z)(z-z0θ(z))cn+γ+1}|z=z0(|z-z0|=|z0|).In particular, if0<c≦1 and θ(z)=(z-z0), then k=0 and the formula (34) reduces to the following form:
(35)f(z)=c∑n=-∞∞‍(z-z0)cn+γΓ(cn+γ+1)Dz-bcn+γ{f(z)}|z=z0.
This last formula (35) is usually referred to as the Taylor-Riemann formula and has been studied in several papers [23, 31–34].We next recall that Tremblay et al. [35] obtained the power series of an analytic function f(z) in terms of the rational expression ((z-z1)/(z-z2)), where z1 and z2 are two arbitrary points inside the region ℛ of analyticity of f(z). In particular, they obtained the following result.Theorem 8.
(i) Letcbe real and positive and let
(36)ω=exp⁡(2πia).
(ii) Letf(z)be analytic in the simply connected region ℛ withz1andz2being interior points of ℛ. (iii) Let the set of curves
(37){C(t):C(t)⊂ℛ,0<t≦r}
be defined by
(38)C(t)=C1(t)∪C2(t)={z:|λt(z1,z2;z)|=|λt(z1,z2;z1+z22)|},
where
(39)λt(z1,z2;z)=[z-z1+z22+t(z1-z22)]·[z-(z1+z22)-t(z1-z22)],
which are the Bernoulli type lemniscates with center located at (z1+z2)/2 and with double-loops in which one loop C1(t) leads around the focus point
(40)z1+z22+(z1-z22)t
and the other loopC2(t)encircles the focus point
(41)z1+z22-(z1-z22)t,
for eachtsuch that0<t≦r. (iv) Let
(42)[(z-z1)(z-z2)]λ=exp⁡(λln⁡⁡(θ((z-z1)(z-z2))))
denote the principal branch of that function which is continuous and insideC(r), cut by the respective two branch lines L± defined by
(43)L±={{z:z=z1+z22±t(z1-z22)}(0≦t≦1),{z:z=z1+z22±it(z1-z22)}(t<0)
such that ln⁡⁡((z-z1)(z-z2)) is real when (z-z1)(z-z2)>0.  (v) Letf(z)satisfy the conditions of Definition 2 for the existence of the fractional derivative of(z-z2)pf(z)of orderαfor z∈ℛ∖{L+∪L-}, denoted byDz-z2α{(z-z2)pf(z)}, whereαandpare real or complex numbers. (vi) Let
(44)K={k:k∈ℕ,arg⁡(λt(z1,z2,z1+z22))<arg⁡(λt(z1,z2,z1+z22))+2πka<arg(λt(z1,z2,z1+z22))+2π}.
Then, for arbitrary complex numbers μ, ν, γ and for z on C1(1) defined by
(45)ξ=z1+z22+z1-z221+eiθ(-π<θ<π),∑k∈K‍c-1ω-γkz1-z2f(ϕ-1(ωkϕ(z)))×[ϕ-1(ϕ(z)ωk)-z1]ν[ϕ-1(ϕ(z)ωk)-z2]μ=∑n=-∞∞‍eiπc(n+1)sin⁡[(μ+cn+γ)π]sin⁡[(μ-c+γ)π]Γ(1-ν+cn+γ)·Dz-z2-ν+cn+γ{(z-z2)μ+cn+γ-1f(z)}|z=z1[ϕ(z)]cn+γ,
where
(46)ϕ(z)=z-z1z-z2.The case0<c≦1of Theorem 8 reduces to the following form:
(47)c-1f(z)(z-z1)ν(z-z2)μz1-z2=∑n=-∞∞‍eiπc(n+1)sin[(μ+cn+γ)π]sin[(μ-c+γ)π]Γ(1-ν+cn+γ)·Dz-z2-ν+cn+γ{(z-z2)μ+cn+γ-1f(z)}|z=z1(z-z1z-z2)cn+γ.Tremblay and Fugère [36] developed the power series of an analytic functionf(z)in terms of the function(z-z1)(z-z2), wherez1andz2are two arbitrary points inside the analyticity region ℛ off(z). Explicitly, they showed the following theorem.Theorem 9.
Under the assumptions of Theorem8, the following expansion formula holds true:
(48)∑k∈K‍c-1ω-γk×[(z2-z1+Δk2)α(z1-z2+Δk2)β·f(z1+z2+Δk2)-eiπ(α-β)sin⁡[(α+c-γ)π]sin⁡[(β+c-γ)π]·(z2-z1-Δk2)α(z1-z2-Δk2)β×f(z1+z2-Δk2)(z2-z1+Δk2)α(z1-z2+Δk2)β]=∑n=-∞∞‍sin⁡[(β-cn-γ)π]sin⁡[(β-c-γ)π]e-iπc(n+1)[θ(z)]cn+γΓ(1-α+cn+γ)·Dz-z2-α+cn+γ{(θ(z)(z-z2)(z-z1))-cn-γ-1(z-z2)β-cn-γ-1×(θ(z)(z-z2)(z-z1))-cn-γ-1×θ′(z)f(z)(θ(z)(z-z2)(z-z1))-cn-γ-1}|z=z1,
where
(49)Δk=(z1-z2)2+4V(ωkθ(z)),V(z)=∑r=1∞‍Dzr-1{[q(z)]-r}|z=0zrr!,θ(z)=(z-z1)(z-z2)q((z-z1)(z-z2)).As special case, if we set0<c≦1, q(z)=1(θ(z)=(z-z1)(z-z2)), andz2=0in (48), we obtain
(50)f(z)=cz-β(z-z1)-α×∑n=-∞∞‍sin⁡[(β-cn-γ)π]sin⁡[(β+c-γ)π]eiπc(n+1)[z(z-z1)]cn+γΓ(1-α+cn+γ)·Dz-α+cn+γ{zβ-cn-γ-1(z+w-z1)f(z)}|z=z1,(w=z).Finally, Osler [21, page 290, Theorem 2] discovered a fundamental relation from which he deduced the generalized chain rule for the fractional derivatives. This result is recalled here as Theorem 10.Theorem 10.
Letf(g-1(z))andf(h-1(z))be defined and analytic in the simply connected region ℛ of the complexz-plane and let the origin be an interior or boundary point of ℛ. Suppose also thatg-1(z)andh-1(z)are regular univalent functions on ℛ and thath-1(0)=g-1(0). Let∮‍f(g-1(z))dzvanish over simple closed contour in ℛ∪{0} through the origin. Then the following relation holds true:
(51)Dg(z)α{f(z)}=Dh(z)α{f(z)g′(z)h′(z)(h(w)-h(z)g(w)-g(z))α+1}|w=z.The relation (51) allows us to obtain very easily known and new summation formulas involving special functions of mathematical physics.By applying relation (51), Gaboury and Tremblay [37] proved the following corollary which will be useful in the next section.Corollary 11.
Under the hypotheses of Theorem10, letpbe a positive integer. Then the following relation holds true:
(52)zpOβα{f(z)}=p(zp-1)-α·zOβα{f(z)(zp-1)α∏s=1p-1(1-zwe-2πis/p)β-α-1}|w=z,
where
(53)g(z)Oβα{⋯}:=Γ(β)Γ(α)[g(z)]1-βDg(z)α-β{[g(z)]α-1⋯}.


## 4. Relations Involving the Extended Multiparameters Hurwitz-Lerch Zeta FunctionΦλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(z,s,a)

In this section, we present the new expansion formulas involving the extended multiparameters Hurwitz-Lerch zeta functionΦλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(z,s,a).Theorem 12.
Under the hypotheses of Theorem5, the following expansion formula holds true:
(54)Φλ1,…,λp,ν;μ1,…,μq,τ(ρ1,…,ρp,κ,σ1,…,σq,κ)(zκ,s,a)=Γ(τ)Γ(1+ν-τ)sin⁡(γπ)π·∑n=-∞∞‍(-1)nΦλ1,…,λp,1;μ1,…,μq,1-γ-n(ρ1,…,ρp,κ,σ1,…,σq,κ)(zκ,s,a)(γ+n)Γ(1+ν-τ-γ-n)Γ(τ+γ+n),
provided that both members of (54) exist.Proof.
Settingu(z)=zν-1andv(z)=Φλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(zκ,s,a)in Theorem 5 withp=q=0andα=ν-τ, we obtain
(55)Dzν-τ{zν-1Φλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(zκ,s,a)}=∑n=-∞∞‍(ν-τγ+n)Dzν-τ-γ-n{zν-1}·Dzγ+n{Φλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(zκ,s,a)},
which, with the help of (22), (26), and (27), yields
(56)Dzν-τ{zν-1Φλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(zκ,s,a)}=Γ(ν)Γ(τ)zτ-1Φλ1,…,λp,ν;μ1,…,μq,τ(ρ1,…,ρp,κ,σ1,…,σq,κ)(zκ,s,a),Dzν-τ-γ-n{zν-1}=Γ(ν)Γ(τ+γ+n)zτ+γ+n-1,Dzγ+n{Φλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(zκ,s,a)}=z-γ-nΓ(1-γ-n)Φλ1,…,λp,1;μ1,…,μq,1-γ-n(ρ1,…,ρp,κ,σ1,…,σq,κ)(zκ,s,a).
Combining (56) with (55) and making some elementary simplifications, the asserted result (54) follows.Theorem 13.
Under the hypotheses of Theorem6, the following expansion formula holds true:(57)Φλ1,…,λp,ν;μ1,…,μq,τ(ρ1,…,ρp,κ,σ1,…,σq,κ)(zκ,s,a)=Γ(τ)Γ(1+ν-τ)Γ(ν)Γ(τ-γ-θ-1)·(sin⁡βπsin⁡[(ν-τ+β-θ)π]sin⁡θπ)×(Γ(1+γ+θ)sin⁡[(ν-τ+β)π]×sin⁡[(β-θ-γ)π]sin⁡[(θ+γ)π])-1·∑n=-∞∞‍((Φλ1,…,λp,θ+n;μ1,…,μq,1+θ+λ(ρ1,…,ρp,κ,σ1,…,σq,κ)(zκ,s,a)Γ(ν-θ-n)Γ(θ+n)×Φλ1,…,λp,θ+n;μ1,…,μq,1+θ+λ(ρ1,…,ρp,κ,σ1,…,σq,κ)(zκ,s,a))×(Γ(2+ν-τ+γ-n)Γ(-γ+n))-1(Φλ1,…,λp,θ+n;μ1,…,μq,1+θ+λ(ρ1,…,ρp,κ,σ1,…,σq,κ)),
provided that both members of (57) exist.Proof.
Upon first substitutingμ↦θandν↦γin Theorem 6 and then setting
(58)α=ν-τ,u(z)=zτ-βv(z)=Φλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(zκ,s,a),
in which bothu(z)andv(z)satisfy the conditions of Theorem 6, we have
(59)Dzν-τ{zν-1Φλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(zκ,s,a)}=zΓ(1+ν-τ)sin⁡βπsin⁡θπsin⁡(ν-τ+β-θ)πsin⁡(ν-τ+β)πsin⁡(β-θ-γ)πsin⁡(θ+γ)π·∑n=-∞∞‍Dzν-τ+γ+1-n{zν-θ-1-n}Γ(2+ν-τ+γ-n)Γ(-γ+n)·Dz-1-γ+n{zθ-1+nΦλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(zκ,s,a)}.
Now, by using (22), (26), and (27), we find that
(60)Dzν-τ{zν-1Φλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(zκ,s,a)}=Γ(ν)Γ(τ)zτ-1Φλ1,…,λp,ν;μ1,…,μq,τ(ρ1,…,ρp,κ,σ1,…,σq,κ)(zκ,s,a),Dzν-τ+γ+1-n{zν-θ-1-n}=Γ(ν-θ-n)Γ(τ-γ-θ-1)zτ-γ-θ-2,Dz-1-γ+n{zθ-1+nΦλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(zκ,s,a)}=Γ(θ+n)Γ(1+θ+γ)zθ+γ×Φλ1,…,λp,θ+n;μ1,…,μq,1+θ+λ(ρ1,…,ρp,κ,σ1,…,σq,κ)(zκ,s,a).
Thus, finally, the result (57) follows by combining (60) and (59).We now shift our focus to the different Taylor-like expansions in terms of different types of functions involving the extended multiparameters Hurwitz-Lerch zeta functionsΦλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(z,s,a).Theorem 14.
Under the assumptions of Theorem7, the following expansion formula holds true:
(61)Φλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(zκ,s,a)=c∑n=-∞∞‍(z0)-cn(z-z0)cnΓ(cn+1)Γ(1-cn)×Φλ1,…,λp,1;μ1,…,μq,1-cn(ρ1,…,ρp,κ,σ1,…,σq,κ)(z0κ,s,a)(|z-z0|=|z0|;λ>0),
provided that both members of (61) exist.Proof.
Settingf(z)=Φλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(zκ,s,a)in Theorem 7 withb=γ=0,0<c≦1, andθ(z)=z-z0, we have
(62)Φλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(zκ,s,a)=c∑n=-∞∞‍Dzcn{Φλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(zκ,s,a)}|z=z0(z-z0)cnΓ(1+cn),
forz0≠0and forzsuch that|z-z0|=|z0|.
Now, by making use of (26) withβ=0andα=cn, we find that
(63)Dzcn{Φλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(zκ,s,a)}|z=z0=z0-cnΓ(1-cn)Φλ1,…,λp,1;μ1,…,μq,1-cn(ρ1,…,ρp,κ,σ1,…,σq,κ)(z0κ,s,a).
By combining (62) and (63), we get result (61) asserted by Theorem 14.Theorem 15.
Under the hypotheses of Theorem8, the following expansion formula holds true:
(64)Φλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(zκ,s,a)=cz-α(z-z1)-βz1α+β·∑n=-∞∞‍eiπc(n+1)sin⁡[(α+cn+γ)π]Γ(α+cn+γ)sin⁡[(α-c+γ)π]Γ(1-β+cn+γ)Γ(α+β)·Φλ1,…,λp,α+cn+γ;μ1,…,μq,α+β(ρ1,…,ρp,κ,σ1,…,σq,κ)(z1κ,s,a)(z-z1z)cn+γ
forλ>0and forzonC1(1)defined by
(65)z=z12+z121+eiθ(-π<θ<π),
provided that both sides of (64) exist.Proof.
By takingf(z)=Φλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(zκ,s,a)in Theorem 8 withz2=0,μ=α,ν=β, and0<c≦1, we find that
(66)Φλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(zκ,s,a)=c(z-z1)-βz-αz1·∑n=-∞∞‍eiπc(n+1)sin⁡[(α+cn+γ)π]sin⁡[(α-c+γ)π]Γ(1-β+cn+γ)×(z-z1z)cn+γ·Dz-β+cn+γ{zα+cn+γ-1Φλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(zκ,s,a)}|z=z1.
Now, with the help of relation (26) withα↦-β+cn+γandβ↦α+cn+γ-1, we have
(67)Dz-β+cn+γ{zα+cn+γ-1Φλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(zκ,s,a)}|z=z1=z1α+β-1Γ(α+cn+γ)Γ(α+β)×Φλ1,…,λp,α+cn+γ;μ1,…,μq,α+β(ρ1,…,ρp,κ,σ1,…,σq,κ)(z1κ,s,a).
Thus, by combining (66) and (67), we are led to assertion (64) of Theorem 15.Theorem 16.
Under the hypotheses of Theorem9, the following expansion formula holds true:
(68)Φλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(zκ,s,a)=cz-β+γ(z-z1)-α+γz1β+α-2γ-1·∑n=-∞∞‍sin⁡[(β-cn-γ)π]eiπc(n+1)[z(z-z1)/z12]cnsin⁡[(β+c-γ)π]Γ(1-α+cn+γ)·Γ(β-cn-γ)Γ(β+α-2cn-2γ)×[(β-cn-γ)z1(α+β-2cn-2γ)(z-z1)×Φλ1,…,λp,β-cn-γ;μ1,…,μq,β+α-2cn-2γ(ρ1,…,ρp,κ,σ1,…,σq,κ)(z1κ,s,a)+(β-cn-γ)z1(α+β-2cn-2γ)×Φλ1,…,λp,1+β-cn-γ;μ1,…,μq,1+β+α-2cn-2γ(ρ1,…,ρp,κ,σ1,…,σq,κ)(z1κ,s,a)(β-cn-γ)z1(α+β-2cn-2γ)]
forλ>0and forzonC1(1)defined by
(69)z=z12+z121+eiθ(-π<θ<π),
provided that both sides of (68) exist.Proof.
Settingf(z)=Φλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(zκ,s,a)in Theorem 9 withz2=0,0<c≦1, q(z)=1, andθ(z)=(z-z1)(z-z2), we find that
(70)Φλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(zκ,s,a)=cz-β(z-z1)-α·∑n=-∞∞‍sin[(β-cn-γ)π]eIπc(n+1)[z(z-z1)]cn+γsin[(β+c-γ)π]Γ(1-α+cn+γ)·Dz-α+cn+γ×{Φλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)zβ-cn-γ-1(z+w-z1)×Φλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(zκ,s,a)}|z=z1,w=z.
With the help of the relation in (26), we have
(71)Dz-α+cn+γ×{Φλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(zκ,s,a)zβ-cn-γ-1(z+w-z1)×Φλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(zκ,s,a)}|z=z1,w=z=(z-z1)Dz-α+cn+γ×{zβ-cn-γ-1Φλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(zκ,s,a)}|z=z1+Dz-α+cn+γ{zβ-cn-γΦλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(zκ,s,a)}|z=z1=z1β+α-2cn-2γ×[(Φλ1,…,λp,β-cn-γ;μ1,…,μq,β+α-2cn-2γ(ρ1,…,ρp,κ,σ1,…,σq,κ)(z-z1)Γ(β-cn-γ)×Φλ1,…,λp,β-cn-γ;μ1,…,μq,β+α-2cn-2γ(ρ1,…,ρp,κ,σ1,…,σq,κ)(z1κ,s,a))×(z1Γ(β+α-2cn-2γ))-1+(Φλ1,…,λp,1+β-cn-γ;μ1,…,μq,1+β+α-2cn-2γ(ρ1,…,ρp,κ,σ1,…,σq,κ)(z1κ,s,a)Γ(1+β-cn-γ)×Φλ1,…,λp,1+β-cn-γ;μ1,…,μq,1+β+α-2cn-2γ(ρ1,…,ρp,κ,σ1,…,σq,κ)(z1κ,s,a))×(Γ(1+β+α-2cn-2γ))-1Φλ1,…,λp,β-cn-γ;μ1,…,μq,β+α-2cn-2γ(ρ1,…,ρp,κ,σ1,…,σq,κ)].
Thus, by combining (70) and (71), we obtain desired result (68).Finally, from Corollary11 given in the preceding section, we obtain the following new relation involving the extended multiparameters Hurwitz-Lerch zeta functionΦλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(z,s,a).Theorem 17.
Under the hypotheses of Corollary11, letkbe a positive integer. Then the following relation holds true:
(72)Φλ1,…,λp,α;μ1,…,μq,β(ρ1,…,ρp,1/k,σ1,…,σq,1/k)(z,s,a)=kΓ(β)Γ(kα)Γ(α)Γ(β+(k-1)α)·∑n=0∞‍∏j=1p‍(λj)nρj∏j=1q‍(μj)nσj(kα)n(β+(k-1)α)nzn(n+a)s·FD(k-1)[e-2(k-1)πi/kkα+n,1+α-β,…,1+α-β;β+(k-1)α+n;e-2πi/k,…,e-2(k-1)πi/k],
where λ>0andFD(n) denotes the Lauricella function of n variables defined by [38, page 60]
(73)FD(n)[a,b1,…,bn;c;x1,…,xn]=∑m1,…,mn=0∞‍(a)m1+⋯+mn(b1)m1⋯(bn)mn(c)m1+⋯+mn×x1m1m1!⋯xnmnmn!(max⁡{|x1|,…,|xn|}<1),
provided that both sides of (72) exist.Proof.
Puttingp=kand lettingf(z)=Φλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(z,s,a)in Corollary 11, we get
(74)zkOβα{Φλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(z,s,a)}=k(zk-1)α·zOβα{∏s=1k-1(1-zwe-2πis/k)‍β-α-1Φλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(z,s,a)(zk-1)α×∏s=1k-1(1-zwe-2πis/k)‍β-α-1}|w=z.
With the help of the definition of zOβαgiven by (53), we find for the left-hand side of (74) that
(75)zOβα{Φλ1,…,λp;μ1,…,μq(ρ1,…,ρp,σ1,…,σq)(z,s,a)}=Φλ1,…,λp,α;μ1,…,μq,β(ρ1,…,ρp,1/k,σ1,…,σq,1/k)(z,s,a).
We now expand each factor in the product in (74) in power series and replace extended multiparameters Hurwitz-Lerch zeta function by its series representation. We thus find for the right-hand side of (74) that
(76)k(zk-1)αzOβα×{((z/w)e-2(k-1)πi/k)mk-1mk-1!∑n=0∞‍∏j=1p‍(λj)nρjn!∏j=1q‍(μj)nσjzn+(k-1)α(n+a)s·∑m1,…,mk-1=0∞‍(1+α-β)m1⋯(1+α-β)mk-1·((z/w)e-2πi/k)m1m1!⋯((z/w)e-2(k-1)πi/k)mk-1mk-1!}|w=z=k(zk-1)α∑n=0∞‍∏j=1p‍(λj)nρjn!∏j=1q‍(μj)nσj(n+a)s·∑m1,…,mk-1=0∞‍(1+α-β)m1⋯(1+α-β)mk-1zm1+⋯+mk-1·(e-2πi/k)m1m1!⋯(e-2(k-1)πi/k)mk-1mk-1!×zOβα{zm1+⋯+mk-1+n+(k-1)α}=k∑n=0∞‍∏j=1p‍(λj)nρjn!∏j=1q‍(μj)nσjzn(n+a)s·∑m1,…,mk-1=0∞‍(1+α-β)m1⋯(1+α-β)mk-1·(e-2πi/k)m1m1!⋯(e-2(k-1)πi/k)mk-1mk-1!×Γ(β)Γ(m1+⋯+mk-1+n+kα)Γ(α)Γ(β+m1+⋯+mk-1+n+kα)=kΓ(β)Γ(kα)Γ(α)Γ(β+(k-1)α)×∑n=0∞‍∏j=1p‍(λj)nρj(kα)nn!∏j=1q‍(μj)nσj(β+(k-1)α)nzn(n+a)s·∑m1,…,mk-1=0∞‍(((β+(k-1)α)m1+⋯+mk-1)-1((n+kα)m1+⋯+mk-1×(1+α-β)m1⋯(1+α-β)mk-1)×((β+(k-1)α)m1+⋯+mk-1)-1)·(e-2πi/k)m1m1!⋯(e-2(k-1)πi/k)mk-1mk-1!.
Finally, by combining (75) and (76), we obtain the result (72) asserted by Theorem 17.Remark 18.
Each of the previous theorems can be written in terms ofH¯-function given in Definition 1. For instance, if we make use of (11), then Theorem 17 becomes(77)H¯p+2,q+31,p+2[-z∣(1-λ1,ρ1;1),…,(1-λp,ρp;1),(1-α,1k,1),(1-a,1;s)(0,1),(1-μ1,σ1;1),…,(1-μp,σp;1),(1-β,1k,1),(-a,1;s)]=∏j=1p‍Γ(λj)∏j=1q‍Γ(μj)kΓ(kα)Γ(β+(k-1)α)∑n=0∞‍∏j=1p‍(λj)nρjn!∏j=1q‍(μj)nσj(kα)n(β+(k-1)α)nzn(n+a)s·FD(k-1)[kα+n,1+α-β,…,1+α-β;β+(k-1)α+n;e-2πi/k,…,e-2(k-1)πi/k].


## 5. Corollaries and Consequences

We conclude this paper by presenting some special cases of the main results. These special cases and consequences are given in the form of the following corollaries.Settingk=3in Theorem 17, we obtain the following corollary.Corollary 19.
Under the hypotheses of Theorem17, the following expansion formula holds true:
(78)Φλ1,…,λp,α;μ1,…,μq,β(ρ1,…,ρp,1/3,σ1,…,σq,1/3)(z,s,a)=3Γ(β)Γ(3α)Γ(α)Γ(β+α)·∑n=0∞‍∏j=1p‍(λj)nρjn!∏j=1q‍(μj)nσj(3α)n(β+α)nzn(n+a)s·F1[3α+n,1+α-β,1+α-β;β+2α+n;-1,1],
whereF1denotes the first Appell function defined by [38, page 22]
(79)F1[a,b1,b2;c;x1,x2]=∑m1,m2=0∞‍(a)m1+m2(b1)m1(b2)m2(c)m1+m2x1m1m1!x2m2m2!(max⁡{|x1|,|x2|}<1),
provided that both sides of (78) exist.Settingp-1=q=1,κ=1, and making the following substitutionsρ1↦ρ,ρ2↦σ,σ1↦η,λ1↦λ,λ2↦μ, andμ1↦νin Theorem 14 lead to the following expansion formula given recently by Srivastava et al. [14].Corollary 20.
Under the hypotheses of Theorem14, the following expansion formula holds true:(80)Φλ,μ;ν(ρ,σ,η)(z,s,a)=c∑n=-∞∞‍(z0)-cn(z-z0)cnΓ(cn+1)Γ(1-cn)Φλ,μ,1;ν,1-cn(ρ,σ,1,η,1)(z0,s,a)
for z such that|z-z0|=|z0|and provided that both members of (80) exist.Remark 21.
The functionΦλ,μ;ν(ρ,σ,κ)(z,s,a), which occurs in Corollary 20 above, as well as its multiparameter generalizations, was introduced and investigated in a series of papers by Srivastava et al. (see [7, page 491, Equation (1.20)]; see also [8, 9, 39]) and is defined as follows:
(81)Φλ,μ;ν(ρ,σ,κ)(z,s,a):=∑n=0∞‍(λ)ρn(μ)σnn!(ν)κnzn(n+a)s(λ,μ∈ℂ;a,ν∈ℂ∖ℤ0-;ρ,σ,κ∈ℝ+;κ-ρ-σ>-1whens,z∈ℂ;κ-ρ-σ=-1,s∈ℂwhen|z|<δ*≔ρ-ρσ-σκκ;κ-ρ-σ=-1,ℜ(s+ν-λ-μ)>1when|z|=δ*).Puttingp-1=q=1,κ=1, and settingρ1=ρ2=σ1=1,λ1=μ, andλ2=μ1 in Theorem 15 reduce to the following expansion formula given recently by Gaboury [15, Equation (4.4)].Corollary 22.
Under the hypotheses of Theorem15, the following expansion formula holds true:
(82)Φμ*(z,s,a)=cz-α(z-z1)-βz1α+β·∑n=-∞∞‍eiπc(n+1)sin⁡[(α+cn+γ)π]Γ(α+cn+γ)sin⁡[(α-c+γ)π]Γ(1-β+cn+γ)Γ(α+β)·Φμ,α+cn+γ;α+β(1,1,1)(z1,s,a)(z-z1z)cn+γ
forzonC1(1)defined by
(83)z=z12+z121+eiθ(-π<θ<π),
provided that both sides of (82) exist.Remark 23.
The functionΦμ*(z,s,a), which occurs in Corollary 22 above, was introduced and studied by Goyal and Laddha [4, page 100, Equation (1.5)] and is defined as follows:
(84)Φμ*(z,s,a):=∑n=0∞‍(μ)nn!zn(n+a)s(μ∈ℂ;a,ν∈ℂ∖ℤ0-;s∈ℂwhen|z|<1;ℜ(s-μ)>1when|z|=1).Settings=0,κ=1,p↦p+1, andq↦q+1with
(85)ρ1=⋯=ρp=1,λp+1=ρp+1=1,σ1=⋯=σq=1,μq+1=β,σq+1=α
in Theorem 16, we deduce the following expansion formula.Corollary 24.
Under the hypotheses of Theorem16, the following formula holds true:
(86)1Γ(β)p+1Ψq+1*[(λ1,1),…,(λp,1),(1,1);(μ1,1),…,(μq,1),(β,α)z]=cz-β+γ(z-z1)-α+γz1β+α-2γ-1·∑n=-∞∞‍sin⁡((β-cn-γ)π)eiπc(n+1)[z(z-z1)/z12]cnsin⁡((β+c-γ)π)Γ(1-α+cn+γ)×Γ(β-cn-γ)Γ(β+α-2cn-2γ)·[(z-z1)Φλ1,…,λp,1,1+β-cn-γ;μ1,…,μq,β,1+β+α-2cn-2γ(1,…,1,1,1,1,…,1,α,1)×(z1,0,a)+z1(β-cn-γ)(α+β-2cn-2γ)×Φλ1,…,λp,1,β-cn-γ;μ1,…,μq,β,β+α-2cn-2γ(1,…,1,1,1,1,…,1,α,1)(z1,0,a)]
forzonC1(1)defined by
(87)z=z12+z121+eiθ(-π<θ<π),
provided that both sides of (86) exist.The functionlΨm*involved in the left-hand side of (86) is the Fox-Wright function(l,m∈ℕ0)withlnumerator parametersa1,…,aland m denominator parametersb1,…,bmsuch that
(88)aj∈ℂ(j=1,…,l),bj∈ℂ∖ℤ0-(j=1,…,m),
defined by (see, for details, [17, 40])
(89)lΨm*[(a1,A1),…,(al,Al);(b1,B1),…,(bm,Bm);z]:=∑n=0∞(a1)A1n⋯(al)Aln(b1)B1n⋯(bm)Bmnznn!(∑j=1l‍Aj>0,(j=1,…,l);Bj>0,(j=1,…,m);1+∑j=1m‍Bj-∑j=1l‍Aj≥0),
where the equality in the convergence condition holds true for suitably bounded values of|z|given by
(90)|z|<(∏j=1l‍Aj-Aj)(∏j=1m‍BjBj).In our series of forthcoming papers, we propose to consider and investigate analogous expansion formulas and other results involving theλ-extensions of the generalized Hurwitz-Lerch zeta functions. These functions have been investigated recently by Srivastava et al. [41] (see also [42]).



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