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

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).

1. Introduction

The generalized Hurwitz zeta function 𝜁(𝑠,𝑎) is defined by [1, page 88 et seq.]𝜁(𝑠,𝑎)∶=^∞_𝑛=01(𝑛+𝑎)^𝑠Re(𝑠)>1;𝑎∈ℂ⧵ℤ⁻₀;ℤ⁻₀∶={0,−1,−2,…}, (1) where 𝜁()(𝑠)=1𝑠,1=𝜁2^𝑠𝜁1−1𝑠,2 (2) 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 function Φ(𝑧,𝑠,𝑎) is defined, as in [1, page 121 et seq.], by Φ(𝑧,𝑠,𝑎)∶=^∞_𝑛=0𝑧^𝑛(𝑛+𝑎)^𝑠𝑎∈ℂ⧵ℤ⁻₀;𝑠∈ℂwhen|𝑧|<1;Re(𝑠)>1when|𝑧|=1). (3) Clearly, we have the following relations: Φ(1,𝑠,𝑎)=𝜁(𝑠,𝑎),Φ(1,𝑠,1)=𝜁(𝑠). (4)

The Hurwitz-Lerch zeta function has the well-known integral representation Φ()=1𝑧,𝑠,𝑎Γ(𝑠)^∞₀𝑡^𝑠−1𝑒^−𝑎𝑡1−𝑧𝑒^−𝑡((𝑎)(𝑠)|𝑧|();(𝑠)).𝑑𝑡Re>0;Re>0when≤1𝑧≠1Re>1when𝑧=1 (5)

Recently, Lin and Srivastava [2] investigated a more general family of Hurwitz-Lerch zeta functions. Explicitly, they introduced the function Φ^(𝜌,𝜎)_𝜇,𝜈(𝑧,𝑠,𝑎) defined by Φ^(𝜌,𝜎)_𝜇,𝜈(𝑧,𝑠,𝑎)∶=^∞_𝑛=0𝜇_𝜌𝑛(𝜈)_𝜎𝑛𝑧^𝑛(𝑎+𝑛)^𝑠𝜇∈ℂ;𝑎,𝜈∈ℂ⧵ℤ⁻₀;𝜌,𝜎∈ℝ⁺;|𝑧||𝑧|,𝜌<𝜎when𝑠,𝑧∈ℂ;𝜌=𝜎,𝑠∈ℂwhen<1;𝜌=𝜎,Re𝑠−𝜇+𝜈>1when=1 (6) where (𝜆)_𝜅 denotes the Pochhammer symbol defined, in terms of the Gamma function, by (𝜆)_𝜅Γ()∶=𝜆+𝜅Γ(𝜆)=𝜆()⋯(),()({0.𝜆+1𝜆+𝜅−1𝜅∈ℕ;𝜆∈ℂ1,𝜅=0;𝜆∈ℂ⧵}) (7) It is easily seen that Φ^(𝜎,𝜎)_𝜈,𝜈(𝑧,𝑠,𝑎)=Φ^(0,0)_𝜇,𝜈()(),Φ𝑧,𝑠,𝑎=Φ𝑧,𝑠,𝑎_(1,1)𝜇,1(𝑧,𝑠,𝑎)=Φ^∗_𝜇(𝑧,𝑠,𝑎)∶=^∞_𝑛=0𝜇_𝑛𝑧𝑛!(𝑛+^𝑛𝑎)^𝑠. (8)(9) The function Φ^∗_𝜇(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 function Φ(𝑧,𝑠,𝑎). The Lerch zeta function defined by 𝑙_𝑠(𝜉)∶=^∞_𝑛=1𝑒^{2𝑛𝜋𝑖𝜉}𝑛^𝑠(𝜉∈ℝ;Re(𝑠)>1) (10) is related to the Hurwitz-Lerch zeta function by the following relation: 𝑙_𝑠(𝜉)=𝑒^{2𝜋𝑖𝜉}Φ𝑒^{2𝜋𝑖𝜉},𝑠,1. (11) 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)]: 𝜙(𝜉,𝑎,𝑠)∶=^∞_𝑛=0𝑒^{2𝑛𝜋𝑖𝜉}(𝑛+𝑎)^𝑠=Φ𝑒^{2𝜋𝑖𝜉},𝑠,𝑎𝑎∈ℂ⧵ℤ⁻₀(𝑠)(𝑠)).;Re>0when𝜉∈ℝ⧵ℤ;Re>1when𝜉∈ℤ (12) Setting 𝑧=exp(2𝜋𝑖𝑝/𝑞) with 𝑝∈ℤ and 𝑞∈ℕ and using the next series identity ^∞_𝑛=0𝑓(𝑛)=^𝑘−1∞_𝑗=0𝑛=0𝑓𝑘𝑛+𝑗, (13) we obtain the following summation formula for the Lipschitz-Lerch zeta function 𝜙(𝜉,𝑎,𝑠): 𝜙𝑝𝑞,𝑎,𝑠=𝑞^𝑞−𝑠_𝑗=1𝜁𝑠,𝑎+𝑗−1𝑞2exp𝑗−1𝑝𝜋𝑖𝑞 (14) 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)]: Φ(𝑧,𝑠,𝑎)=𝑖𝑧^−𝑎(2𝜋)^𝑠−1Γ()×−11−𝑠exp2𝑠𝜋𝑖Φ𝑒^{−2𝜋𝑖𝑎},1−𝑠,log𝑧12𝜋𝑖−exp2𝑎+2𝑠𝑒𝜋𝑖×Φ^{2𝜋𝑖𝑎},1−𝑠,1−log𝑧(𝑠)(𝑧)2𝜋𝑖Re<0;arg−logmod2𝜋𝑖≤𝜋;𝑎∈ℂ⧵ℤ⁻₀, (15) 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: 𝑛−1𝑘−1∶=1,𝑛=𝑘=1,0,𝑛≥1,𝑘=0,𝑛−1𝑘−1,otherwise. (16)

2. Expansion Formulas for Φ^∗_𝜇(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 Φ^∗_𝜇(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 Φ^∗_𝜇(𝑧,𝑠,𝑎): Φ^∗_𝜇(𝑧,𝑠,𝑎)=𝑖𝑧^−𝑎Γ(1−𝑠)^∞_𝑗=0𝑗−𝑎+1_{𝜇−𝑗−1}Γ×𝜇−𝑗𝑗!_𝑗𝑘=0()𝑗−1𝑘−11−𝑠_𝑘𝐵^(𝑗)_𝑗−𝑘⋅(2𝜋)^{𝑠−𝑘−1}−1exp2()𝑒𝑠−𝑘𝜋𝑖×Φ^{−2𝜋𝑖𝑎},1−𝑠+𝑘,log𝑧12𝜋𝑖−exp2𝑎+2()𝑒𝑠−𝑘𝜋𝑖×Φ^{2𝜋𝑖𝑎},1−𝑠+𝑘,1−log𝑧,2𝜋𝑖 (17) with Re(𝜇)>0, 𝑠∈ℂ, |arg(−log(𝑧)mod2𝜋𝑖)|≤𝜋, 𝑎∈ℂ⧵ℤ⁻₀ and where 𝐵_𝑛^(𝛼)∶=𝐵_𝑛^(𝛼)(0) are the generalized Bernoulli numbers [9] given by the following generating function: 𝑧𝑒^𝑧−1^𝛼𝑒^𝑥𝑧=^∞_𝑛=0𝐵_𝑛^(𝛼)(𝑥)𝑧^𝑛|𝑧|𝑛!<2𝜋;1^𝛼∶=1. (18)

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 ℬ_𝑛^(𝛼)(𝑥;𝜆) (which will be defined in Section 3): ℬ_𝑛^(𝑙)𝑝𝑞;𝑒^{2𝜋𝑖𝜉}=𝑖(−1)^𝑙𝑛!(𝑙−1)!^𝑙−1_𝑗=0𝑗𝑙−1𝑝𝑗−𝑞+1_{𝑙−𝑗−1}×^𝑗_𝑘=0()𝑗−1𝑘−11−𝑙+𝑛_𝑘×𝐵^(𝑗)_𝑗−𝑘2𝜋𝑞^{𝑙−𝑛−𝑘−1}⋅^𝑞_𝑟=1𝜁1−𝑙+𝑛+𝑘,𝜉+𝑟−1𝑞−1⋅exp2()+2()𝑝𝑙−𝑛−𝑘𝜉+𝑟−1𝑞−𝜋𝑖_𝑞𝑟=1𝜁1−𝑙+𝑛+𝑘,𝑟−𝜉𝑞1⋅exp2()+2()𝑝𝑙−𝑛−𝑘𝑟−𝜉𝑞.𝜋𝑖𝑛,𝑙∈ℕ;𝜉∈ℝ;0<𝜉<1;𝑝∈ℤ;𝑞∈ℕ (19) 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 Φ^∗_𝜇(𝑧,𝑠,𝑎). Explicitly, they proved the following theorem.

Theorem 1. The following expansion holds for Φ^∗_𝜇(𝑧,𝑠,𝑎): Φ^∗_𝜇()=()𝑧𝑧,𝑠,𝑎𝑖Γ1−𝑠^−𝑎sin𝛽𝜋sin𝛽+𝜇−𝜔𝜋Γ𝜇Γ()𝜋⋅1−𝜔−𝑎sin𝜇+𝛽𝜋sin𝛽−𝜔_{∞𝑚𝑚=0𝑘𝑘=0𝑗=0}𝑘𝑗×Γ𝑚−1𝑘−1𝜇−𝑎−𝜔−𝑚Γ𝐵𝜇−𝑚𝑚!^{(𝑚)𝑚−𝑘}×(2𝜋)^{𝑠−1−𝑗}(1−𝑠)_𝑗⋅(𝜔−1+𝑚)_𝑘−𝑗×exp−𝑠−𝑗𝜋𝑖2×Φ𝑒^{−2𝜋𝑖𝑎},1−𝑠+𝑗,log𝑧2𝜋𝑖−exp2𝑎+𝑠−𝑗2𝑒𝜋𝑖×Φ^{2𝜋𝑖𝑎},1−𝑠+𝑗,1−log𝑧2𝜋𝑖𝜇∈ℂ⧵ℤ⁻₀(𝑧);𝜔∈ℂ;Re1−𝛽>0;𝑠∈ℂ;arg−logmod2𝜋𝑖≤𝜋;𝑎∈ℂ⧵ℤ⁻₀. (20)

More interestingly, setting 𝑧=𝑒^{2𝜋𝑖𝜉}𝑝,𝑎=𝑞𝑝∈ℤ;𝑞∈ℕ;𝜉∈ℝ, (21) 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: Φ^∗_𝜇𝑒^{2𝜋𝑖𝜉}𝑝,𝜇−𝑠,𝑞=𝑖Γ1−𝜇+𝑠sin𝛽𝜋sin𝛽+𝜇−𝜔𝜋Γ𝜇Γ𝜋×1−𝜔−𝑝/𝑞sin𝜇+𝛽𝜋sin𝛽−𝜔_{∞𝑚𝑚=0𝑘𝑘=0𝑗=0}𝑘𝑗⋅Γ𝑚−1𝑘−1𝜇−𝑝/𝑞−𝜔−𝑚Γ𝜇−𝑚𝑚!×𝐵^(𝑚)_𝑚−𝑘2𝜋𝑞^{𝜇−𝑠−1−𝑗}×1−𝜇+𝑠_𝑗(𝜔−1+𝑚)_𝑘−𝑗⋅^𝑞_𝑟=1𝜁1−𝜇+𝑠+𝑗,𝜉+𝑟−1𝑞×exp−𝑖𝜋𝜇−𝑠−𝑗2+2(𝑟−1+𝜉)𝑝𝑞−^𝑞_𝑟=1𝜁1−𝜇+𝑠+𝑗,𝑟−𝜉𝑞×exp𝑖𝜋𝜇−𝑠−𝑗2+2(𝑟−𝜉)𝑝𝑞𝜇∈ℂ⧵ℤ⁻₀;𝜔∈ℂ;Re1−𝛽>0;𝑠∈ℂ;0<𝜉<1;𝑎∈ℂ⧵ℤ⁻₀. (22)

Remark 3. It is worthy to mention that (20) can be interpreted as the Fourier series expansion of Φ^∗_𝜇(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, Φ(𝑧,𝑠,𝑎)=𝑧^−𝑎Γ(1−𝑠)_𝑛∈ℤ𝑒^2𝜋ina2𝜋𝑖𝑛−log𝑧^1−𝑠. (23)

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 polynomials ℬ_𝑛^(𝛼)(𝑥;𝜆) are defined, for 𝜆,𝑥∈ℂ, by the following generating function [16, 17]: 𝑡𝜆𝑒^𝑡−1^𝛼𝑒^𝑥𝑡=^∞_𝑛=0ℬ_𝑛^(𝛼)()𝑡𝑥;𝜆^𝑛,(𝜆().𝑛!𝑡+log)<2𝜋,𝛼∈ℂ,if𝜆≠1;𝛼∈ℕif𝜆=1 (24)

Definition 5. The generalized Apostol-Euler polynomials ℰ_𝑛^(𝛼)(𝑥;𝜆) are defined, for 𝜆,𝛼 and 𝑥∈ℂ, by the following generating function [18]: 2𝜆𝑒^𝑡+1^𝛼𝑒^𝑥𝑡=^∞_𝑛=0ℰ_𝑛^(𝛼)()𝑡𝑥;𝜆^𝑛,().𝑛!𝑡+log𝜆<𝜋,𝛼∈ℂ,𝜆≠−1 (25)

Definition 6. The generalized Apostol-Genocchi polynomials 𝒢_𝑛^(𝛼)(𝑥;𝜆) are defined, for 𝜆,𝑥∈ℂ, by the following generating function [17, 19]: 2𝑡𝜆𝑒^𝑡+1^𝛼𝑒^𝑥𝑡=^∞_𝑛=0𝒢_𝑛^(𝛼)()𝑡𝑥;𝜆^𝑛,(𝜆().𝑛!𝑡+log)<𝜋,𝛼∈ℂ,if𝜆≠−1;𝛼∈ℕif𝜆=−1 (26)

Definition 7. The generalized Apostol-Frobenius-Euler polynomials ℋ_𝑛^(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]: 1−𝑢𝜆𝑒^𝑡−𝑢^𝛼𝑒^𝑥𝑡=^∞_𝑛=0ℋ_𝑛^(𝛼)()𝑡𝑥;𝜆∣𝑢^𝑛,𝜆𝑛!𝑡+log𝑢<𝜋. (27)
Setting 𝜆=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 ℰ_𝑛^(𝛼)(𝑥;𝜆) and the generalized Hurwitz-Lerch zeta function Φ^∗_𝛼(𝑧,𝑠,𝑎).

Theorem 8. Let 𝜆 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 ℰ_𝑛^(𝛼)(𝑥;𝜆)=2^𝛼Φ^∗_𝛼(−𝜆,−𝑛,𝑥). (28)

From this last theorem, we can find the relationships between the generalized Hurwitz-Lerch zeta function Φ^∗_𝛼(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 1−𝑢𝜆𝑒^𝑡−𝑢^𝛼𝑒^𝑥𝑡=𝑢−12𝑢^𝛼2(−𝜆/𝑢)𝑒^𝑡+1^𝛼𝑒^𝑥𝑡 (29) and thus, ℋ_𝑛^(𝛼)(𝑥;𝜆∣𝑢)=𝑢−12𝑢^𝛼ℰ_𝑛^(𝛼)𝑥;−𝜆𝑢. (30)

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 ℬ_𝑛^(𝑙)()𝑛𝑙𝑥;𝜆=𝑙!−12^𝑙ℰ^(𝑙)_𝑛−𝑙(𝑥;−𝜆)𝑙∈ℕ₀,𝒢,𝜆≠1_𝑛(𝑙)()𝑛𝑙𝑥;𝜆=𝑙!ℰ^(𝑙)_𝑛−𝑙(𝑥;𝜆)𝑙∈ℕ₀. (31) 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: ℬ_𝑛^(𝑙)()𝑛l𝑥;𝜆=𝑙!(−1)^𝑙Φ^∗_𝑙()𝜆|𝜆,𝑙−𝑛,𝑥|≤1;𝜆≠1;𝑙∈ℕ₀,𝒢_𝑛^(𝑙)(𝑥;𝜆)=2^𝑙𝑛𝑙𝑙!Φ^∗_𝑙()|𝜆|−𝜆,𝑙−𝑛,𝑥≤1;𝜆≠−1;𝑙∈ℕ₀,ℋ_𝑛^(𝛼)(𝑥;𝜆∣𝑢)=𝑢−1𝑢^𝛼Φ^∗_𝛼𝜆𝑢𝜆,−𝑛,𝑥𝑢≠0,1;𝑢≤1;𝜆≠𝑢. (32)(33)(34)

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 Section 3. 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: ℬ_𝑛^(𝑙)𝑝𝑞;𝑒^{2𝜋𝑖𝜉}=𝑖(−1)^𝑙𝑛!Γ(𝑙)Γ×1−𝜔−𝑝/𝑞_{𝑙−1𝑚𝑚=0𝑘𝑘=0𝑗=0}𝑘𝑗⋅Γ𝑚−1𝑘−1𝑙−𝑝/𝑞−𝜔−𝑚Γ()𝐵𝑙−𝑚𝑚!^{(𝑚)𝑚−𝑘}2𝜋𝑞^{𝑙−𝑛−1−𝑗}×(1−𝑙+𝑛)_𝑗(𝜔−1+𝑚)_𝑘−𝑗⋅^𝑞_𝑟=1𝜁1−𝑙+𝑛+𝑗,𝜉+𝑟−1𝑞×exp−𝑖𝜋𝑙−𝑛−𝑗2+2(𝑟−1+𝜉)𝑝𝑞−^𝑞_𝑟=1𝜁1−𝑙+𝑛+𝑗,𝑟−𝜉𝑞×exp𝑖𝜋𝑙−𝑛−𝑗2+2(𝑟−𝜉)𝑝𝑞.𝑝𝑙,𝑛∈ℕ;𝜔∈ℂ;Re1−𝛽>0;𝜉∈ℝ,0<𝜉<1;𝑞∈ℚ⧵ℤ⁻₀. (35)

Proof. Setting 𝜇=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: ℰ_𝑛^(𝛼)𝑝𝑞;𝑒^{2𝜋𝑖𝜉}=𝑖Γ(1+𝑛)2^𝛼sin𝛽𝜋sin𝛽+𝛼−𝜔𝜋Γ(𝛼)Γ𝜋×1−𝜔−𝑝/𝑞sin𝛼+𝛽𝜋sin𝛽−𝜔_{∞𝑚𝑚=0𝑘𝑘=0𝑗=0}𝑘𝑗⋅𝑚−1Γ𝑘−1𝛼−𝑝/𝑞−𝜔−𝑚Γ(𝛼−𝑚)𝑚!×𝐵^(𝑚)_𝑚−𝑘2𝜋𝑞^{−𝑛−1−𝑗}(1+𝑛)_𝑗(𝜔−1+𝑚)_𝑘−𝑗⋅^𝑞_𝑟=1𝜁1+𝑛+𝑗,2𝜉+2𝑟−12𝑞×exp−𝑖𝜋−𝑛−𝑗2+(2𝑟+2𝜉−1)𝑝𝑞−^𝑞_𝑟=1𝜁1+𝑛+𝑗,2𝑟−2𝜉−12𝑞×exp𝑖𝜋−𝑛−𝑗2+(2𝑟−2𝜉−1)𝑝𝑞𝛼∈ℂ⧵ℤ⁻₀1;𝜔∈ℂ;Re1−𝛽>0;𝜉∈ℝ,0≤𝜉≤1,𝜉≠2;𝑝𝑞∈ℚ⧵ℤ⁻₀. (36)

Proof. Replacing 𝑧 by −𝑒^{2𝜋𝑖𝜉}(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: 𝒢_𝑛^(𝑙)𝑝𝑞;𝑒^{2𝜋𝑖𝜉}=𝑖2^𝑙𝑛!Γ(𝑙)Γ×1−𝜔−𝑝/𝑞_{𝑙−1𝑚𝑚=0𝑘𝑘=0𝑗=0}𝑘𝑗⋅Γ𝑚−1𝑘−1𝑙−𝑝/𝑞−𝜔−𝑚Γ()𝐵𝑙−𝑚𝑚!^{(𝑚)𝑚−𝑘}2𝜋𝑞^{𝑙−𝑛−1−𝑗}×(1−𝑙+𝑛)_𝑗(𝜔−1+𝑚)_𝑘−𝑗⋅^𝑞_𝑟=1𝜁1−𝑙+𝑛+𝑗,2𝜉+2𝑟−12𝑞×exp−𝑖𝜋𝑙−𝑛−𝑗2+(2𝑟+2𝜉−1)𝑝𝑞−^𝑞_𝑟=1𝜁1−𝑙+𝑛+𝑗,2𝑟−2𝜉−12𝑞×exp𝑖𝜋𝑙−𝑛−𝑗2+(2𝑟−2𝜉−1)𝑝𝑞1𝑙,𝑛∈ℕ;𝜔∈ℂ;Re1−𝛽>0;𝜉∈ℝ,0≤𝜉≤1,𝜉≠2;𝑝𝑞∈ℚ⧵ℤ⁻₀. (37)

Proof. Substituting 𝑧 by −𝑒^{2𝜋𝑖𝜉}(𝜉∈ℝ), 𝑎 by (𝑝/𝑞)(𝑝∈ℤ;𝑞∈ℕ) and 𝑠 by 𝜇−𝑠(𝜇∈ℂ) in (20) gives Φ^∗_𝜇−𝑒^{2𝜋𝑖𝜉}𝑝,𝜇−𝑠,𝑞;=𝑖Γ1−𝜇+𝑠sin𝛽𝜋sin𝛽+𝜇−𝜔𝜋Γ𝜇Γ𝜋×1−𝜔−𝑝/𝑞sin𝜇+𝛽𝜋sin𝛽−𝜔_{∞𝑚𝑚=0𝑘𝑘=0𝑗=0}⋅𝑘𝑗𝑚−1Γ𝑘−1𝜇−𝑝/𝑞−𝜔−𝑚Γ𝜇−𝑚𝑚!×𝐵^(𝑚)_𝑚−𝑘2𝜋𝑞^{𝜇−𝑠−1−𝑗}1−𝜇+𝑠_𝑗(𝜔−1+𝑚)_𝑘−𝑗⋅^𝑞_𝑟=1𝜁1−𝜇+𝑠+𝑗,2𝜉+2𝑟−12𝑞×exp−𝑖𝜋𝜇−𝑠−𝑗2+(2𝑟+2𝜉−1)𝑝𝑞−^𝑞_𝑟=1𝜁1−𝜇−𝑠+𝑗,2𝑟−2𝜉−12𝑞×exp𝑖𝜋𝜇−𝑠−𝑗2+(2𝑟−2𝜉−1)𝑝𝑞. (38) 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: ℋ_𝑛^(𝛼)𝑝𝑞;𝑒^{2𝜋𝑖𝜉}∣𝑒^{2𝜋𝑖𝜃}=𝑖1−𝑒^{−2𝜋𝑖𝜃𝛼}𝑛!sin𝛽𝜋sin𝛽+𝛼−𝜔𝜋Γ(𝛼)Γ𝜋×1−𝜔−𝑝/𝑞sin𝛼+𝛽𝜋sin𝛽−𝜔_{∞𝑚𝑚=0𝑘𝑘=0𝑗=0}⋅𝑘𝑗𝑚−1Γ𝑘−1𝛼−𝑝/𝑞−𝜔−𝑚Γ(𝛼−𝑚)𝑚!×𝐵^(𝑚)_𝑚−𝑘2𝜋𝑞^{−𝑛−1−𝑗}(1+𝑛)_𝑗(𝜔−1+𝑚)_𝑘−𝑗⋅^𝑞_𝑟=1𝜁()1+𝑛+𝑗,𝜉−𝜃+𝑟−1𝑞×exp−𝑖𝜋−𝑛−𝑗2+2(𝑟−1+(𝜉−𝜃))𝑝𝑞−^𝑞_𝑟=1𝜁()1+𝑛+𝑗,𝑟−𝜉−𝜃𝑞×exp𝑖𝜋−𝑛−𝑗2+2(𝑟−(𝜉−𝜃))𝑝𝑞.p𝜉,𝜃∈ℝ,0≤𝜉≤1,0<𝜃<1,0<𝜉−𝜃<1,𝜉≠𝜃;q∈ℚ⧵ℤ⁻₀;𝛼∈ℂ⧵ℤ⁻₀;𝜔∈ℂ;Re1−𝛽>0. (39)

Proof. The result follows by setting 𝑧=𝑒^{2𝜋𝑖(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=𝑒^{2𝜋𝑖(𝜉)} and 𝑢=𝑒^{2𝜋𝑖(𝜃)}.

It is worthy to note that setting 𝑙=1 in (35) gives ℬ_𝑛𝑝𝑞;𝑒^{2𝜋𝑖𝜉}=−𝑛!2𝜋𝑞^𝑛𝑞_𝑟=1𝜁𝑛,𝜉+𝑟−1𝑞×exp−𝑖𝜋−𝑛2+2(𝑟−1+𝜉)𝑝𝑞−^𝑞_𝑟=1𝜁𝑛,𝑟−𝜉𝑞×exp𝑖𝜋−𝑛2+2(𝑟−𝜉)𝑝𝑞𝑝𝑛∈ℕ;𝑛≥1𝜉∈ℝ,0<𝜉<1;𝑞∈ℚ⧵ℤ⁻₀, (40) a result given by Srivastava [4, page 84, Equation (4.6)].

Moreover, if we replace 𝛼 by 1 in (36), we recover a result obtained by Luo [11, page 345, Equation (4.3)], namely, ℰ_𝑛𝑝𝑞;𝑒^{2𝜋𝑖𝜉}=2𝑛!2𝜋𝑞^𝑛+1𝑞_𝑟=1𝜁1+𝑛,2𝜉+2𝑟−12𝑞×exp−𝑖𝜋−𝑛−12+(2𝑟+2𝜉−1)𝑝𝑞−^𝑞_𝑟=1𝜁1+𝑛,2𝑟−2𝜉−12𝑞×exp𝑖𝜋−𝑛+12+(2𝑟−2𝜉−1)𝑝𝑞1𝜉∈ℝ,0≤𝜉≤1,𝜉≠2𝑝;𝑝∈ℤ;𝑞∈ℕ;𝑞∈ℚ⧵ℤ⁻₀. (41)