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

(1)

Citation for this paper:

Srivastava, H.M., Gaboury, S., & Tremblay, R. (2014). New Relations Involving an

Extended Multiparameter Hurwitz-Lerch Zeta Function with Applications.

International Journal of Analysis, Vol. 2014, Article ID 680850.

UVicSPACE: Research & Learning Repository

_____________________________________________________________

Faculty of Science

Faculty Publications

_____________________________________________________________

New Relations Involving an Extended Multiparameter Hurwitz-Lerch Zeta Function

with Applications

H.M. Srivastava, Sébastien Gaboury, & Richard Tremblay

2014

the Creative Commons Attribution License. http://creativecommons.org/licenses/by/3.0

This article was originally published at:

http://dx.doi.org/10.1155/2014/680850

(2)

Research Article

New Relations Involving an Extended Multiparameter

Hurwitz-Lerch Zeta Function with Applications

H. M. Srivastava,

1

Sébastien Gaboury,

2

and Richard Tremblay

2

1_{Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada V8W 3R4}

2_{Department of Mathematics and Computer Science, University of Quebec at Chicoutimi, Chicoutimi, QC, Canada G7H 2B1} Correspondence should be addressed to S´ebastien Gaboury; s1gabour@uqac.ca

Received 28 February 2014; Accepted 16 April 2014; Published 13 May 2014 Academic Editor: Shamsul Qamar

Copyright © 2014 H. M. Srivastava et al. 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. 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.

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 121 et seq.]; see also [2] and [3, page 194 et seq.]) Φ (𝑧, 𝑠, 𝑎) :=∑∞ 𝑛=0 𝑧𝑛 (𝑛 + 𝑎)𝑠, (𝑎 ∈ C \ Z−₀; 𝑠 ∈ C when |𝑧| < 1; R (𝑠) > 1 when |𝑧| = 1) . (1)

The Hurwitz-Lerch zeta function contains, as its special cases, the Riemann zeta function𝜁(𝑠), the Hurwitz zeta function 𝜁(𝑠, 𝑎), and the Lerch zeta function ℓ𝑠(𝜉) defined by

𝜁 (𝑠) :=∑∞ 𝑛=1 1 𝑛𝑠 = Φ (1, 𝑠, 1) = 𝜁 (𝑠, 1) (R (𝑠) > 1) , (2) 𝜁 (𝑠, 𝑎) :=∑∞ 𝑛=0 1 (𝑛 + 𝑎)𝑠 = Φ (1, 𝑠, 𝑎) (R (𝑠) > 1; 𝑎 ∈ C \ Z−0) , (3) ℓ𝑠(𝜉) := ∞ ∑ 𝑛=1 𝑒2𝑛𝜋𝑖𝜉 (𝑛 + 1)𝑠 = Φ (𝑒2𝜋𝑖𝜉, 𝑠, 1) (R (𝑠) > 1; 𝜉 ∈ R) , (4) respectively.

The Hurwitz-Lerch zeta function is connected with other special functions of analytic number theory such as the polylogarithmic function (or de Jonqui`ere’s function)𝐿𝑖_𝑠(𝑧):

𝐿𝑖_𝑠(𝑧) :=∑∞ 𝑛=1 𝑧𝑛 𝑛𝑠 = 𝑧Φ (𝑧, 𝑠, 1) (𝑠 ∈ C when |𝑧| < 1; R (𝑠) > 1 when |𝑧| = 1) (5)

and the Lipschitz-Lerch zeta function𝜙(𝜉, 𝑎, 𝑠) (see [1, page 122, Equation 2.5(11)]) 𝜙 (𝜉, 𝑎, 𝑠) :=∑∞ 𝑛=0 𝑒2𝑛𝜋𝑖𝜉 (𝑛 + 𝑎)𝑠 = Φ (𝑒2𝜋𝑖𝜉, 𝑠, 𝑎) (𝑎 ∈ C \ Z−₀; R (𝑠) > 0 when 𝜉 ∈ R \ Z; R (𝑠) > 1 when 𝜉 ∈ Z) . (6)

The Hurwitz-Lerch zeta functionΦ(𝑧, 𝑠, 𝑎) defined in (7) can be continued meromorphically to the whole complex𝑠-plane,

Volume 2014, Article ID 680850, 14 pages http://dx.doi.org/10.1155/2014/680850

(3)

except for a simple pole at𝑠 = 1 with its residue 1. It is well known that Φ (𝑧, 𝑠, 𝑎) = _{Γ (𝑠)}1 ∫∞ 0 𝑡𝑠−1_𝑒−𝑎𝑡 1 − 𝑧𝑒−𝑡𝑑𝑡 (R (𝑎) > 0; R (𝑠) > 0 when |𝑧| ≦ 1 (𝑧 ̸= 1) ; R (𝑠) > 1 when 𝑧 = 1) . (7)

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 functionΦ(𝑧, 𝑠, 𝑎) defined by (7) (see also [9]). In particular, they considered the following functions:

Φ(𝜌1,...,𝜌𝑝,𝜎1,...,𝜎𝑞) 𝜆1,...,𝜆𝑝;𝜇1,...,𝜇𝑞 (𝑧, 𝑠, 𝑎) := ∞ ∑ 𝑛=0 ∏𝑝_𝑗=1(𝜆𝑗)_𝑛𝜌_𝑗 𝑛!∏𝑞_𝑗=1(𝜇_𝑗)_𝑛𝜎 𝑗 𝑧𝑛 (𝑛 + 𝑎)𝑠 (𝑝, 𝑞 ∈ N0; 𝜆𝑗∈ C (𝑗 = 1, . . . , 𝑝) ; 𝑎, 𝜇_𝑗∈ C \ Z−₀(𝑗 = 1, . . . , 𝑞) ; 𝜌_𝑗, 𝜎_𝑘 ∈ R+_{(𝑗 = 1, . . . , 𝑝; 𝑘 = 1, . . . , 𝑞) ;} Δ > −1 when 𝑠, 𝑧 ∈ C; Δ = −1, 𝑠 ∈ C when |𝑧| < ∇∗; Δ = −1, R (Ξ) > 1₂ when |𝑧| = ∇∗) (8) with ∇∗:= (∏𝑝 𝑗=1𝜌 −𝜌𝑗 𝑗 ) ⋅ ( 𝑞 ∏ 𝑗=1𝜎 𝜎𝑗 𝑗 ) , Δ :=∑𝑞 𝑗=1 𝜎_𝑗−∑𝑝 𝑗=1 𝜌_𝑗, Ξ := 𝑠 +∑𝑞 𝑗=1 𝜇_𝑗−∑𝑝 𝑗=1 𝜆_𝑗+𝑝 − 𝑞 2 . (9)

Here, and for the remainder of this paper, (𝜆)_𝜅 denotes the Pochhammer symbol defined, in terms of the gamma function, by

(𝜆)𝜅:= Γ (𝜆 + 𝜅)_{Γ (𝜆)}

= {𝜆 (𝜆 + 1) ⋅ ⋅ ⋅ (𝜆 + 𝑛 − 1) (𝜅 = 𝑛 ∈ N; 𝜆 ∈ C)₁ _{(𝜅 = 0; 𝜆 ∈ C \ {0}) ;}

(10) it is being understood conventionally that (0)₀ := 1 and assumed tacitly that theΓ-quotient exists (see, for details, [10, page 21 et seq.]).

In their work, Srivastava et al. [7, page 504, Theo-rem 8] also proved the following relation for the function Φ(𝜌1,...,𝜌𝑝,𝜎1,...,𝜎𝑞) 𝜆1,...,𝜆𝑝;𝜇1,...,𝜇𝑞 (𝑧, 𝑠, 𝑎): Φ(𝜌1,...,𝜌𝑝,𝜎1,...,𝜎𝑞) 𝜆1,...,𝜆𝑝;𝜇1,...,𝜇𝑞 (𝑧, 𝑠, 𝑎) = ∏𝑞_𝑗=1Γ (𝜇𝑗) ∏𝑝_𝑗=1Γ (𝜆_𝑗) ⋅ 𝐻1,𝑝+1_{𝑝+1,𝑞+2}[−𝑧 | _{(0, 1) , (1 − 𝜇}(1 − 𝜆1, 𝜌1; 1) , . . . , (1 − 𝜆𝑝, 𝜌𝑝; 1) , (1 − 𝑎, 1; 𝑠) 1, 𝜎1; 1) , . . . , (1 − 𝜇𝑝, 𝜎𝑝; 1) , (−𝑎, 1; 𝑠)] (11)

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] 𝐻 (𝑧) = 𝐻(𝑧)𝑚,𝑛_p,q[ [ 𝑧 | (𝑎𝑗, 𝐴𝑗; 𝛼𝑗) 𝑛 𝑗=1, (𝑎𝑗, 𝐴𝑗) p 𝑗=𝑛+1 (𝑏_𝑗, 𝐵_𝑗)𝑚_𝑗=1, (𝑏_𝑗, 𝐵_𝑗; 𝛽_𝑗)q_𝑗=𝑚+1 ]_] := _2𝜋𝑖1 ∫ L𝜒 (𝑠) 𝑧 𝑠_𝑑𝑠 (𝑧 ̸= 0; 𝑖 = √−1; 𝜒 (𝑠) := (∏𝑚 𝑗=1 Γ (𝑏_𝑗− 𝐵_𝑗𝑠) ⋅∏𝑛 𝑗=1 {Γ (1 − 𝑎_𝑗+ 𝐴_𝑗𝑠)}𝛼𝑗₎ × (∏p 𝑗=𝑛+1 Γ (𝑎_𝑗− 𝐴_𝑗𝑠) ⋅ ∏q 𝑗=𝑚+1{Γ (1 − 𝑏𝑗+ 𝐵𝑗𝑠)} 𝛽𝑗₎ −1 ) . (12) Here the parameters

(4)

and the exponents

𝛼_𝑗 (𝑗 = 1, . . . , p) , 𝛽_𝑗 (𝑗 = 1, . . . , q) (14) can take noninteger values andL = L_𝑖𝜏;∞is a Mellin-Barnes type contour starting at the point𝜏 − 𝑖∞ and terminating at the point𝜏 + 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

Λ :=∑𝑚 𝑗=1 𝐵_𝑗+∑𝑛 𝑗=1󵄨󵄨󵄨󵄨󵄨𝛼𝑗󵄨󵄨󵄨󵄨󵄨 𝐴𝑗 − ∑q 𝑗=𝑚+1󵄨󵄨󵄨󵄨󵄨𝛽𝑗󵄨󵄨󵄨󵄨󵄨 𝐵𝑗 − ∑p 𝑗=𝑛+1 𝐴_𝑗> 0 (15)

and the region of absolute convergence is

󵄨󵄨󵄨󵄨arg(𝑧)󵄨󵄨󵄨󵄨 < 1₂𝜋Λ. (16) Note that when

𝛼₁= ⋅ ⋅ ⋅ = 𝛼_𝑛= 1, 𝛽_𝑚+1= ⋅ ⋅ ⋅ = 𝛽_q = 1, (17) the𝐻-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 Φ(𝜌_𝜆1,...,𝜌𝑝,𝜎1,...,𝜎𝑞)

1,...,𝜆𝑝;𝜇1,...,𝜇𝑞(𝑧, 𝑠, 𝑎) introduced and studied by Srivastava et al. [7]. InSection 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,

D𝛼_𝑧{𝑧𝑝𝑓 (𝑧)} = 1 Γ (−𝛼)∫ 𝑧 0 𝑓 (𝜉) 𝜉 𝑝_{(𝜉 − 𝑧)}−𝛼−1_𝑑𝜉 (R (𝛼) < 0; R (𝑝) > 1) , (18)

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

D𝛼_𝑧{𝑧𝑝𝑓 (𝑧)} = _𝑑𝑧𝑑𝑚_𝑚{D𝛼−𝑚_𝑧 {𝑧𝑝𝑓 (𝑧)}} . (19)

Branch line for

C1 C2 C3 C4 z Re(𝜉) g−1(0) a exp[−(a + 1)ln(g(𝜉) − g(z))] exp[p(ln(g(𝜉)] Im(𝜉)

Figure 1: Pochhammer’s contour.

This allows us to modify the restrictionR(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 frac-tional 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. Let 𝑓(𝑧) be analytic in a simply connected

region R of the complex 𝑧-plane. Let 𝑔(𝑧) be regular and univalent onR and let 𝑔−1(0) be an interior point of R. Then, if𝛼 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)𝑝_{𝑓(𝑧) with respect to 𝑔(𝑧) by} 𝐷𝛼_𝑔(𝑧){[𝑔 (𝑧)]𝑝𝑓 (𝑧)} =𝑒−𝑖𝜋𝑝Γ (1 + 𝛼) 4𝜋 sin (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+1 𝑑𝜉. (20) For nonintegers𝛼 and 𝑝, the functions 𝑔(𝜉)𝑝 and [𝑔(𝜉) − 𝑔(𝑧)]−𝛼−1in the integrand have two branch lines which begin, respectively, at𝜉 = 𝑧 and 𝜉 = 𝑔−1(0), and both branches pass through the point𝜉 = 𝑎 without crossing the Pochhammer contour𝑃(𝑎) = {𝐶₁ ∪ 𝐶₂∪ 𝐶₃ ∪ 𝐶₄} at any other point as shown inFigure 1. Here𝐹(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 function𝐹(𝜉) (see

Figure 2).

Remark 3. InDefinition 2, the function𝑓(𝑧) must be analytic at𝜉 = 𝑔−1(0). However, it is interesting to note here that if

(5)

Branch line for

Re(𝜉) z z2 z1 C2 C1 c = z1+ z2 2 (𝜉 − z2)𝜇𝜃(𝜉)a−𝛾−1 Im(𝜉) (𝜉 − z1)𝜃(𝜉)a−𝛾−1

Figure 2: Multiloops contour.

we could also allow𝑓(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 at 𝜉 = 𝑔−1(0) and 𝜉 = 𝑧 in (20), then we know that the integral is analytic for all𝑝 and for all𝛼 and for 𝑧 in R \ {𝑔−1(0)}. Indeed, in this case, the only possible singularities of𝐷𝛼_𝑔(𝑧){𝑔(𝑧)𝑝𝑓(𝑧)} are 𝛼 = −1, −2, . . . and 𝑝 = 0, ±1, ±2, . . ., which can directly be identified from the coefficient of the integral (20). However, by integrating by parts𝑁 times the integral in (20) by two different ways, we can show that𝛼 = −1, −2, . . . and 𝑝 = 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)]:

𝐷]_𝑧{𝑧𝜆−1𝐻𝑚,𝑛_𝑝,𝑞(𝜔𝑧𝜅)} = 𝑧𝜆−]−1𝐻(𝑧)𝑚,𝑛+1_{𝑝+1,𝑞+1} × [ [ 𝜔𝑧𝜅 | (1 − 𝜆, 𝜅; 1) , (𝑎𝑗, 𝐴𝑗; 𝛼𝑗) 𝑛 𝑗=1, (𝑎𝑗, 𝐴𝑗) 𝑝 𝑗=𝑛+1 (𝑏_𝑗, 𝐵_𝑗)𝑚_𝑗=1, (𝑎_𝑗, 𝐴_𝑗; 𝛽_𝑗)𝑞_𝑗=𝑚+1, (1 − 𝜆 + ], 𝜅; 1)]_] (R (𝜆) > 0; 𝜅 > 0) , (21)

in order to derive the following important fractional deriva-tive formula for this work:

𝐷]−𝜏_𝑧 {𝑧]−1Φ(𝜌1,...,𝜌𝑝,𝜎1,...,𝜎𝑞) 𝜆1,...,𝜆𝑝;𝜇1,...,𝜇𝑞 (𝑧 𝜅_{, 𝑠, 𝑎)}} = ∏ 𝑞 𝑗=1Γ (𝜇𝑗) ∏𝑝_𝑗=1Γ (𝜆_𝑗)𝑧 𝜏−1 ⋅ 𝐻1,𝑝+2_{𝑝+2,𝑞+3}[−𝑧𝜅| _{(0, 1) , (1 − 𝜇}(1 − 𝜆1, 𝜌1; 1) , . . . , (1 − 𝜆𝑝, 𝜌𝑝; 1) , (1 − ], 𝜅; 1) , (1 − 𝑎, 1; 𝑠) 1, 𝜎1; 1) , . . . , (1 − 𝜇𝑝, 𝜎𝑝; 1) , (1 − 𝜏, 𝜅; 1) , (−𝑎, 1; 𝑠)] =Γ (]) Γ (𝜏)𝑧𝜏−1Φ (𝜌1,...,𝜌𝑝,𝜅,𝜎1,...,𝜎𝑞,𝜅) 𝜆1,...,𝜆𝑝,];𝜇1,...,𝜇𝑞,𝜏 (𝑧 𝜅_{, 𝑠, 𝑎) (R (]) > 0; 𝜅 > 0) .} (22)

This fractional calculus formula was obtained by using the Riemann-Liouville representation for the fractional deriva-tive. Adopting the Pochhammer based representation for the fractional derivative, these last restrictions become𝜅 + ] − 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 = 1 in (22) and making the following substitutions𝜌₁ 󳨃→ 𝜌, 𝜌₂ 󳨃→ 𝜎, 𝜎₁ 󳨃→ 𝜅, 𝜆₁ 󳨃→ 𝜆, 𝜆₂󳨃→ 𝜇, and 𝜇₁󳨃→ ] lead to 𝐷]−𝜏 𝑧 {𝑧]−1Φ(𝜌,𝜎,𝜅)𝜆,𝜇;] (𝑧𝜅, 𝑠, 𝑎)} = Γ (]) Γ (𝜆) Γ (𝜇)𝑧𝜏−1𝐻 1,4 4,4 ×[−𝑧𝜅| (1 − 𝜆,𝜌;1),(1 − 𝜇,𝜎;1),(1 − ], 𝜅; 1),(1 − 𝑎,1;𝑠)_{(0, 1),(1 − ],𝜅; 1) , (1 − 𝜏, 𝜅; 1) , (−𝑎, 1; 𝑠)} ] = Γ (]) Γ (𝜆) Γ (𝜇)𝑧𝜏−1𝐻 1,3 3,3 × [−𝑧𝜅 | (1 − 𝜆, 𝜌; 1) , (1 − 𝜇, 𝜎; 1) , (1 − 𝑎, 1; 𝑠)_{(0, 1) , (1 − 𝜏, 𝜅; 1) , (−𝑎, 1; 𝑠)} ] = Γ (]) Γ (𝜏)𝑧𝜏−1Φ (𝜌,𝜎,𝜅) 𝜆,𝜇;𝜏 (𝑧𝜅, 𝑠, 𝑎)

(] not a negative integer; 𝜅 > 0) . (23)

(6)

Furthermore, if we put𝜌 = 𝜎 = 𝜅 = 1 in (23), then we obtain 𝐷]−𝜏_𝑧 {𝑧]−1Φ(1,1,1)_𝜆,𝜇;] (𝑧, 𝑠, 𝑎)} = Γ (]) Γ (𝜏)𝑧𝜏−1Φ𝜆,𝜇;𝜏(𝑧, 𝑠, 𝑎) = Γ (]) Γ (𝜆) Γ (𝜇)𝑧𝜏−1𝐻 1,3 3,3 × [−𝑧 | (1 − 𝜆, 1; 1) , (1 − 𝜇, 1; 1) , (1 − 𝑎, 1; 𝑠)_{(0, 1) , (1 − 𝜏, 1; 1) , (−𝑎, 1; 𝑠)} ] (] not a negative integer; 𝜅 > 0) .

(24)

Finally, letting𝜆 = ] in (24), this yields after elementary cal-culations Φ∗_𝜇(𝑧, 𝑠, 𝑎) = Γ (𝜏) Γ (]) Γ (𝜇)𝐻 1,3 3,3 × [−𝑧 | (1 − ], 1; 1) , (1 − 𝜇, 1; 1) , (1 − 𝑎, 1; 𝑠)_{(0, 1) , (1 − 𝜏, 1; 1) , (−𝑎, 1; 𝑠)} ] . (25)

Another fractional derivative formula that will be very useful in this work is given by the following formula:

𝐷𝛼 𝑧{𝑧𝛽Φ(𝜌𝜆11,...,𝜆,...,𝜌𝑝𝑝;𝜇,𝜎11,...,𝜇,...,𝜎𝑞𝑞)(𝑧 𝜅_{, 𝑠, 𝑎)}} = Γ (1 + 𝛽) Γ (1 + 𝛽 − 𝛼)𝑧𝛽−𝛼 ⋅ Φ(𝜌1,...,𝜌𝑝,𝜅,𝜎1,...,𝜎𝑞,𝜅) 𝜆1,...,𝜆𝑝,1+𝛽;𝜇1,...,𝜇𝑞,1+𝛽−𝛼(𝑧 𝜅_{, 𝑠, 𝑎)}

(𝛽 not a negative integer, 𝜅 > 0) .

(26)

This last result can be established with the help of the following well known formula [28, page 83, Equation(2.4)]:

𝐷𝛼_𝑧{𝑧𝑝} = Γ (1 + 𝑝)

Γ (1 + 𝑝 − 𝛼)𝑧𝑝−𝛼 (R (𝑝) > −1) . (27) Adopting the Pochhammer based representation for the fractional derivative modifies the restriction to the case when𝑝 is 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.Theorem 5is a slightly modified the-orem obtained in 1970 by Osler [22].Theorem 6was 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) Let R be a simply connected region containing

the origin. (ii) Let 𝑢(𝑧) and V(𝑧) satisfy the conditions of

Definition 2for the existence of the fractional derivative. Then, forR(𝑝 + 𝑞) > −1 and 𝛾 ∈ C, the following Leibniz rule holds true: 𝐷𝛼_𝑧{𝑧𝑝+𝑞𝑢 (𝑧) V (𝑧)} = ∑∞ 𝑛=−∞( 𝛼𝛾 + 𝑛) 𝐷 𝛼−𝛾−𝑛 𝑧 × {𝑧𝑝𝑢 (𝑧)} 𝐷𝛾+𝑛𝑧 {𝑧𝑞V (𝑧)} . (28)

Theorem 6. (i) Let R be a simply connected region containing

the origin. (ii) Let 𝑢(𝑧) and V(𝑧) satisfy the conditions of

Definition 2for the existence of the fractional derivative. (iii) LetU ⊂ R be the region of analyticity of the function 𝑢(𝑧) and letV ⊂ R be the region of analyticity of the function V(𝑧). Then, for

𝑧 ̸= 0, 𝑧 ∈ U ∩ V, R (1 − 𝛽) > 0, (29)

the following product rule holds true:

𝐷𝛼_𝑧{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 − ]) 𝜋] sin [(𝜇 + ]) 𝜋] ⋅ ∑∞

𝑛=−∞

𝐷𝛼+]+1−𝑛_𝑧 {𝑧𝛼+𝛽−𝜇−1−𝑛𝑢 (𝑧)} 𝐷−1−]+𝑛𝑧 {𝑧𝜇−1+𝑛V (𝑧)}

Γ (2 + 𝛼 + ] − 𝑛) Γ (−] + 𝑛) . (30) Next, in 1971, Osler [30] established the following gen-eralized Taylor-like series expansion involving fractional derivatives.

Theorem 7. Let 𝑓(𝑧) be an analytic function in a simply

con-nected regionR. Let 𝛼 and 𝛾 be arbitrary complex numbers and

𝜃 (𝑧) = (𝑧 − 𝑧0) 𝑞 (𝑧) (31)

with𝑞(𝑧) a regular and univalent function without any zero in

R. Let 𝑎 be a positive real number and 𝐾 = {0, 1, . . . , [𝑐]

([𝑐] the largest integer not greater than 𝑐)} . (32)

Let𝑏 and 𝑧₀be two points inR such that 𝑏 ̸= 𝑧₀ and let

(7)

Then the following relationship holds true: ∑ 𝑘∈𝐾 𝑐−1𝜔−𝛾𝑘𝑓 (𝜃−1(𝜃 (𝑧) 𝜔𝑘)) = ∑∞ 𝑛=−∞ [𝜃 (𝑧)]𝑐𝑛+𝛾 Γ (𝑐𝑛 + 𝛾 + 1) ⋅ 𝐷𝑐𝑛+𝛾_𝑧−𝑏 {𝑓 (𝑧) 𝜃󸀠(𝑧) (𝑧 − 𝑧0 𝜃 (𝑧)) 𝑐𝑛+𝛾+1 }󵄨󵄨󵄨󵄨_{󵄨󵄨󵄨󵄨} 󵄨𝑧=𝑧0 (󵄨󵄨󵄨󵄨𝑧 − 𝑧0󵄨󵄨󵄨󵄨 = 󵄨󵄨󵄨󵄨𝑧0󵄨󵄨󵄨󵄨). (34)

In particular, if0 < 𝑐 ≦ 1 and 𝜃(𝑧) = (𝑧 − 𝑧₀), then 𝑘 = 0 and the formula (34) reduces to the following form:

𝑓 (𝑧) = 𝑐 ∑∞

𝑛=−∞

(𝑧 − 𝑧₀)𝑐𝑛+𝛾

Γ (𝑐𝑛 + 𝛾 + 1)𝐷𝑐𝑛+𝛾𝑧−𝑏 {𝑓 (𝑧)}󵄨󵄨󵄨󵄨󵄨𝑧=𝑧0. (35) 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 function𝑓(𝑧) in terms of the rational expression((𝑧−𝑧₁)/(𝑧−𝑧₂)), where 𝑧₁and𝑧₂are two arbitrary points inside the regionR of analyticity of 𝑓(𝑧). In particular, they obtained the following result.

Theorem 8. (i) Let 𝑐 be real and positive and let

𝜔 = exp (2𝜋𝑖_𝑎 ) . (36) (ii) Let 𝑓(𝑧) be analytic in the simply connected region R

with𝑧₁ and𝑧₂being interior points ofR. (iii) Let the set of curves {𝐶 (𝑡) : 𝐶 (𝑡) ⊂ R, 0 < 𝑡 ≦ 𝑟} (37) be defined by 𝐶 (𝑡) = 𝐶1(𝑡) ∪ 𝐶2(𝑡) = {𝑧 : 󵄨󵄨󵄨󵄨𝜆𝑡(𝑧1, 𝑧2; 𝑧)󵄨󵄨󵄨󵄨 =󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝜆𝑡(𝑧1, 𝑧2;𝑧1+ 𝑧₂ 2)󵄨󵄨󵄨󵄨󵄨󵄨󵄨}, (38) where 𝜆_𝑡(𝑧₁, 𝑧₂; 𝑧) = [𝑧 −𝑧1+ 𝑧2 2 + 𝑡 ( 𝑧₁− 𝑧₂ 2 )] ⋅ [𝑧 − (𝑧1+ 𝑧2 2 ) − 𝑡 ( 𝑧₁− 𝑧₂ 2 )] , (39)

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

𝑧₁+ 𝑧₂ 2 + (

𝑧₁− 𝑧₂

2 ) 𝑡 (40)

and the other loop𝐶₂(𝑡) encircles the focus point

𝑧1+ 𝑧2

2 − ( 𝑧1− 𝑧2

2 ) 𝑡, (41)

for each𝑡 such that 0 < 𝑡 ≦ 𝑟. (iv) Let

[(𝑧 − 𝑧₁) (𝑧 − 𝑧₂)]𝜆= exp (𝜆 ln (𝜃 ((𝑧 − 𝑧₁) (𝑧 − 𝑧₂)))) (42)

denote the principal branch of that function which is continu-ous and inside𝐶(𝑟), cut by the respective two branch lines 𝐿_± defined by 𝐿_±= { { { { { { { {𝑧 : 𝑧 = 𝑧1+ 𝑧2 2 ± 𝑡 ( 𝑧₁− 𝑧₂ 2 )} (0 ≦ 𝑡 ≦ 1) , {𝑧 : 𝑧 = 𝑧1+ 𝑧2 2 ± 𝑖𝑡 ( 𝑧₁− 𝑧₂ 2 )} (𝑡 < 0) (43)

such that ln((𝑧 − 𝑧₁)(𝑧 − 𝑧₂)) is real when (𝑧 − 𝑧₁)(𝑧 −

𝑧₂) > 0. (v) Let 𝑓(𝑧) satisfy the conditions ofDefinition 2for the existence of the fractional derivative of(𝑧 − 𝑧₂)𝑝𝑓(𝑧) of order𝛼 for 𝑧 ∈ R\{𝐿₊∪𝐿₋}, denoted by 𝐷𝛼_𝑧−𝑧₂{(𝑧−𝑧₂)𝑝𝑓(𝑧)}, where𝛼 and 𝑝 are real or complex numbers. (vi) Let

𝐾 = {𝑘 : 𝑘 ∈ N, arg (𝜆_𝑡(𝑧₁, 𝑧₂,𝑧1+ 𝑧2 2 )) < arg (𝜆_𝑡(𝑧₁, 𝑧₂,𝑧1+ 𝑧2 2 )) + 2𝜋𝑘 𝑎 < arg (𝜆_𝑡(𝑧₁, 𝑧₂,𝑧1+ 𝑧2 2 )) + 2𝜋} . (44)

Then, for arbitrary complex numbers𝜇, ], 𝛾 and for 𝑧 on 𝐶₁(1) defined by 𝜉 = 𝑧1+ 𝑧2 2 + 𝑧₁− 𝑧₂ 2 √1 + 𝑒𝑖𝜃 (−𝜋 < 𝜃 < 𝜋) , ∑ 𝑘∈𝐾 𝑐−1𝜔−𝛾𝑘 𝑧₁− 𝑧₂𝑓 (𝜙−1(𝜔𝑘𝜙 (𝑧))) × [𝜙−1(𝜙 (𝑧) 𝜔𝑘) − 𝑧₁]][𝜙−1(𝜙 (𝑧) 𝜔𝑘) − 𝑧₂]𝜇 = ∑∞ 𝑛=−∞ 𝑒𝑖𝜋𝑐(𝑛+1)_sin_{[(𝜇 + 𝑐𝑛 + 𝛾) 𝜋]} sin[(𝜇 − 𝑐 + 𝛾) 𝜋] Γ (1 − ] + 𝑐𝑛 + 𝛾) ⋅ 𝐷−]+𝑐𝑛+𝛾_𝑧−𝑧₂ {(𝑧 − 𝑧₂)𝜇+𝑐𝑛+𝛾−1𝑓 (𝑧)}󵄨󵄨󵄨󵄨_󵄨_𝑧=𝑧 1[𝜙 (𝑧)] 𝑐𝑛+𝛾_, (45) where 𝜙 (𝑧) = 𝑧 − 𝑧1 𝑧 − 𝑧₂. (46) The case0 < 𝑐 ≦ 1 ofTheorem 8reduces to the following form: 𝑐−1_{𝑓 (𝑧) (𝑧 − 𝑧} 1)](𝑧 − 𝑧2)𝜇 𝑧₁− 𝑧₂ = ∑∞ 𝑛=−∞ 𝑒𝑖𝜋𝑐(𝑛+1)_sin_{[(𝜇 + 𝑐𝑛 + 𝛾) 𝜋]} sin[(𝜇 − 𝑐 + 𝛾) 𝜋] Γ (1 − ] + 𝑐𝑛 + 𝛾) ⋅ 𝐷−]+𝑐𝑛+𝛾_𝑧−𝑧₂ {(𝑧 − 𝑧₂)𝜇+𝑐𝑛+𝛾−1_{𝑓(𝑧)}󵄨󵄨󵄨󵄨󵄨}_𝑧=𝑧 1( 𝑧 − 𝑧₁ 𝑧 − 𝑧₂) 𝑐𝑛+𝛾 . (47)

(8)

Tremblay and Fug`ere [36] developed the power series of an analytic function𝑓(𝑧) in terms of the function (𝑧 − 𝑧₁)(𝑧 − 𝑧₂), where 𝑧₁ and 𝑧₂are two arbitrary points inside the analyticity regionR of 𝑓(𝑧). Explicitly, they showed the following theorem.

Theorem 9. Under the assumptions ofTheorem 8, the follow-ing expansion formula holds true:

∑ 𝑘∈𝐾 𝑐−1𝜔−𝛾𝑘 × [(𝑧2− 𝑧1₂+ √Δ𝑘) 𝛼 (𝑧1− 𝑧2₂+ √Δ𝑘) 𝛽 ⋅ 𝑓 (𝑧1+ 𝑧2+ √Δ𝑘 2 ) − 𝑒𝑖𝜋(𝛼−𝛽) sin[(𝛼 + 𝑐 − 𝛾) 𝜋] sin[(𝛽 + 𝑐 − 𝛾) 𝜋] ⋅ (𝑧2− 𝑧1₂− √Δ𝑘) 𝛼 (𝑧1− 𝑧2₂− √Δ𝑘) 𝛽 × 𝑓 (𝑧1+ 𝑧2₂− √Δ𝑘)] = ∑∞ 𝑛=−∞ sin[(𝛽 − 𝑐𝑛 − 𝛾) 𝜋] sin[(𝛽 − 𝑐 − 𝛾) 𝜋] 𝑒−𝑖𝜋𝑐(𝑛+1)[𝜃 (𝑧)]𝑐𝑛+𝛾 Γ (1 − 𝛼 + 𝑐𝑛 + 𝛾) ⋅ 𝐷−𝛼+𝑐𝑛+𝛾_𝑧−𝑧₂ {(𝑧 − 𝑧₂)𝛽−𝑐𝑛−𝛾−1 × ( 𝜃 (𝑧) (𝑧 − 𝑧2) (𝑧 − 𝑧1)) −𝑐𝑛−𝛾−1 × 𝜃󸀠(𝑧) 𝑓 (𝑧) }󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨𝑧=𝑧1 , (48) where Δ_𝑘 = (𝑧₁− 𝑧₂)2+ 4𝑉 (𝜔𝑘𝜃 (𝑧)) , 𝑉 (𝑧) =∑∞ 𝑟=1𝐷 𝑟−1 𝑧 {[𝑞 (𝑧)]−𝑟}󵄨󵄨󵄨󵄨󵄨_𝑧=0 𝑧 𝑟 𝑟!, 𝜃 (𝑧) = (𝑧 − 𝑧1) (𝑧 − 𝑧2) 𝑞 ((𝑧 − 𝑧1) (𝑧 − 𝑧2)) . (49)

As special case, if we set0 < 𝑐 ≦ 1, 𝑞(𝑧) = 1 (𝜃(𝑧) = (𝑧 − 𝑧₁)(𝑧 − 𝑧₂)), and 𝑧₂= 0 in (48), we obtain 𝑓 (𝑧) = 𝑐𝑧−𝛽_{(𝑧 − 𝑧} 1)−𝛼 × ∑∞ 𝑛=−∞ sin[(𝛽 − 𝑐𝑛 − 𝛾) 𝜋] sin[(𝛽 + 𝑐 − 𝛾) 𝜋] 𝑒𝑖𝜋𝑐(𝑛+1)_{[𝑧 (𝑧 − 𝑧} 1)]𝑐𝑛+𝛾 Γ (1 − 𝛼 + 𝑐𝑛 + 𝛾) ⋅ 𝐷−𝛼+𝑐𝑛+𝛾 𝑧 {𝑧𝛽−𝑐𝑛−𝛾−1(𝑧 + 𝑤 − 𝑧1) 𝑓 (𝑧)}󵄨󵄨󵄨󵄨_󵄨_𝑧=𝑧₁_,(𝑤=𝑧). (50)

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 asTheorem 10.

Theorem 10. Let 𝑓(𝑔−1_{(𝑧)) and 𝑓(ℎ}−1_{(𝑧)) be defined and}

analytic in the simply connected regionR of the complex 𝑧-plane and let the origin be an interior or boundary point of R. Suppose also that 𝑔−1(𝑧) and ℎ−1(𝑧) are regular univalent functions on R and that ℎ−1(0) = 𝑔−1(0). Let∮ 𝑓(𝑔−1(𝑧))𝑑𝑧 vanish over simple closed contour in

R ∪ {0} through the origin. Then the following relation holds

true: 𝐷𝛼_𝑔(𝑧){𝑓 (𝑧)} = 𝐷𝛼ℎ(𝑧){𝑓 (𝑧) 𝑔 󸀠_(𝑧) ℎ󸀠_(𝑧) ( ℎ(𝑤) − ℎ(𝑧) 𝑔(𝑤) − 𝑔(𝑧)) 𝛼+1 }󵄨󵄨󵄨󵄨_{󵄨󵄨󵄨󵄨} 󵄨𝑤=𝑧 . (51)

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 ofTheorem 10, let𝑝 be a positive integer. Then the following relation holds true:

𝑧𝑝𝑂_𝛽𝛼{𝑓 (𝑧)} = 𝑝(𝑧𝑝−1)−𝛼 ⋅_𝑧𝑂_𝛽𝛼{𝑓 (𝑧) (𝑧𝑝−1)𝛼𝑝−1∏ 𝑠=1 (1 − 𝑧 𝑤𝑒−2𝜋𝑖𝑠/𝑝) 𝛽−𝛼−1 }󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨𝑤=𝑧 , (52) where 𝑔(𝑧)𝑂𝛼𝛽{⋅ ⋅ ⋅ } := Γ (𝛽)_{Γ (𝛼)}[𝑔 (𝑧)]1−𝛽𝐷𝛼−𝛽𝑔(𝑧){[𝑔 (𝑧)]𝛼−1 ⋅ ⋅ ⋅ } . (53)

4. Relations Involving the Extended

Multiparameters Hurwitz-Lerch Zeta

Function

Φ

(𝜌_𝜆1,...,𝜌𝑝,𝜎1,...,𝜎𝑞)

1,...,𝜆𝑝;𝜇1,...,𝜇𝑞

(𝑧,𝑠,𝑎)

In this section, we present the new expansion formulas involving the extended multiparameters Hurwitz-Lerch zeta functionΦ(𝜌_𝜆1,...,𝜌𝑝,𝜎1,...,𝜎𝑞)

(9)

Theorem 12. Under the hypotheses ofTheorem 5, the following expansion formula holds true:

Φ(𝜌1,...,𝜌𝑝,𝜅,𝜎1,...,𝜎𝑞,𝜅) 𝜆1,...,𝜆𝑝,];𝜇1,...,𝜇𝑞,𝜏 (𝑧 𝜅_{, 𝑠, 𝑎)} = Γ (𝜏) Γ (1 + ] − 𝜏) sin (𝛾𝜋) 𝜋 ⋅ ∑∞ 𝑛=−∞ (−1)𝑛 Φ(𝜌1,...,𝜌𝑝,𝜅,𝜎1,...,𝜎𝑞,𝜅) 𝜆1,...,𝜆𝑝,1;𝜇1,...,𝜇𝑞,1−𝛾−𝑛(𝑧 𝜅_{, 𝑠, 𝑎)} (𝛾 + 𝑛) Γ (1 + ] − 𝜏 − 𝛾 − 𝑛) Γ (𝜏 + 𝛾 + 𝑛), (54)

provided that both members of (54) exist.

Proof. Setting𝑢(𝑧) = 𝑧]−1 and V(𝑧) = Φ(𝜌_𝜆1,...,𝜌𝑝,𝜎1,...,𝜎𝑞) 1,...,𝜆𝑝;𝜇1,...,𝜇𝑞 (𝑧

𝜅_{, 𝑠,}

𝑎) inTheorem 5with𝑝 = 𝑞 = 0 and 𝛼 = ] − 𝜏, we obtain 𝐷_𝑧]−𝜏{𝑧]−1 Φ(𝜌1,...,𝜌𝑝,𝜎1,...,𝜎𝑞) 𝜆1,...,𝜆𝑝;𝜇1,...,𝜇𝑞 (𝑧 𝜅_{, 𝑠, 𝑎)}} = ∑∞ 𝑛=−∞(] − 𝜏𝛾 + 𝑛) 𝐷 ]−𝜏−𝛾−𝑛 𝑧 {𝑧]−1} ⋅ 𝐷𝛾+𝑛_𝑧 {Φ(𝜌1,...,𝜌𝑝,𝜎1,...,𝜎𝑞) 𝜆1,...,𝜆𝑝;𝜇1,...,𝜇𝑞 (𝑧 𝜅_{, 𝑠, 𝑎)} ,} (55)

which, with the help of (22), (26), and (27), yields 𝐷_𝑧]−𝜏{𝑧]−1 Φ(𝜌1,...,𝜌𝑝,𝜎1,...,𝜎𝑞) 𝜆1,...,𝜆𝑝;𝜇1,...,𝜇𝑞 (𝑧 𝜅_{, 𝑠, 𝑎)}} = Γ (]) Γ (𝜏) 𝑧𝜏−1Φ (𝜌1,...,𝜌𝑝,𝜅,𝜎1,...,𝜎𝑞,𝜅) 𝜆1,...,𝜆𝑝,];𝜇1,...,𝜇𝑞,𝜏 (𝑧 𝜅_{, 𝑠, 𝑎) ,} 𝐷]−𝜏−𝛾−𝑛_𝑧 {𝑧]−1} = Γ (]) Γ (𝜏 + 𝛾 + 𝑛)𝑧𝜏+𝛾+𝑛−1, 𝐷𝛾+𝑛_𝑧 {Φ(𝜌1,...,𝜌𝑝,𝜎1,...,𝜎𝑞) 𝜆1,...,𝜆𝑝;𝜇1,...,𝜇𝑞 (𝑧 𝜅_{, 𝑠, 𝑎)}} = 𝑧−𝛾−𝑛 Γ (1 − 𝛾 − 𝑛) Φ (𝜌1,...,𝜌𝑝,𝜅,𝜎1,...,𝜎𝑞,𝜅) 𝜆1,...,𝜆𝑝,1;𝜇1,...,𝜇𝑞,1−𝛾−𝑛(𝑧 𝜅_{, 𝑠, 𝑎) .} (56)

Combining (56) with (55) and making some elementary simplifications, the asserted result (54) follows.

Φ(𝜌1,...,𝜌𝑝,𝜅,𝜎1,...,𝜎𝑞,𝜅)

𝜆1,...,𝜆𝑝,];𝜇1,...,𝜇𝑞,𝜏 (𝑧

𝜅_{, 𝑠, 𝑎)}

= Γ (𝜏) Γ (1 + ] − 𝜏) Γ (]) Γ (𝜏 − 𝛾 − 𝜃 − 1)

⋅ (sin 𝛽𝜋 sin [(] − 𝜏 + 𝛽 − 𝜃) 𝜋] sin 𝜃𝜋) × (Γ (1 + 𝛾 + 𝜃) sin [(] − 𝜏 + 𝛽) 𝜋] × sin [(𝛽 − 𝜃 − 𝛾) 𝜋] sin [(𝜃 + 𝛾) 𝜋])−1 ⋅ ∑∞ 𝑛=−∞((Γ (] − 𝜃 − 𝑛) Γ (𝜃 + 𝑛) × Φ(𝜌1,...,𝜌𝑝,𝜅,𝜎1,...,𝜎𝑞,𝜅) 𝜆1,...,𝜆𝑝,𝜃+𝑛;𝜇1,...,𝜇𝑞,1+𝜃+𝜆(𝑧 𝜅_{, 𝑠, 𝑎))} × (Γ (2 + ] − 𝜏 + 𝛾 − 𝑛) Γ (−𝛾 + 𝑛))−1) , (57)

provided that both members of (57) exist.

Proof. Upon first substituting𝜇 󳨃→ 𝜃 and ] 󳨃→ 𝛾 in Theorem 6and then setting

𝛼 = ] − 𝜏, 𝑢 (𝑧) = 𝑧𝜏−𝛽 V (𝑧) = Φ(𝜌1,...,𝜌𝑝,𝜎1,...,𝜎𝑞)

𝜆1,...,𝜆𝑝;𝜇1,...,𝜇𝑞 (𝑧

𝜅_{, 𝑠, 𝑎) ,} (58)

in which both𝑢(𝑧) and V(𝑧) satisfy the conditions of

Theorem 6, we have 𝐷]−𝜏_𝑧 {𝑧]−1Φ(𝜌1,...,𝜌𝑝,𝜎1,...,𝜎𝑞)

𝜆1,...,𝜆𝑝;𝜇1,...,𝜇𝑞 (𝑧

𝜅_{, 𝑠, 𝑎)}}

=𝑧Γ (1 + ] − 𝜏) sin 𝛽𝜋 sin 𝜃𝜋 sin (] − 𝜏 + 𝛽 − 𝜃) 𝜋 sin(] − 𝜏 + 𝛽) 𝜋 sin (𝛽 − 𝜃 − 𝛾) 𝜋 sin (𝜃 + 𝛾) 𝜋 ⋅ ∑∞ 𝑛=−∞ 𝐷]−𝜏+𝛾+1−𝑛 𝑧 {𝑧]−𝜃−1−𝑛} Γ (2 + ] − 𝜏 + 𝛾 − 𝑛) Γ (−𝛾 + 𝑛) ⋅ 𝐷−1−𝛾+𝑛_𝑧 {𝑧𝜃−1+𝑛Φ(𝜌1,...,𝜌𝑝,𝜎1,...,𝜎𝑞) 𝜆1,...,𝜆𝑝;𝜇1,...,𝜇𝑞 (𝑧 𝜅_{, 𝑠, 𝑎)} .} (59)

Now, by using (22), (26), and (27), we find that 𝐷]−𝜏_𝑧 {𝑧]−1Φ(𝜌1,...,𝜌𝑝,𝜎1,...,𝜎𝑞) 𝜆1,...,𝜆𝑝;𝜇1,...,𝜇𝑞 (𝑧 𝜅_{, 𝑠, 𝑎)}} = Γ (]) Γ (𝜏)𝑧𝜏−1Φ (𝜌1,...,𝜌𝑝,𝜅,𝜎1,...,𝜎𝑞,𝜅) 𝜆1,...,𝜆𝑝,];𝜇1,...,𝜇𝑞,𝜏 (𝑧 𝜅_{, 𝑠, 𝑎) ,} 𝐷]−𝜏+𝛾+1−𝑛_𝑧 {𝑧]−𝜃−1−𝑛} = _{Γ (𝜏 − 𝛾 − 𝜃 − 1)}Γ (] − 𝜃 − 𝑛) 𝑧𝜏−𝛾−𝜃−2, 𝐷−1−𝛾+𝑛_𝑧 {𝑧𝜃−1+𝑛Φ(𝜌1,...,𝜌𝑝,𝜎1,...,𝜎𝑞) 𝜆1,...,𝜆𝑝;𝜇1,...,𝜇𝑞 (𝑧 𝜅_{, 𝑠, 𝑎)}} = Γ (𝜃 + 𝑛) Γ (1 + 𝜃 + 𝛾)𝑧𝜃+𝛾 × Φ(𝜌1,...,𝜌𝑝,𝜅,𝜎1,...,𝜎𝑞,𝜅) 𝜆1,...,𝜆𝑝,𝜃+𝑛;𝜇1,...,𝜇𝑞,1+𝜃+𝜆(𝑧 𝜅_{, 𝑠, 𝑎) .} (60)

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 functions Φ(𝜌1,...,𝜌𝑝,𝜎1,...,𝜎𝑞)

𝜆1,...,𝜆𝑝;𝜇1,...,𝜇𝑞 (𝑧, 𝑠, 𝑎).

Theorem 14. Under the assumptions ofTheorem 7, the follow-ing expansion formula holds true:

Φ(𝜌1,...,𝜌𝑝,𝜎1,...,𝜎𝑞) 𝜆1,...,𝜆𝑝;𝜇1,...,𝜇𝑞 (𝑧 𝜅_{, 𝑠, 𝑎)} = 𝑐 ∑∞ 𝑛=−∞ (𝑧₀)−𝑐𝑛(𝑧 − 𝑧₀)𝑐𝑛 Γ (𝑐𝑛 + 1) Γ (1 − 𝑐𝑛) × Φ(𝜌1,...,𝜌𝑝,𝜅,𝜎1,...,𝜎𝑞,𝜅) 𝜆1,...,𝜆𝑝,1;𝜇1,...,𝜇𝑞,1−𝑐𝑛(𝑧 𝜅 0, 𝑠, 𝑎) (󵄨󵄨󵄨󵄨𝑧 − 𝑧0󵄨󵄨󵄨󵄨 = 󵄨󵄨󵄨󵄨𝑧0󵄨󵄨󵄨󵄨;𝜆 > 0), (61)

(10)

Proof. Setting𝑓(𝑧) = Φ(𝜌_𝜆1,...,𝜌𝑝,𝜎1,...,𝜎𝑞)

1,...,𝜆𝑝;𝜇1,...,𝜇𝑞 (𝑧

𝜅_{, 𝑠, 𝑎) in}_{Theorem 7}

with𝑏 = 𝛾 = 0, 0 < 𝑐 ≦ 1 , and 𝜃(𝑧) = 𝑧 − 𝑧₀, we have Φ(𝜌1,...,𝜌𝑝,𝜎1,...,𝜎𝑞) 𝜆1,...,𝜆𝑝;𝜇1,...,𝜇𝑞 (𝑧 𝜅_{, 𝑠, 𝑎)} = 𝑐 ∑∞ 𝑛=−∞ 𝐷𝑐𝑛 𝑧 {Φ(𝜌𝜆11,...,𝜆,...,𝜌𝑝𝑝;𝜇,𝜎11,...,𝜇,...,𝜎𝑞𝑞)(𝑧 𝜅_{, 𝑠, 𝑎)}}󵄨󵄨󵄨󵄨 󵄨󵄨𝑧=𝑧0 (𝑧 − 𝑧₀)𝑐𝑛 Γ (1 + 𝑐𝑛) , (62) for𝑧₀ ̸= 0 and for 𝑧 such that |𝑧 − 𝑧₀| = |𝑧₀|.

Now, by making use of (26) with𝛽 = 0 and 𝛼 = 𝑐𝑛, we find that 𝐷𝑐𝑛_𝑧 {Φ(𝜌1,...,𝜌𝑝,𝜎1,...,𝜎𝑞) 𝜆1,...,𝜆𝑝;𝜇1,...,𝜇𝑞 (𝑧 𝜅_{, 𝑠, 𝑎)}}󵄨󵄨󵄨󵄨 󵄨󵄨𝑧=𝑧0 = 𝑧−𝑐𝑛0 Γ (1 − 𝑐𝑛)Φ (𝜌1,...,𝜌𝑝,𝜅,𝜎1,...,𝜎𝑞,𝜅) 𝜆1,...,𝜆𝑝,1;𝜇1,...,𝜇𝑞,1−𝑐𝑛(𝑧 𝜅 0, 𝑠, 𝑎) . (63)

By combining (62) and (63), we get result (61) asserted by

Theorem 14.

Φ(𝜌1,...,𝜌𝑝,𝜎1,...,𝜎𝑞) 𝜆1,...,𝜆𝑝;𝜇1,...,𝜇𝑞 (𝑧 𝜅_{, 𝑠, 𝑎)} = 𝑐𝑧−𝛼(𝑧 − 𝑧₁)−𝛽𝑧𝛼+𝛽₁ ⋅ ∑∞ 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) ⋅ Φ(𝜌1,...,𝜌𝑝,𝜅,𝜎1,...,𝜎𝑞,𝜅) 𝜆1,...,𝜆𝑝,𝛼+𝑐𝑛+𝛾;𝜇1,...,𝜇𝑞,𝛼+𝛽(𝑧 𝜅 1, 𝑠, 𝑎) (𝑧 − 𝑧_𝑧 1) 𝑐𝑛+𝛾 (64)

for𝜆 > 0 and for 𝑧 on 𝐶₁(1) defined by

𝑧 = 𝑧1

2 + 𝑧₁

2√1 + 𝑒𝑖𝜃 (−𝜋 < 𝜃 < 𝜋) , (65)

provided that both sides of (64) exist.

Proof. By taking𝑓(𝑧) = Φ(𝜌_𝜆1,...,𝜌𝑝,𝜎1,...,𝜎𝑞)

1,...,𝜆𝑝;𝜇1,...,𝜇𝑞 (𝑧

𝜅_{, 𝑠, 𝑎) in}

Theorem 8with𝑧₂ = 0, 𝜇 = 𝛼, ] = 𝛽 , and 0 < 𝑐 ≦ 1, we find that Φ(𝜌1,...,𝜌𝑝,𝜎1,...,𝜎𝑞) 𝜆1,...,𝜆𝑝;𝜇1,...,𝜇𝑞 (𝑧 𝜅_{, 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 − 𝑧1 𝑧 ) 𝑐𝑛+𝛾 ⋅ 𝐷_𝑧−𝛽+𝑐𝑛+𝛾{𝑧𝛼+𝑐𝑛+𝛾−1Φ(𝜌1,...,𝜌𝑝,𝜎1,...,𝜎𝑞) 𝜆1,...,𝜆𝑝;𝜇1,...,𝜇𝑞 (𝑧 𝜅, 𝑠, 𝑎)}󵄨󵄨󵄨󵄨 󵄨󵄨𝑧=𝑧1 . (66)

Now, with the help of relation (26) with𝛼 󳨃→ −𝛽 + 𝑐𝑛 + 𝛾 and 𝛽 󳨃→ 𝛼 + 𝑐𝑛 + 𝛾 − 1, we have 𝐷−𝛽+𝑐𝑛+𝛾_𝑧 {𝑧𝛼+𝑐𝑛+𝛾−1Φ(𝜌1,...,𝜌𝑝,𝜎1,...,𝜎𝑞) 𝜆1,...,𝜆𝑝;𝜇1,...,𝜇𝑞 (𝑧 𝜅, 𝑠, 𝑎)}󵄨󵄨󵄨󵄨 󵄨󵄨𝑧=𝑧1 = 𝑧𝛼+𝛽−1₁ Γ (𝛼 + 𝑐𝑛 + 𝛾) Γ (𝛼 + 𝛽) × Φ(𝜌1,...,𝜌𝑝,𝜅,𝜎1,...,𝜎𝑞,𝜅) 𝜆1,...,𝜆𝑝,𝛼+𝑐𝑛+𝛾;𝜇1,...,𝜇𝑞,𝛼+𝛽(𝑧 𝜅 1, 𝑠, 𝑎) . (67)

Thus, by combining (66) and (67), we are led to assertion (64) ofTheorem 15.

Φ(𝜌1,...,𝜌𝑝,𝜎1,...,𝜎𝑞) 𝜆1,...,𝜆𝑝;𝜇1,...,𝜇𝑞 (𝑧 𝜅_{, 𝑠, 𝑎)} = 𝑐𝑧−𝛽+𝛾(𝑧 − 𝑧₁)−𝛼+𝛾𝑧𝛽+𝛼−2𝛾−1₁ ⋅ ∑∞ 𝑛=−∞ sin[(𝛽 − 𝑐𝑛 − 𝛾) 𝜋] 𝑒𝑖𝜋𝑐(𝑛+1)[𝑧 (𝑧 − 𝑧₁) /𝑧2₁]𝑐𝑛 sin[(𝛽 + 𝑐 − 𝛾) 𝜋] Γ (1 − 𝛼 + 𝑐𝑛 + 𝛾) ⋅ _{Γ (𝛽 + 𝛼 − 2𝑐𝑛 − 2𝛾)}Γ (𝛽 − 𝑐𝑛 − 𝛾) × [ (𝑧 − 𝑧₁) × Φ(𝜌1,...,𝜌𝑝,𝜅,𝜎1,...,𝜎𝑞,𝜅) 𝜆1,...,𝜆𝑝,𝛽−𝑐𝑛−𝛾;𝜇1,...,𝜇𝑞,𝛽+𝛼−2𝑐𝑛−2𝛾(𝑧 𝜅 1, 𝑠, 𝑎) + (𝛽 − 𝑐𝑛 − 𝛾) 𝑧1 (𝛼 + 𝛽 − 2𝑐𝑛 − 2𝛾) × Φ(𝜌1,...,𝜌𝑝,𝜅,𝜎1,...,𝜎𝑞,𝜅) 𝜆1,...,𝜆𝑝,1+𝛽−𝑐𝑛−𝛾;𝜇1,...,𝜇𝑞,1+𝛽+𝛼−2𝑐𝑛−2𝛾(𝑧 𝜅 1, 𝑠, 𝑎) ] (68)

for𝜆 > 0 and for 𝑧 on 𝐶₁(1) defined by

𝑧 = 𝑧1

2 + 𝑧₁

2√1 + 𝑒𝑖𝜃 (−𝜋 < 𝜃 < 𝜋) , (69)

Proof. Setting𝑓(𝑧) = Φ(𝜌_𝜆1,...,𝜌𝑝,𝜎1,...,𝜎𝑞) 1,...,𝜆𝑝;𝜇1,...,𝜇𝑞 (𝑧 𝜅_{, 𝑠, 𝑎) in}_{Theorem 9} with𝑧₂= 0, 0 < 𝑐 ≦ 1, 𝑞(𝑧) = 1 , and 𝜃(𝑧) = (𝑧−𝑧₁)(𝑧−𝑧₂), we find that Φ(𝜌1,...,𝜌𝑝,𝜎1,...,𝜎𝑞) 𝜆1,...,𝜆𝑝;𝜇1,...,𝜇𝑞 (𝑧 𝜅_{, 𝑠, 𝑎)} = 𝑐𝑧−𝛽(𝑧 − 𝑧₁)−𝛼 ⋅ ∑∞ 𝑛=−∞ sin[(𝛽 − 𝑐𝑛 − 𝛾) 𝜋] 𝑒𝐼𝜋𝑐(𝑛+1)[𝑧 (𝑧 − 𝑧₁)]𝑐𝑛+𝛾 sin[(𝛽 + 𝑐 − 𝛾) 𝜋] Γ (1 − 𝛼 + 𝑐𝑛 + 𝛾) ⋅𝐷−𝛼+𝑐𝑛+𝛾_𝑧 × {𝑧𝛽−𝑐𝑛−𝛾−1_{(𝑧 + 𝑤 − 𝑧} 1) × Φ(𝜌1,...,𝜌𝑝,𝜎1,...,𝜎𝑞) 𝜆1,...,𝜆𝑝;𝜇1,...,𝜇𝑞 (𝑧 𝜅, 𝑠, 𝑎)}󵄨󵄨󵄨󵄨 󵄨󵄨𝑧=𝑧1,𝑤=𝑧. (70)

(11)

With the help of the relation in (26), we have 𝐷−𝛼+𝑐𝑛+𝛾 𝑧 × {𝑧𝛽−𝑐𝑛−𝛾−1(𝑧 + 𝑤 − 𝑧₁) ×Φ(𝜌1,...,𝜌𝑝,𝜎1,...,𝜎𝑞) 𝜆1,...,𝜆𝑝;𝜇1,...,𝜇𝑞 (𝑧 𝜅, 𝑠, 𝑎)}󵄨󵄨󵄨󵄨 󵄨󵄨𝑧=𝑧1,𝑤=𝑧 = (𝑧 − 𝑧₁) 𝐷−𝛼+𝑐𝑛+𝛾 𝑧 × {𝑧𝛽−𝑐𝑛−𝛾−1Φ(𝜌1,...,𝜌𝑝,𝜎1,...,𝜎𝑞) 𝜆1,...,𝜆𝑝;𝜇1,...,𝜇𝑞 (𝑧 𝜅, 𝑠, 𝑎)}󵄨󵄨󵄨󵄨 󵄨󵄨𝑧=𝑧1 +𝐷−𝛼+𝑐𝑛+𝛾_𝑧 {𝑧𝛽−𝑐𝑛−𝛾Φ(𝜌1,...,𝜌𝑝,𝜎1,...,𝜎𝑞) 𝜆1,...,𝜆𝑝;𝜇1,...,𝜇𝑞 (𝑧 𝜅, 𝑠, 𝑎)}󵄨󵄨󵄨󵄨 󵄨󵄨𝑧=𝑧1 = 𝑧₁𝛽+𝛼−2𝑐𝑛−2𝛾 × [((𝑧 − 𝑧₁) Γ (𝛽 − 𝑐𝑛 − 𝛾) × Φ(𝜌1,...,𝜌𝑝,𝜅,𝜎1,...,𝜎𝑞,𝜅) 𝜆1,...,𝜆𝑝,𝛽−𝑐𝑛−𝛾;𝜇1,...,𝜇𝑞,𝛽+𝛼−2𝑐𝑛−2𝛾(𝑧 𝜅 1, 𝑠, 𝑎)) × (𝑧₁ Γ (𝛽 + 𝛼 − 2𝑐𝑛 − 2𝛾))−1 + (Γ (1 + 𝛽 − 𝑐𝑛 − 𝛾) × Φ(𝜌1,...,𝜌𝑝,𝜅,𝜎1,...,𝜎𝑞,𝜅) 𝜆1,...,𝜆𝑝,1+𝛽−𝑐𝑛−𝛾;𝜇1,...,𝜇𝑞,1+𝛽+𝛼−2𝑐𝑛−2𝛾(𝑧 𝜅 1, 𝑠, 𝑎)) × (Γ (1 + 𝛽 + 𝛼 − 2𝑐𝑛 − 2𝛾))−1] . (71) Thus, by combining (70) and (71), we obtain desired result (68).

Finally, from Corollary 11 given in the preceding section, we obtain the following new relation involving the extended multiparameters Hurwitz-Lerch zeta functionΦ(𝜌_𝜆1,...,𝜌𝑝,𝜎1,...,𝜎𝑞)

1,...,𝜆𝑝;𝜇1,...,𝜇𝑞 (𝑧, 𝑠, 𝑎).

Theorem 17. Under the hypotheses ofCorollary 11, let𝑘 be a positive integer. Then the following relation holds true:

Φ(𝜌1,...,𝜌𝑝,1/𝑘,𝜎1,...,𝜎𝑞,1/𝑘) 𝜆1,...,𝜆𝑝,𝛼;𝜇1,...,𝜇𝑞,𝛽 (𝑧, 𝑠, 𝑎) = 𝑘 Γ (𝛽) Γ (𝑘𝛼) Γ (𝛼) Γ (𝛽 + (𝑘 − 1) 𝛼) ⋅∑∞ 𝑛=0 ∏𝑝_𝑗=1(𝜆_𝑗)_𝑛𝜌 𝑗 ∏𝑞_𝑗=1(𝜇_𝑗)_𝑛𝜎 𝑗 (𝑘𝛼)_𝑛 (𝛽 + (𝑘 − 1) 𝛼)_𝑛 𝑧𝑛 (𝑛 + 𝑎)𝑠 ⋅ 𝐹_𝐷(𝑘−1)[𝑘𝛼 + 𝑛, 1 + 𝛼 − 𝛽, . . . , 1 + 𝛼 − 𝛽; 𝛽 + (𝑘 − 1) 𝛼 + 𝑛; 𝑒−2𝜋𝑖/𝑘, . . . , 𝑒−2(𝑘−1)𝜋𝑖/𝑘] , (72)

where𝜆 > 0 and 𝐹_𝐷(𝑛) denotes the Lauricella function of 𝑛 variables defined by [38, page 60]

𝐹_𝐷(𝑛)[𝑎, 𝑏₁, . . . , 𝑏_𝑛; 𝑐; 𝑥₁, . . . , 𝑥_𝑛] = ∑∞ 𝑚1,...,𝑚𝑛=0 (𝑎)𝑚1+⋅⋅⋅+𝑚𝑛(𝑏1)𝑚1⋅ ⋅ ⋅ (𝑏𝑛)𝑚𝑛 (𝑐)_𝑚₁_{+⋅⋅⋅+𝑚}_𝑛 × 𝑥 𝑚1 1 𝑚₁!⋅ ⋅ ⋅ 𝑥𝑚𝑛 𝑛 𝑚_𝑛! (max {󵄨󵄨󵄨󵄨𝑥1󵄨󵄨󵄨󵄨,...,󵄨󵄨󵄨󵄨𝑥𝑛󵄨󵄨󵄨󵄨} < 1), (73)

Proof. Putting𝑝 = 𝑘 and letting 𝑓(𝑧) = Φ(𝜌_𝜆1,...,𝜌𝑝,𝜎1,...,𝜎𝑞)

1,...,𝜆𝑝;𝜇1,...,𝜇𝑞 (𝑧, 𝑠, 𝑎) inCorollary 11, we get 𝑧𝑘𝑂_𝛽𝛼{Φ(𝜌_𝜆1,...,𝜌𝑝,𝜎1,...,𝜎𝑞) 1,...,𝜆𝑝;𝜇1,...,𝜇𝑞 (𝑧, 𝑠, 𝑎)} = 𝑘 (𝑧𝑘−1₎𝛼 ⋅_𝑧𝑂_𝛽𝛼{_{ { Φ(𝜌1,...,𝜌𝑝,𝜎1,...,𝜎𝑞) 𝜆1,...,𝜆𝑝;𝜇1,...,𝜇𝑞 (𝑧, 𝑠, 𝑎) (𝑧 𝑘−1₎𝛼 × 𝑘−1∏ 𝑠=1(1 − 𝑧 𝑤𝑒−2𝜋𝑖𝑠/𝑘) 𝛽−𝛼−1 } } } 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑤=𝑧 . (74)

With the help of the definition of _𝑧𝑂_𝛽𝛼 given by (53), we find for the left-hand side of (74) that

𝑧𝑂𝛼𝛽{Φ(𝜌𝜆11,...,𝜆,...,𝜌𝑝𝑝;𝜇,𝜎11,...,𝜇,...,𝜎𝑞𝑞)(𝑧, 𝑠, 𝑎)} = Φ(𝜌1,...,𝜌𝑝,1/𝑘,𝜎1,...,𝜎𝑞,1/𝑘)

𝜆1,...,𝜆𝑝,𝛼;𝜇1,...,𝜇𝑞,𝛽 (𝑧, 𝑠, 𝑎) .

(75)

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

𝑘 (𝑧𝑘−1₎𝛼 𝑧𝑂 𝛼 𝛽 ×{_{ { ∞ ∑ 𝑛=0 ∏𝑝_𝑗=1(𝜆_𝑗)_𝑛𝜌 𝑗 𝑛!∏𝑞_𝑗=1(𝜇_𝑗)_𝑛𝜎 𝑗 𝑧𝑛+(𝑘−1)𝛼 (𝑛 + 𝑎)𝑠 ⋅ ∑∞ 𝑚1,...,𝑚𝑘−1=0 (1 + 𝛼 − 𝛽)_𝑚₁⋅ ⋅ ⋅ (1 + 𝛼 − 𝛽)_𝑚_𝑘−1 ⋅ ((𝑧/𝑤) 𝑒 −2𝜋𝑖/𝑘₎𝑚1 𝑚1! ⋅ ⋅ ⋅((𝑧/𝑤) 𝑒 −2(𝑘−1)𝜋𝑖/𝑘₎𝑚𝑘−1 𝑚𝑘−1! } } } 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑤=𝑧

(12)

= 𝑘 (𝑧𝑘−1₎𝛼 ∞ ∑ 𝑛=0 ∏𝑝_𝑗=1(𝜆𝑗)_𝑛𝜌_𝑗 𝑛!∏𝑞_𝑗=1(𝜇_𝑗)_𝑛𝜎 𝑗(𝑛 + 𝑎) 𝑠 ⋅ ∑∞ 𝑚1,...,𝑚𝑘−1=0 (1 + 𝛼 − 𝛽)_𝑚 1⋅ ⋅ ⋅ (1 + 𝛼 − 𝛽)𝑚𝑘−1 𝑧𝑚1+⋅⋅⋅+𝑚𝑘−1 ⋅ (𝑒 −2𝜋𝑖/𝑘₎𝑚1 𝑚₁! ⋅ ⋅ ⋅ (𝑒−2(𝑘−1)𝜋𝑖/𝑘₎𝑚𝑘−1 𝑚_𝑘−1! × _𝑧𝑂_𝛽𝛼{𝑧𝑚1+⋅⋅⋅+𝑚𝑘−1+𝑛+(𝑘−1)𝛼} = 𝑘∑∞ 𝑛=0 ∏𝑝_𝑗=1(𝜆_𝑗)_𝑛𝜌 𝑗 𝑛!∏𝑞_𝑗=1(𝜇_𝑗)_𝑛𝜎 𝑗 𝑧𝑛 (𝑛 + 𝑎)𝑠 ⋅ ∑∞ 𝑚1,...,𝑚𝑘−1=0 (1 + 𝛼 − 𝛽)_𝑚 1⋅ ⋅ ⋅ (1 + 𝛼 − 𝛽)𝑚𝑘−1 ⋅ (𝑒 −2𝜋𝑖/𝑘₎𝑚1 𝑚₁! ⋅ ⋅ ⋅ (𝑒−2(𝑘−1)𝜋𝑖/𝑘)𝑚𝑘−1 𝑚_𝑘−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 ∏𝑝_𝑗=1(𝜆_𝑗)_𝑛𝜌 𝑗(𝑘𝛼)𝑛 𝑛!∏𝑞_𝑗=1(𝜇_𝑗)_𝑛𝜎 𝑗(𝛽 + (𝑘 − 1) 𝛼)𝑛 𝑧𝑛 (𝑛 + 𝑎)𝑠 ⋅ ∑∞ 𝑚1,...,𝑚𝑘−1=0 ( ((𝑛 + 𝑘𝛼)𝑚1+⋅⋅⋅+𝑚𝑘−1 × (1 + 𝛼 − 𝛽)_𝑚₁⋅ ⋅ ⋅ (1 + 𝛼 − 𝛽)_𝑚_𝑘−1) ×((𝛽 + (𝑘 − 1) 𝛼)_𝑚₁_{+⋅⋅⋅+𝑚}_𝑘−1)−1) ⋅ (𝑒 −2𝜋𝑖/𝑘₎𝑚1 𝑚₁! ⋅ ⋅ ⋅ (𝑒−2(𝑘−1)𝜋𝑖/𝑘)𝑚𝑘−1 𝑚_𝑘−1! . (76) Finally, by combining (75) and (76), we obtain the result (72) asserted byTheorem 17.

Remark 18. Each of the previous theorems can be written in

terms of 𝐻-function given inDefinition 1. For instance, if we make use of (11), thenTheorem 17becomes

𝐻1,𝑝+2_{𝑝+2,𝑞+3}[[ [ −𝑧 | (1 − 𝜆1, 𝜌1; 1) , . . . , (1 − 𝜆𝑝, 𝜌𝑝; 1) , (1 − 𝛼, 1 𝑘, 1) , (1 − 𝑎, 1; 𝑠) (0, 1) , (1 − 𝜇₁, 𝜎₁; 1) , . . . , (1 − 𝜇_𝑝, 𝜎_𝑝; 1) , (1 − 𝛽,1 𝑘, 1) , (−𝑎, 1; 𝑠) ] ] ] =∏ 𝑝 𝑗=1Γ (𝜆𝑗) ∏𝑞_𝑗=1Γ (𝜇𝑗) 𝑘Γ (𝑘𝛼) Γ (𝛽 + (𝑘 − 1) 𝛼) ∞ ∑ 𝑛=0 ∏𝑝_𝑗=1(𝜆𝑗)_𝑛𝜌_𝑗 𝑛!∏𝑞_𝑗=1(𝜇𝑗)_𝑛𝜎_𝑗 (𝑘𝛼)𝑛 (𝛽 + (𝑘 − 1) 𝛼)_𝑛 𝑧𝑛 (𝑛 + 𝑎)𝑠 ⋅ 𝐹_𝐷(𝑘−1)[𝑘𝛼 + 𝑛, 1 + 𝛼 − 𝛽, . . . , 1 + 𝛼 − 𝛽; 𝛽 + (𝑘 − 1) 𝛼 + 𝑛; 𝑒−2𝜋𝑖/𝑘, . . . , 𝑒−2(𝑘−1)𝜋𝑖/𝑘] . (77)

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 = 3 in Theorem 17, we obtain the following corollary.

Corollary 19. Under the hypotheses ofTheorem 17, the follow-ing expansion formula holds true:

Φ(𝜌1,...,𝜌𝑝,1/3,𝜎1,...,𝜎𝑞,1/3) 𝜆1,...,𝜆𝑝,𝛼;𝜇1,...,𝜇𝑞,𝛽 (𝑧, 𝑠, 𝑎) = 3Γ (𝛽) Γ (3𝛼) Γ (𝛼) Γ (𝛽 + 𝛼) ⋅∑∞ 𝑛=0 ∏𝑝_𝑗=1(𝜆_𝑗)_𝑛𝜌 𝑗 𝑛!∏𝑞_𝑗=1(𝜇_𝑗)_𝑛𝜎 𝑗 (3𝛼)_𝑛 (𝛽 + 𝛼)_𝑛 𝑧𝑛 (𝑛 + 𝑎)𝑠 ⋅ 𝐹1[3𝛼 + 𝑛, 1 + 𝛼 − 𝛽, 1 + 𝛼 − 𝛽; 𝛽 + 2𝛼 + 𝑛; −1, 1] , (78)

where𝐹₁ denotes the first Appell function defined by [38, page 22] 𝐹₁[𝑎, 𝑏₁, 𝑏₂; 𝑐; 𝑥₁, 𝑥₂] = ∑∞ 𝑚1,𝑚2=0 (𝑎)_𝑚₁_+𝑚₂(𝑏₁)_𝑚₁(𝑏₂)_𝑚₂ (𝑐)_𝑚₁_+𝑚₂ 𝑥𝑚1 1 𝑚₁! 𝑥𝑚2 2 𝑚₂! (max {󵄨󵄨󵄨󵄨𝑥1󵄨󵄨󵄨󵄨,󵄨󵄨󵄨󵄨𝑥2󵄨󵄨󵄨󵄨} < 1), (79)

Setting𝑝 − 1 = 𝑞 = 1, 𝜅 = 1, and making the following substitutions𝜌₁ 󳨃→ 𝜌, 𝜌₂ 󳨃→ 𝜎, 𝜎₁ 󳨃→ 𝜂, 𝜆₁ 󳨃→ 𝜆, 𝜆₂ 󳨃→ 𝜇, and𝜇₁ 󳨃→ ] inTheorem 14lead to the following expansion formula given recently by Srivastava et al. [14].

(13)

Corollary 20. Under the hypotheses ofTheorem 14, the follow-ing expansion formula holds true:

Φ(𝜌,𝜎,𝜂)_𝜆,𝜇;] (𝑧, 𝑠, 𝑎) = 𝑐 ∑∞ 𝑛=−∞ (𝑧₀)−𝑐𝑛(𝑧 − 𝑧₀)𝑐𝑛 Γ (𝑐𝑛 + 1) Γ (1 − 𝑐𝑛)Φ (𝜌,𝜎,1,𝜂,1) 𝜆,𝜇,1;],1−𝑐𝑛(𝑧0, 𝑠, 𝑎) (80)

for𝑧 such that |𝑧 − 𝑧₀| = |𝑧₀| and provided that both members of (80) exist.

Remark 21. The functionΦ(𝜌,𝜎,𝜅)_𝜆,𝜇;] (𝑧, 𝑠, 𝑎), which occurs in Corollary 20above, as well as its multiparameter generaliza-tions, 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:

Φ(𝜌,𝜎,𝜅)_𝜆,𝜇;] (𝑧, 𝑠, 𝑎) :=∑∞ 𝑛=0 (𝜆)_𝜌𝑛(𝜇)_𝜎𝑛 𝑛!(])𝜅𝑛 𝑧𝑛 (𝑛 + 𝑎)𝑠 (𝜆, 𝜇 ∈ C; 𝑎, ] ∈ C \ Z−₀; 𝜌, 𝜎, 𝜅 ∈ R+; 𝜅 − 𝜌 − 𝜎 > −1 when 𝑠, 𝑧 ∈ C; 𝜅 − 𝜌 − 𝜎 = −1, 𝑠 ∈ C when |𝑧| < 𝛿∗_{:= 𝜌}−𝜌_𝜎−𝜎_𝜅𝜅_; 𝜅 − 𝜌 − 𝜎 = −1, R (𝑠 + ] − 𝜆 − 𝜇) > 1 when |𝑧| = 𝛿∗) . (81) Putting 𝑝 − 1 = 𝑞 = 1, 𝜅 = 1, and setting 𝜌₁= 𝜌₂= 𝜎₁= 1, 𝜆₁= 𝜇, and 𝜆₂= 𝜇₁inTheorem 15reduce to the following expansion formula given recently by Gaboury [15, Equation (4.4)].

Corollary 22. Under the hypotheses ofTheorem 15, the follow-ing expansion formula holds true:

Φ∗_𝜇(𝑧, 𝑠, 𝑎) = 𝑐𝑧−𝛼(𝑧 − 𝑧₁)−𝛽𝑧𝛼+𝛽₁ ⋅ ∑∞ 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) ⋅ Φ(1,1,1)_{𝜇,𝛼+𝑐𝑛+𝛾;𝛼+𝛽}(𝑧₁, 𝑠, 𝑎) (𝑧 − 𝑧1 𝑧 ) 𝑐𝑛+𝛾 (82) for𝑧 on 𝐶₁(1) defined by 𝑧 = 𝑧1 2 + 𝑧₁ 2√1 + 𝑒𝑖𝜃 (−𝜋 < 𝜃 < 𝜋) , (83)

Remark 23. The functionΦ∗_𝜇(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: Φ∗_𝜇(𝑧, 𝑠, 𝑎) :=∑∞ 𝑛=0 (𝜇)_𝑛 𝑛! 𝑧𝑛 (𝑛 + 𝑎)𝑠 (𝜇 ∈ C; 𝑎, ] ∈ C \ Z−₀; 𝑠 ∈ C when |𝑧| < 1; R (𝑠 − 𝜇) > 1 when |𝑧| = 1) . (84)

Setting𝑠 = 0, 𝜅 = 1, 𝑝 󳨃→ 𝑝 + 1, and 𝑞 󳨃→ 𝑞 + 1 with 𝜌₁= ⋅ ⋅ ⋅ = 𝜌_𝑝 = 1, 𝜆_𝑝+1= 𝜌_𝑝+1= 1,

𝜎₁= ⋅ ⋅ ⋅ = 𝜎_𝑞= 1, 𝜇_𝑞+1= 𝛽, 𝜎_𝑞+1= 𝛼 (85) inTheorem 16, we deduce the following expansion formula.

Corollary 24. Under the hypotheses ofTheorem 16, the follow-ing formula holds true:

1 Γ (𝛽)𝑝+1Ψ𝑞+1∗ [(𝜆_(𝜇1, 1) , . . . , (𝜆𝑝, 1) , (1, 1) ; 1, 1) , . . . , (𝜇𝑞, 1) , (𝛽, 𝛼) 𝑧] = 𝑐𝑧−𝛽+𝛾(𝑧 − 𝑧₁)−𝛼+𝛾𝑧𝛽+𝛼−2𝛾−1₁ ⋅ ∑∞ 𝑛=−∞ sin((𝛽 − 𝑐𝑛 − 𝛾) 𝜋) 𝑒𝑖𝜋𝑐(𝑛+1)[𝑧 (𝑧 − 𝑧₁) /𝑧2₁]𝑐𝑛 sin((𝛽 + 𝑐 − 𝛾) 𝜋) Γ (1 − 𝛼 + 𝑐𝑛 + 𝛾) × Γ (𝛽 − 𝑐𝑛 − 𝛾) Γ (𝛽 + 𝛼 − 2𝑐𝑛 − 2𝛾) ⋅ [(𝑧 − 𝑧₁) Φ(1,...,1,1,1,1,...,1,𝛼,1)_𝜆 1,...,𝜆𝑝,1,1+𝛽−𝑐𝑛−𝛾;𝜇1,...,𝜇𝑞,𝛽,1+𝛽+𝛼−2𝑐𝑛−2𝛾 × (𝑧₁, 0, 𝑎) + 𝑧1(𝛽 − 𝑐𝑛 − 𝛾) (𝛼 + 𝛽 − 2𝑐𝑛 − 2𝛾) × Φ(1,...,1,1,1,1,...,1,𝛼,1)_𝜆₁_,...,𝜆_𝑝_{,1,𝛽−𝑐𝑛−𝛾;𝜇}₁_,...,𝜇_𝑞_{,𝛽,𝛽+𝛼−2𝑐𝑛−2𝛾}(𝑧₁, 0, 𝑎)] (86) for𝑧 on 𝐶₁(1) defined by 𝑧 = 𝑧1 2 + 𝑧₁ 2√1 + 𝑒𝑖𝜃 (−𝜋 < 𝜃 < 𝜋) , (87)

The function _𝑙Ψ_𝑚∗ involved in the left-hand side of (86) is the Fox-Wright function(𝑙, 𝑚 ∈ N₀) with 𝑙 numerator parameters 𝑎₁, . . . , 𝑎_𝑙 and𝑚 denomi-nator parameters𝑏₁, . . . , 𝑏_𝑚 such that

𝑎_𝑗 ∈ C (𝑗 = 1, . . . , 𝑙) ,

(14)

defined by (see, for details, [17,40]) 𝑙Ψ𝑚∗[(𝑎_(𝑏₁1_{, 𝐵}, 𝐴₁1_{) , . . . , (𝑏}) , . . . , (𝑎_𝑚𝑙_{, 𝐵}, 𝐴_𝑚𝑙) ;_{) ; 𝑧]} :=∑∞ 𝑛=0 (𝑎₁)_𝐴₁_𝑛⋅ ⋅ ⋅ (𝑎_𝑙)_𝐴_𝑙_𝑛 (𝑏₁)_𝐵₁_𝑛⋅ ⋅ ⋅ (𝑏_𝑚)_𝐵_𝑚_𝑛 𝑧𝑛 𝑛! (𝐴_𝑗> 0, (𝑗 = 1, . . . , 𝑙) ; 𝐵_𝑗 > 0, (𝑗 = 1, . . . , 𝑚) ; 1 +∑𝑚 𝑗=1𝐵𝑗− 𝑙 ∑ 𝑗=1𝐴𝑗 ≥ 0) , (89)

where the equality in the convergence condition holds true for suitably bounded values of|𝑧| given by

|𝑧| < (∏𝑙 𝑗=1 𝐴−𝐴𝑗 𝑗 ) ( 𝑚 ∏ 𝑗=1 𝐵𝐵𝑗 𝑗 ) . (90)

In our series of forthcoming papers, we propose to consider and investigate analogous expansion formulas and other results involving the 𝜆-extensions of the generalized Hurwitz-Lerch zeta functions. These functions have been investigated recently by Srivastava et al. [41] (see also [42]).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

[1] H. M. Srivastava and J. Choi, Series Associated with the Zeta and

Related Functions, Kluwer Academic Publishers, Dordrecht,

The Netherlands, 2001.

[2] H. M. Srivastava, “Some formulas for the Bernoulli and Euler polynomials at rational arguments,” Mathematical Proceedings

of the Cambridge Philosophical Society, vol. 129, no. 1, pp. 77–84,

2000.

[3] H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and

Associated Series and Integrals, Elsevier Science Publishers,

Amsterdam, The Netherlands, 2012.

[4] S. P. Goyal and R. K. Laddha, “On the generalized Riemann zeta functions and the generalized Lambert transform,” Ganita

Sandesh, vol. 11, no. 2, pp. 99–108, 1997.

[5] S.-D. Lin and H. M. Srivastava, “Some families of the Hurwitz-Lerch zeta functions and associated fractional derivative and other integral representations,” Applied Mathematics and

Com-putation, vol. 154, no. 3, pp. 725–733, 2004.

[6] M. Carg, K. Jain, and S. L. Kalla, “A further study of general Hurwitz-Lerch zeta function,” Algebras, Groups and Geometries, vol. 25, no. 3, pp. 311–319, 2008.

[7] H. M. Srivastava, R. K. Saxena, T. K. Pog´any, and R. Saxena, “Integral and computational representations of the extended Hurwitz-Lerch zeta function,” Integral Transforms and Special

Functions, vol. 22, no. 7, pp. 487–506, 2011.

[8] H. M. Srivastava, “Some families of mixed multilateral generat-ing relations,” Journal of Rajasthan Academy of Physical Sciences, vol. 1, no. 1, pp. 1–6, 2002.

[9] H. M. Srivastava, “Some generalizations and basic (or 𝑞-) extensions of the Bernoulli, Euler and Genocchi polynomials,”

Applied Mathematics & Information Sciences, vol. 5, no. 3, pp.

390–444, 2011.

[10] H. M. Srivastava and H. L. Manocha, A treatise on generating

functions, Ellis Horwood Series: Mathematics and Its

Applica-tions, Ellis Horwood, Chichester, UK; John Wiley & Sons, New York, NY, USA, 1984.

[11] A. A. Inayat-Hussain, “New properties of hypergeometric series derivable from Feynman integrals. II. A generalisation of the𝐻 function,” Journal of Physics. A. Mathematical and General, vol. 20, no. 13, pp. 4119–4128, 1987.

[12] R. G. Buschman and H. M. Srivastava, “The H function asso-ciated with a certain class of Feynman integrals,” Journal of

Physics A: Mathematical and General, vol. 23, no. 20, pp. 4707–

4710, 1990.

[13] H. M. Srivastava, K. C. Gupta, and S. P. Goyal, The H-Functions

of One and Two Variables with Applications, South Asian

Publishers, New Delhi, India, 1982.

[14] H. M. Srivastava, S. Gaboury, and A. Bayad, “Expansion formu-las for an extended Hurwitz-Lerch zeta function obtained via fractionalcalculus,” preprint 2014.

[15] S. Gaboury, “Some relations involving generalized Hurwitz-Lerch zeta function obtained by means of fractional derivatives with applications to Apostol-type polynomials,” Advances in

Difference Equations, vol. 2013, article 361, pp. 1–13, 2013.

[16] S. Gaboury and A. Bayad, “Series representations at special values of generalized Hurwitz-Lerch zeta function,” Abstract

and Applied Analysis, vol. 2013, Article ID 975615, 8 pages, 2013.

[17] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory

and Applications of Fractional Differential Equations, vol. 204

of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2006.

[18] A. Erd´elyi, “An integral equation involving Legendre functions,”

SIAM Journal on Applied Mathematics, vol. 12, pp. 15–30, 1964.

[19] J. Liouville, “Mémoire sur le calcul des différentielles à indices quelconques,” Journal de l’ École Polytechnique, vol. 13, pp. 71–

162, 1832.

[20] M. Riesz, “L’int´egrale de Riemann-Liouville et le probl`eme de Cauchy,” Acta Mathematica, vol. 81, pp. 1–223, 1949.

[21] T. J. Osler, “The fractional derivative of a composite function,”

SIAM Journal on Mathematical Analysis, vol. 1, pp. 288–293,

1970.

[22] T. J. Osler, “Leibniz rule for fractional derivatives generalized and an application to infinite series,” SIAM Journal on Applied

Mathematics, vol. 18, pp. 658–674, 1970.

[23] T. J. Osler, Leibniz rule, the chain rule and Taylor’s theorem for

fractional derivatives [Ph.D. thesis], New York University, 1970.

[24] T. J. Osler, “Mathematical notes: fractional derivatives and Leibniz rule,” The American Mathematical Monthly, vol. 78, no. 6, pp. 645–649, 1971.

[25] J.-L. Lavoie, R. Tremblay, and T. J. Osler, “Fundamental prop-erties of fractional derivatives via Pochhammer integrals,” in

Fractional Calculus and Its Applications (Proc. Internat. Conf., Univ. New Haven, West Haven, Conn., 1974), B. Ross, Ed., vol.

457 of Lecture Notes in Math., pp. 323–356, Springer, Berlin, Germany, 1975.

(15)

[26] R. Tremblay, Une Contribution à la Théorie de la Dérivée

Frac-tionnaire [Ph.D. thesis], Laval University, Qu´ebec, Canada, 1974.

[27] H. M. Srivastava, S.-D. Lin, and P.-Y. Wang, “Some fractional-calculus results for the𝐻-function associated with a class of Feynman integrals,” Russian Journal of Mathematical Physics, vol. 13, no. 1, pp. 94–100, 2006.

[28] K. S. Miller and B. Ross, An Introduction to the Fractional

Calculus and Fractional Differential Equations, John Wiley &

Sons, New York, NY, USA, 1993.

[29] R. Tremblay, S. Gaboury, and B.-J. Fug`ere, “A new Leibniz rule and its integral analogue for fractional derivatives,” Integral

Transforms and Special Functions, vol. 24, no. 2, pp. 111–128, 2013.

[30] T. J. Osler, “Taylor’s series generalized for fractional derivatives and applications,” SIAM Journal on Mathematical Analysis, vol. 2, pp. 37–48, 1971.

[31] G. H. Hardy, “Riemann’s form of Taylor’s series,” Journal of the

London Mathematical Society, vol. 20, pp. 48–57, 1945.

[32] O. Heaviside, Electromagnetic Theory, vol. 2, Dover Publishing, New York, NY, USA, 1950.

[33] B. Riemann, Versuch einer allgemeinen Auffasung der Integration

und Differentiation, The Collected Works of Bernhard Riemann,

Dover Publishing Company, New York, NY, USA, 1953. [34] Y. Watanabe, “Zum Riemanschen binomischen Lehrsatz,”

Pro-ceedings of the Physico-Mathematical Society of Japan, vol. 14, pp.

22–35, 1932.

[35] R. Tremblay, S. Gaboury, and B.-J. Fug`ere, “Taylor-like expan-sion in terms of a rational function obtained by means of frac-tional derivatives,” Integral Transforms and Special Functions, vol. 24, no. 1, pp. 50–64, 2013.

[36] R. Tremblay and B.-J. Fug`ere, “The use of fractional derivatives to expand analytical functions in terms of quadratic functions with applications to special functions,” Applied Mathematics

and Computation, vol. 187, no. 1, pp. 507–529, 2007.

[37] S. Gaboury and R. Tremblay, “Summation formulas obtained by means of the generalized chain rule for fractional derivatives,” preprint 2014.

[38] H. M. Srivastava and P. W. Karlsson, Multiple Gaussian

Hyper-geometric Series, Ellis Horwood Series: Mathematics and Its

Applications, Ellis Horwood, Chichester, UK; John Wiley & Sons, New York, NY, USA, 1985.

[39] H. M. Srivastava, D. Jankov, T. K. Pog´any, and R. K. Saxena, “Two-sided inequalities for the extended Hurwitz-Lerch zeta function,” Computers & Mathematics with Applications, vol. 62, no. 1, pp. 516–522, 2011.

[40] A. Erd´elyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi,

Higher Transcendental Functions. Vols. I, II, McGraw-Hill Book,

New York, NY, USA, 1953.

[41] H. M. Srivastava, M.-J. Luo, and R. K. Raina, “New results involving a class of generalized Hurwitz-Lerch zeta functions and their applications,” Turkish Journal of Analysis and Number

Theory, vol. 1, pp. 26–35, 2013.

[42] H. M. Srivastava, “A new family of the𝜆-generalized Hurwitz-Lerch zeta functions with applications,” Applied Mathematics &