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MATHEMATICS OF COMPUTATION Volume 78, Number 268, October 2009, Pages 2193–2208 S 0025-5718(09)02230-3 Article electronically published on June 12, 2009

FOURIER EXPANSIONS AND INTEGRAL REPRESENTATIONS FOR THE APOSTOL-BERNOULLI AND APOSTOL-EULER POLYNOMIALS QIU-MING LUO Abstract. We investigate Fourier expansions for the Apostol-Bernoulli and Apostol-Euler polynomials using the Lipschitz summation formula and obtain their integral representations. We give some explicit formulas at rational arguments for these polynomials in terms of the Hurwitz zeta function. We also derive the integral representations for the classical Bernoulli and Euler polynomials and related known results.

1. Introduction The classical Bernoulli polynomials and Euler polynomials are defined by means of the following generating functions (see [1, pp. 804-806] or [18, pp. 25-32]) (1.1)

∞ ∑ 𝑧𝑒𝑥𝑧 𝑧𝑛 = 𝐵 (𝑥) 𝑛 𝑒𝑧 − 1 𝑛=0 𝑛!

(∣𝑧∣ < 2𝜋)

∞ ∑ 2𝑒𝑥𝑧 𝑧𝑛 = 𝐸 (𝑥) 𝑛 𝑒𝑧 + 1 𝑛=0 𝑛!

(∣𝑧∣ < 𝜋),

and (1.2)

( ) respectively. Obviously, 𝐵𝑛 := 𝐵𝑛 (0), 𝐸𝑛 := 2𝑛 𝐸𝑛 12 are the Bernoulli numbers and Euler numbers respectively. Some interesting analogues of the classical Bernoulli polynomials and numbers were first investigated by Apostol [2, p. 165, Eq. (3.1)] and (more recently) by Srivastava [20, pp. 83-84]. We begin by recalling here Apostol’s definitions as follows: Received by the editor June 3, 2008 and, in revised form, September 26, 2008. 2000 Mathematics Subject Classification. Primary 11B68; Secondary 42A16, 11M35. Key words and phrases. Lipschitz summation formula, Fourier expansion, integral representation, Apostol-Bernoulli and Apostol-Euler polynomials and numbers, Bernoulli and Euler polynomials and numbers, Hurwitz Zeta function, Lerch’s functional equation, rational arguments. The author expresses his sincere gratitude to the referee for valuable suggestions and comments. The author thanks Professor Chi-Wang Shu who helped with the submission of this manuscript to the Web submission system of the AMS.. The present investigation was supported in part by the PCSIRT Project of the Ministry of Education of China under Grant #IRT0621, Innovation Program of Shanghai Municipal Education Committee of China under Grant #08ZZ24 and Henan Innovation Project For University Prominent Research Talents of China under Grant #2007KYCX0021. c ⃝2009 American Mathematical Society Reverts to public domain 28 years from publication

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Definition 1.1 (Apostol [2]; see also Srivastava [20]). The Apostol-Bernoulli polynomials ℬ𝑛 (𝑥; 𝜆) in 𝑥 are defined by means of the generating function ∞ ∑ 𝑧𝑒𝑥𝑧 𝑧𝑛 (1.3) = ℬ (𝑥; 𝜆) 𝑛 𝜆𝑒𝑧 − 1 𝑛=0 𝑛! (∣𝑧∣ < 2𝜋 when 𝜆 = 1; ∣𝑧∣ < ∣log 𝜆∣ when 𝜆 ∕= 1) with, of course, 𝐵𝑛 (𝑥) = ℬ𝑛 (𝑥; 1)

and

ℬ𝑛 (𝜆) := ℬ𝑛 (0; 𝜆) ,

where ℬ𝑛 (𝜆) denotes the so-called Apostol-Bernoulli numbers (in fact, it is a function in 𝜆). Recently, Luo and Srivastava introduced the Apostol-Euler polynomials as follows: Definition 1.2 (Luo [14]; see also Luo and Srivastava [13]). The Apostol-Euler polynomials ℰ𝑛 (𝑥; 𝜆) in 𝑥 are defined by means of the generating function ∞ ∑ 2𝑒𝑥𝑧 𝑧𝑛 (1.4) = (∣𝑧∣ < ∣log(−𝜆)∣) , ℰ𝑛 (𝑥; 𝜆) 𝑧 𝜆𝑒 + 1 𝑛=0 𝑛! with, of course, 𝐸𝑛 (𝑥) = ℰ𝑛 (𝑥; 1)

and

𝑛

ℰ𝑛 (𝜆) := 2 ℰ𝑛

(

) 1 ;𝜆 , 2

where ℰ𝑛 (𝜆) denote the so-called Apostol-Euler numbers (in fact, it is a function in 𝜆). Remark 1.3. In Definition 1.1 and Definition 1.2, the original constraints ∣𝑧 + log 𝜆∣ < 2𝜋 and ∣𝑧 + log 𝜆∣ < 𝜋, respectively, should be replaced by the conditions ∣𝑧∣ < 2𝜋 when 𝜆 = 1; ∣𝑧∣ < ∣log 𝜆∣ when 𝜆 ∕= 1 and ∣𝑧∣ < ∣log(−𝜆)∣ for the referee’s clear and detailed argumentation. Hence, the corresponding constraints in References [13], [14], [15] and [20] should also be such. The Apostol-Bernoulli and Apostol-Euler polynomials have been investigated by many people (see, e.g., [2], [4], [5], [9], [13]–[17], [20] and [22]). D. H. Lehmer [11] gave a new approach to Bernoulli polynomials, starting from a function equation (Rabbe’s multiplication theorem). H. Haruki and T. M. Rassias [10] provided the new integral representations for the Bernoulli and Euler polynomials as well as using a similar function equation. Recently, D. Cvijović [7] reproduced the results of H. Haruki and T. M. Rassias in a different way and showed several different integral representations for the Bernoulli and Euler polynomials. In the present paper, we first investigate Fourier expansions for the ApostolBernoulli and Apostol-Euler polynomials based on the Lipschitz summation formula, and then provide their integral representations. We obtain some explicit formulas for the Apostol-Bernoulli and Apostol-Euler polynomials at rational arguments in terms of the Hurwitz zeta function. We also deduce the corresponding uniform integral representations for the classical Bernoulli and Euler polynomials. We will see that the results of Cvijović or H. Haruki and T. M. Rassias are the corresponding direct consequences of our formulas. The paper is organized as follows. In the first section we rewrite the definitions of Apostol-Bernoulli and Apostol-Euler polynomials. In the second section we derive

FOURIER EXPANSIONS AND INTEGRAL REPRESENTATIONS

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Fourier expansions for the Apostol-Bernoulli and Apostol-Euler polynomials. In the third section we show their integral representations. In the fourth section we obtain their explicit formulas at rational arguments in terms of the Hurwitz zeta function. In the fifth section we deduce the corresponding uniform integral representations for the classical Bernoulli and Euler polynomials and related results of Cvijović or H. Haruki and T. M. Rassias. In the sixth section we give some applications 𝑛−1 (2𝜋)2𝑛 𝐵2𝑛 is and remarks; for example, the classical Euler formula 𝜁(2𝑛) = (−1)2(2𝑛)! obtained according to our method. 2. Fourier expansions for the Apostol-Bernoulli and Apostol-Euler polynomials In this section we investigate Fourier expansions for the Apostol-Bernoulli and Apostol-Euler polynomials by applying the Lipschitz summation formula. First we recall the Lipschitz summation formula (see [12] or [19]) as follows: ∑ 𝑒2𝜋𝑖(𝑛+𝜇)𝜏 Γ(𝛼) ∑ 𝑒−2𝜋𝑖𝑘𝜇 (2.1) = , 1−𝛼 (𝑛 + 𝜇) (−2𝜋𝑖)𝛼 (𝜏 + 𝑘)𝛼 𝑛+𝜇>0 𝑘∈ℤ

where 𝜇 ∈ ℤ and ℜ(𝛼) > 1 or 𝜇 ∈ ℝ ∖ ℤ and ℜ(𝛼) > 0; 𝜏 ∈ 𝐻 is the complex upper half plane and Γ denotes the Gamma function. Theorem 2.1. For 𝑛 = 1, 0 < 𝑥 < 1 and 𝑛 > 1, 0 ≤ 𝑥 ≤ 1, 𝜆 ∈ ℂ ∖ {0}, we have 𝑒2𝜋𝑖𝑘𝑥 𝑛! ∑′ (2.2) ℬ𝑛 (𝑥; 𝜆) = −𝛿𝑛 (𝑥; 𝜆) − 𝑥 𝜆 (2𝜋𝑖𝑘 − log 𝜆)𝑛 [ ∞ ) ] [( 𝑛!𝑖𝑛 ∑ exp −2𝜋𝑘𝑥 + 𝑛𝜋 2 𝑖 = −𝛿𝑛 (𝑥; 𝜆) − 𝑥 (2.3) 𝜆 (2𝜋𝑖𝑘 + log 𝜆)𝑛 𝑘=1 ) ]] [( ∞ ∑ exp 2𝜋𝑘𝑥 − 𝑛𝜋 2 𝑖 + , (2𝜋𝑖𝑘 − log 𝜆)𝑛 where the symbol

∑′

𝑘=1

denotes the standard convention of a sum over the integers

that omits 0; 𝛿𝑛 (𝑥; 𝜆) = 0 or

(−1)𝑛 𝑛! 𝜆𝑥 log𝑛 𝜆

according as 𝜆 = 1 or 𝜆 ∕= 1, respectively.

Proof. For 0 ≤ 𝑥 ≤ 1, by (1.3) and the generalized binomial theorem, we have ∞ ∞ ∑ ∑ 𝑒2𝜋𝑖𝜏 𝑥 (2𝜋𝑖𝜏 )𝑘−1 = 2𝜋𝑖𝜏 =− (2.4) ℬ𝑘 (𝑥; 𝜆) 𝜆𝑘 𝑒2𝜋𝑖(𝑘+𝑥)𝜏 𝑘! 𝜆𝑒 −1 𝑘=0 𝑘=0 ( ) log ∣𝜆∣ ∣log 𝜆∣ when 𝜆 ∕= 1; ℑ𝜏 > ∣𝜏 ∣ < 1 when 𝜆 = 1; ∣𝜏 ∣ < . 2𝜋 2𝜋 We differentiate both sides of (2.4) with respect to the variable 𝜏 , by 𝑛 − 1 times and noting that ℬ0 (𝑥; 𝜆) = 𝛿1,𝜆 (see [13, p. 301]). Then we get (2.5)

∞ ∑ 𝑘=𝑛

ℬ𝑘 (𝑥; 𝜆)

(−1)𝑛−1 (𝑛 − 1)! (2𝜋𝑖)𝑘−1𝜏 𝑘−𝑛 + 𝛿1,𝜆 𝑘(𝑘 − 𝑛)! 2𝜋𝑖𝜏 𝑛 = −(2𝜋𝑖)𝑛−1

∞ ∑ 𝑘=0

where 𝛿1,𝜆 is the Kronecker symbol.

𝜆𝑘 (𝑘 + 𝑥)𝑛−1 𝑒2𝜋𝑖(𝑘+𝑥)𝜏 ,

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On the other hand, letting 𝛼 = 𝑛, 𝜇 → 𝑥, 𝜏 → 𝜏 + (−1)𝑛 (𝑛 − 1)!

(2.6)

∑ 𝑘∈ℤ

log 𝜆 2𝜋𝑖

in (2.1), we find that

∞

∑ 𝑒−2𝜋𝑖𝑘𝑥 𝜆𝑘+𝑥 (𝑘 + 𝑥)𝑛−1 𝑒2𝜋𝑖(𝑘+𝑥)𝜏 . 𝑛 = [2𝜋𝑖(𝜏 + 𝑘) + log 𝜆] 𝑘=0

Combining (2.5) and (2.6), we obtain 𝜆𝑥

∞ ∑


𝑘=𝑛

(−1)𝑛−1 (𝑛 − 1)! (2𝜋𝑖)𝑘−1𝜏 𝑘−𝑛 + 𝜆𝑥 𝛿1,𝜆 𝑘(𝑘 − 𝑛)! 2𝜋𝑖𝜏 𝑛 = (−1)𝑛−1 (𝑛 − 1)!(2𝜋𝑖)𝑛−1

∑ 𝑘∈ℤ

𝑒−2𝜋𝑖𝑘𝑥 . [2𝜋𝑖(𝜏 + 𝑘) + log 𝜆]𝑛

Separating this 𝑘 = 0 term in the above sum on the right side yields that (2.7)

𝜆𝑥

∞ ∑


𝑘=𝑛

×

∑′

(2𝜋𝑖)𝑘−1𝜏 𝑘−𝑛 = (−1)𝑛−1 (𝑛 − 1)!(2𝜋𝑖)𝑛−1 𝑘(𝑘 − 𝑛)!

𝑒−2𝜋𝑖𝑘𝑥 (−1)𝑛−1 (𝑛 − 1)!(2𝜋𝑖)𝑛−1 (1 − 𝛿1,𝜆 ). 𝑛 + [2𝜋𝑖(𝜏 + 𝑘) + log 𝜆] (2𝜋𝑖𝜏 + log 𝜆)𝑛

Letting 𝜏 → 0 in (2.7) we are led at once to the assertion (2.2) of Theorem 2.1. 𝑛𝜋𝑖 Noting that 𝑖𝑛 = 𝑒 2 , (−1)𝑛 = 𝑒−𝑛𝜋𝑖 and via a simple calculation, then the assertion (2.3) of Theorem 2.1 is a direct consequence of (2.2). This completes our proof. □ In the same manner, we may prove the following. Theorem 2.2. For 𝑛 = 0, 0 < 𝑥 < 1 and 𝑛 > 0, 0 ≤ 𝑥 ≤ 1, 𝜆 ∈ ℂ ∖ {0, −1}, we have (2.8) (2.9)

𝑒(2𝑘−1)𝜋𝑖𝑥 2 ⋅ 𝑛! ∑ 𝜆𝑥 [(2𝑘 − 1)𝜋𝑖 − log 𝜆]𝑛+1 𝑘∈ℤ [ ∞ [( ) ] 2 ⋅ 𝑛!𝑖𝑛+1 ∑ exp 𝑛+1 2 𝜋 − (2𝑘 + 1)𝜋𝑥 𝑖 = 𝜆𝑥 [(2𝑘 + 1)𝜋𝑖 + log 𝜆]𝑛+1 𝑘=0 [( ) ]] ∞ ∑ exp − 𝑛+1 2 𝜋 + (2𝑘 + 1)𝜋𝑥 𝑖 + . 𝑛+1 [(2𝑘 + 1)𝜋𝑖 − log 𝜆] 𝑘=0

ℰ𝑛 (𝑥; 𝜆) =

By Theorem 2.1 and Theorem 2.2, we can deduce respectively the Fourier expansions for the classical Bernoulli and Euler polynomials as follows: Corollary 2.3. For 𝑛 = 1, 0 < 𝑥 < 1 and 𝑛 > 1, 0 ≤ 𝑥 ≤ 1, we have (2.10) (2.11)

𝐵𝑛 (𝑥) = − =−

𝑛! ∑′ 𝑒2𝜋𝑖𝑘𝑥 (2𝜋𝑖)𝑛 𝑘𝑛 ( ∞ 2 ⋅ 𝑛! ∑ cos 2𝜋𝑘𝑥 − (2𝜋)𝑛

𝑘=1

𝑘𝑛

𝑛𝜋 2

) .


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Corollary 2.4. For 𝑛 = 0, 0 < 𝑥 < 1 and 𝑛 > 0, 0 ≤ 𝑥 ≤ 1, we have

(2.12) (2.13)

2 ⋅ 𝑛! ∑ 𝑒(2𝑘−1)𝜋𝑖𝑥 (𝜋𝑖)𝑛+1 (2𝑘 − 1)𝑛+1 𝑘∈ℤ [ ∞ 4 ⋅ 𝑛! ∑ sin (2𝑘 + 1)𝜋𝑥 − = 𝑛+1 𝜋 (2𝑘 + 1)𝑛+1

𝐸𝑛 (𝑥) =

𝑛𝜋 2

] .

𝑘=0

𝜆 1 Remark 2.5. Replacing 𝜏 by 𝜏 + log 2𝜋𝑖 + 2 in (2.1) and applying ℰ0 (𝑥; 𝜆) = for details, [13]–[15]) when we prove the assertion (2.8) of Theorem 2.2.

2 𝜆+1

(see,

Remark 2.6. We define the 𝑛-th Apostol-Bernoulli function as (2.14)

ℬˆ𝑛 (𝑥; 𝜆) := ℬ𝑛 (𝑥; 𝜆) (0 ≤ 𝑥 < 1),

ℬˆ𝑛 (𝑥 + 1; 𝜆) = 𝜆−1 ℬˆ𝑛 (𝑥; 𝜆),

which is also called the quasi-periodicity Apostol-Bernoulli polynomials. For any 𝑥 ∈ ℝ, 𝑟 ∈ ℤ, we have (2.15)

ℬˆ𝑛 (𝑥; 𝜆) = 𝜆−[𝑥] ℬ𝑛 ({𝑥}; 𝜆),

ℬˆ𝑛 (𝑥 + 𝑟; 𝜆) = 𝜆−𝑟 ℬˆ𝑛 (𝑥; 𝜆).

Here the notation {𝑥} denotes the fractional part of 𝑥, and the notation [𝑥] denotes the greatest integer not exceeding 𝑥. Clearly, the Apostol-Bernoulli polynomials ℬ𝑛 (𝑥; 𝜆) (0 ≤ 𝑥 < 1) are the quasiperiodicity functions in 𝑥 with period 1. One of the special cases of the quasiperiodicity Apostol-Bernoulli polynomials is just Carlitz’s periodic Bernoulli function [3, p. 661] for 𝜆 = 1. Remark 2.7. We define the 𝑛-th Apostol-Euler function as (2.16)

ℰˆ𝑛 (𝑥; 𝜆) := ℰ𝑛 (𝑥; 𝜆) (0 ≤ 𝑥 < 1),

ℰˆ𝑛 (𝑥 + 1; 𝜆) = −𝜆−1 ℰˆ𝑛 (𝑥; 𝜆),

which is called the quasi-periodicity Apostol-Euler polynomials. For any 𝑥 ∈ ℝ, 𝑟 ∈ ℤ, we have (2.17)

ℰˆ𝑛 (𝑥; 𝜆) = (−1)[𝑥] 𝜆−[𝑥] ℰ𝑛 ({𝑥}; 𝜆),

ℰˆ𝑛 (𝑥 + 𝑟; 𝜆) = (−1)𝑟 𝜆−𝑟 ℰˆ𝑛 (𝑥; 𝜆).

Obviously, the Apostol-Euler polynomials ℰ𝑛 (𝑥; 𝜆) (0 ≤ 𝑥 < 1) are the quasiperiodicity functions in 𝑥 with period 1. One of the special cases of the quasiperiodicity Apostol-Euler polynomials is just Carlitz’s periodic Euler function [3, p. 661] for 𝜆 = 1.

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Remark 2.8. We employ a useful relationship [15, p. 636, Eq. (38)] (2.18)

ℰ𝑛 (𝑥; 𝜆) =

(𝑥 )] 2 [ ℬ𝑛+1 (𝑥; 𝜆) − 2𝑛+1 ℬ𝑛+1 ; 𝜆2 𝑛+1 2

to (2.2) and (2.3), respectively; we can also arrive at the corresponding (2.8) and (2.9). Remark 2.9. Throughout this paper, we take the principal value of the logarithm log 𝜆, i.e., log 𝜆 = log ∣𝜆∣+𝑖 arg 𝜆 (−𝜋 < arg 𝜆 ≤ 𝜋) when 𝜆 ∕= 1; We choose log 1 = 0 when 𝜆 = 1.

3. Integral representations for the Apostol-Bernoulli and Apostol-Euler polynomials In this section we give the integral representations for the Apostol-Bernoulli and Apostol-Euler polynomials with their Fourier expansions. For convenience, we take 𝜆 = 𝑒2𝜋𝑖𝜉 (𝜉 ∈ ℝ, ∣𝜉∣ < 1) in this section. Theorem 3.1. For 𝑛 = 1, 2, . . . , 0 ≤ ℜ(𝑥) ≤ 1, ∣𝜉∣ < 1, 𝜉 ∈ ℝ, we have (3.1)

ℬ𝑛 (𝑥; 𝑒2𝜋𝑖𝜉 ) = −Δ𝑛 (𝑥; 𝜉) ∫ ∞ 𝑈 (𝑛; 𝑥, 𝑡) cosh(2𝜋𝜉𝑡) + 𝑖 𝑉 (𝑛; 𝑥, 𝑡) sinh(2𝜋𝜉𝑡) 𝑛−1 − 𝑛𝑒−2𝜋𝑖𝑥𝜉 𝑡 d𝑡, cosh 2𝜋𝑡 − cos 2𝜋𝑥 0

where Δ𝑛 (𝑥; 𝜉) = 0 or

(−1)𝑛 𝑛! 𝑒2𝜋𝑖𝑥𝜉 (2𝜋𝑖𝜉)𝑛

according as 𝜉 = 0 or 𝜉 ∕= 0, respectively, and

] ( 𝑛𝜋 ) [ ( 𝑛𝜋 ) − cos 𝑒−2𝜋𝑡 , 𝑈 (𝑛; 𝑥, 𝑡) = cos 2𝜋𝑥 − 2 2) [ ( ] ( 𝑛𝜋 𝑛𝜋 ) 𝑉 (𝑛; 𝑥, 𝑡) = sin 2𝜋𝑥 − + sin 𝑒−2𝜋𝑡 . 2 2 Proof. Returning to (2.2) and setting 𝜆 = 𝑒2𝜋𝑖𝜉 , 𝑘 → −𝑘 yields ℬ𝑛 (𝑥; 𝑒2𝜋𝑖𝜉 ) = −Δ𝑛 (𝑥; 𝜉) −

𝑛!𝑒−2𝜋𝑖𝑥𝜉 ∑′ 𝑒−2𝜋𝑖𝑘𝑥 . (−2𝜋𝑖)𝑛 (𝑘 + 𝜉)𝑛

Using the known integral formula ∫ (3.2)

0

∞

𝑡𝑛 𝑒−𝑎𝑡 d𝑡 =

𝑛! 𝑎𝑛+1

(𝑛 = 0, 1, . . . ; ℜ(𝑎) > 0),

( )𝑛 𝑛𝜋𝑖 and noting that − 1𝑖 = 𝑒 2 and (−1)𝑛 = 𝑒−𝑛𝜋𝑖 , then we have


2𝜋𝑖𝜉

ℬ𝑛 (𝑥; 𝑒

𝑛𝑒−2𝜋𝑖𝑥𝜉 = − Δ𝑛 (𝑥; 𝜉) − (−2𝜋𝑖)𝑛

)

{

∞ ∑

𝑛

{∫ 0

{∫

{∫

𝑛𝑒−2𝜋𝑖𝑥𝜉 2(2𝜋)𝑛 ∫ +

2𝜋𝑖𝑘𝑥

𝑒

∫

∞

0

∞

∫

∞

∞ ∑

} 𝑡

𝑛−1 −(𝑘−𝜉)𝑡

𝑒

d𝑡

𝑒−(2𝜋𝑖𝑥+𝑡)𝑘 d𝑡

𝑘=1 ∞ ∑ 𝜉𝑡 𝑛−1 (2𝜋𝑖𝑥−𝑡)𝑘

𝑒 𝑡

𝑒

} d𝑡

𝑘=1

𝑒𝑡 ∫

𝑡𝑛−1 𝑒−(𝑘+𝜉)𝑡 d𝑡

0

𝑒−𝜉𝑡 𝑡𝑛−1

∞

0

∞

0

∞ ∑

∞

+ (−1)𝑛 = − Δ𝑛 (𝑥; 𝜉) −

𝑒

𝑘=1

+ (−1)𝑛 𝑛𝑒−2𝜋𝑖𝑥𝜉 = − Δ𝑛 (𝑥; 𝜉) − (−2𝜋𝑖)𝑛

∫

𝑘=1

+ (−1) 𝑛𝑒−2𝜋𝑖𝑥𝜉 = − Δ𝑛 (𝑥; 𝜉) − (−2𝜋𝑖)𝑛

−2𝜋𝑖𝑘𝑥

2199

𝑒−2𝜋𝑖𝑥 𝑒−𝜉𝑡 𝑡𝑛−1 d𝑡 − 𝑒−2𝜋𝑖𝑥

∞

0

𝑒2𝜋𝑖𝑥 𝑒𝜉𝑡 𝑡𝑛−1 d𝑡 𝑒𝑡 − 𝑒2𝜋𝑖𝑥

}

𝑛𝜋𝑖 2

(𝑒−2𝜋𝑖𝑥 − 𝑒−𝑡 ) −𝜉𝑡 𝑛−1 𝑒 𝑡 d𝑡 cosh 𝑡 − cos 2𝜋𝑥 0 } ∞ − 𝑛𝜋𝑖 𝑒 2 (𝑒2𝜋𝑖𝑥 − 𝑒−𝑡 ) 𝜉𝑡 𝑛−1 𝑒 𝑡 d𝑡 . cosh 𝑡 − cos 2𝜋𝑥 0 𝑒

It follows that we make the transformation 𝑡 = 2𝜋𝑢, and after simplification we obtain the desired (3.1) immediately. This completes the proof. □ We can obtain the following integral representations for the Apostol-Euler polynomials by a similar method. Theorem 3.2. For 𝑛 = 1, 2, . . . , 0 ≤ ℜ(𝑥) ≤ 1, ∣𝜉∣ < 12 , 𝜉 ∈ ℝ, we have (3.3)

where

ℰ𝑛 (𝑥; 𝑒2𝜋𝑖𝜉 ) = 2𝑒−2𝜋𝑖𝑥𝜉 ∫ ∞ 𝑋(𝑛; 𝑥, 𝑡) cosh(2𝜋𝜉𝑡) + 𝑖 𝑌 (𝑛; 𝑥, 𝑡) sinh(2𝜋𝜉𝑡) 𝑛 𝑡 d𝑡, × cosh 2𝜋𝑡 − cos 2𝜋𝑥 0 [ ( ( 𝑛𝜋 ) 𝑛𝜋 )] 𝑋(𝑛; 𝑥, 𝑡) = 𝑒−𝜋𝑡 sin 𝜋𝑥 + + 𝑒𝜋𝑡 sin 𝜋𝑥 − , 2 ) 2 )] [ ( ( 𝑛𝜋 𝑛𝜋 𝑌 (𝑛; 𝑥, 𝑡) = 𝑒−𝜋𝑡 cos 𝜋𝑥 + − 𝑒𝜋𝑡 cos 𝜋𝑥 − . 2 2

On the other hand, we can also arrive at the following different integral representations for the Apostol-Bernoulli and Apostol-Euler polynomials. Theorem 3.3. For 𝑛 = 1, 2, . . . , 0 ≤ ℜ(𝑥) ≤ 1, ∣𝜉∣ < 1, 𝜉 ∈ ℝ, we have (3.4)

ℬ𝑛 (𝑥; 𝑒2𝜋𝑖𝜉 ) = −Δ𝑛 (𝑥; 𝜉) + ∫ ×

0

1

2𝑛𝑒−2𝜋𝑖𝑥𝜉 (−2𝜋)𝑛

𝑈 ′ (𝑛; 𝑥, 𝑡) cosh(𝜉 log 𝑡) − 𝑖 𝑉 ′ (𝑛; 𝑥, 𝑡) sinh(𝜉 log 𝑡) (log 𝑡)𝑛−1 d𝑡, 𝑡2 − 2𝑡 cos 2𝜋𝑥 + 1

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(−1)𝑛 𝑛! according as 𝜉 = 0 or 𝜉 ∕= 0, respectively, and 𝑒2𝜋𝑖𝑥𝜉 (2𝜋𝑖𝜉)𝑛 [ ( ( 𝑛𝜋 )] 𝑛𝜋 ) 𝑈 ′ (𝑛; 𝑥, 𝑡) = cos 2𝜋𝑥 − − 𝑡 cos , 2) 2 )] ( [ ( 𝑛𝜋 𝑛𝜋 + 𝑡 sin . 𝑉 ′ (𝑛; 𝑥, 𝑡) = sin 2𝜋𝑥 − 2 2

where Δ𝑛 (𝑥; 𝜉) = 0 or

Proof. First we substitute cosh 2𝜋𝑡 = (3.5)

𝑒2𝜋𝑡 +𝑒−2𝜋𝑡 2

into (3.1). Then we see that

ℬ𝑛 (𝑥; 𝑒2𝜋𝑖𝜉 ) = −Δ𝑛 (𝑥; 𝜉) − 2𝑛𝑒−2𝜋𝑖𝑥𝜉 ∫ ∞ 𝑈 (𝑛; 𝑥, 𝑡) cosh(2𝜋𝜉𝑡) + 𝑖 𝑉 (𝑛; 𝑥, 𝑡) sinh(2𝜋𝜉𝑡) 𝑛−1 𝑡 × d𝑡. 𝑒2𝜋𝑡 + 𝑒−2𝜋𝑡 − 2 cos 2𝜋𝑥 0

Then making the transformation 𝑢 = 𝑒−2𝜋𝑡 in (3.5), we easily obtain formula (3.4). This completes the proof. □ Similarly, we obtain Theorem 3.4. For 𝑛 = 1, 2, . . . , 0 ≤ ℜ(𝑥) ≤ 1, ∣𝜉∣ < 12 , 𝜉 ∈ ℝ, we have (3.6)

where

4𝑒−2𝜋𝑖𝑥𝜉 ℰ𝑛 (𝑥; 𝑒2𝜋𝑖𝜉 ) = (−1)𝑛 𝜋 𝑛+1 ∫ 1 ′ 𝑋 (𝑛; 𝑥, 𝑡) cosh(2𝜉 log 𝑡) − 𝑖 𝑌 ′ (𝑛; 𝑥, 𝑡) sinh(2𝜉 log 𝑡) (log 𝑡)𝑛 d𝑡, × 4 − 2𝑡2 cos 2𝜋𝑥 + 1 𝑡 0 ( [ ( 𝑛𝜋 )] 𝑛𝜋 ) + sin 𝜋𝑥 − , 𝑋 ′ (𝑛; 𝑥, 𝑡) = 𝑡2 sin 𝜋𝑥 + 2 ) 2 )] ( [ ( 𝑛𝜋 𝑛𝜋 𝑌 ′ (𝑛; 𝑥, 𝑡) = 𝑡2 cos 𝜋𝑥 + − cos 𝜋𝑥 − . 2 2

Remark 3.5. For any integers ℓ, we see easily that ℬ𝑛 (𝑥; 𝑒2𝜋𝑖(ℓ+𝜉) ) = ℬ𝑛 (𝑥; 𝑒2𝜋𝑖𝜉 ), ℰ𝑛 (𝑥; 𝑒2𝜋𝑖(ℓ+𝜉) ) = ℰ𝑛 (𝑥; 𝑒2𝜋𝑖𝜉 ). Therefore, the Apostol-Bernoulli polynomials ℬ𝑛 (𝑥; 𝑒2𝜋𝑖𝜉 ) and the Apostol-Euler polynomials ℰ𝑛 (𝑥; 𝑒2𝜋𝑖𝜉 ) are the periodicity functions in 𝜉 with period 2𝜋. In view of this observation we say that 𝜉 may take any real numbers in Theorem 3.1–Theorem 3.4. Remark 3.6. We can also prove Theorem 2.1 and Theorem 2.2 by Theorem 3.1 and Theorem 3.2, respectively, in an inverse process. 4. Explicit formulas for the Apostol-Bernoulli and Apostol-Euler polynomials at rational arguments In this section we obtain some explicit formulas for the Apostol-Bernoulli and Apostol-Euler polynomials at rational arguments. We can see that some known formulas of Cvijović and Klinowski are the corresponding special cases of our formulas. The Hurwitz-Lerch zeta function Φ(𝑧, 𝑠, 𝑎) defined by (cf., e.g., [21, p. 121, et seq.]) (4.1)

Φ(𝑧, 𝑠, 𝑎) :=

∞ ∑

𝑧𝑛 (𝑛 + 𝑎)𝑠 𝑛=0

( ) 𝑎 ∈ ℂ ∖ ℤ− 0 ; 𝑠 ∈ ℂ when ∣𝑧∣ < 1; ℜ(𝑠) > 1 when ∣𝑧∣ = 1


2201

contains, as its special cases, not only the Riemann and Hurwitz zeta functions (4.2)

𝜁(𝑠) := Φ(1, 𝑠, 1) = 𝜁 (𝑠, 1) =

( ) ∑ ∞ 1 1 1 𝜁 𝑠, = 𝑠 2 −1 2 𝑛𝑠 𝑛=1

and (4.3)

𝜁(𝑠, 𝑎) := Φ(1, 𝑠, 𝑎) =

∞ ∑

1 𝑠 (𝑛 + 𝑎) 𝑛=0

( ) ℜ (𝑠) > 1; 𝑎 ∈ / ℤ− 0

and the Lerch zeta function (or periodic zeta function) (4.4)

∞ ∑ ( ) 𝑒2𝑛𝜋𝑖𝜉 = 𝑒2𝜋𝑖𝜉 Φ 𝑒2𝜋𝑖𝜉 , 𝑠, 1 𝑠 𝑛 𝑛=1

𝑙𝑠 (𝜉) :=

(𝜉 ∈ ℝ; ℜ(𝑠) > 1) , but also such other functions as the polylogarithmic function (4.5)

Li𝑠 (𝑧) := ( 𝑠∈ℂ

when

∞ ∑ 𝑧𝑛 = 𝑧 Φ(𝑧, 𝑠, 1) 𝑛𝑠 𝑛=1

∣𝑧∣ < 1; ℜ(𝑠) > 1 when

) ∣𝑧∣ = 1

and the Lipschitz-Lerch zeta function (cf. [21, p. 122, Eq. 2.5 (11)]) (4.6)

𝜙(𝜉, 𝑎, 𝑠) :=

∞ ∑ ( ) 𝑒2𝑛𝜋𝑖𝜉 = Φ 𝑒2𝜋𝑖𝜉 , 𝑠, 𝑎 =: 𝐿 (𝜉, 𝑠, 𝑎) 𝑠 (𝑛 + 𝑎) 𝑛=0

( 𝑎 ∈ ℂ ∖ ℤ− 0 ; ℜ(𝑠) > 0 when 𝜉 ∈ ℝ ∖ ℤ; ℜ(𝑠) > 1 when

) 𝜉∈ℤ ,

which was first studied by Rudolf Lipschitz (1832-1903) and Maty´ aˇs Lerch (18601922) in connection with Dirichlet’s famous theorem on primes in arithmetic progressions. Recently, H. M. Srivastava made use of Apostol’s formula (see [2, p. 164]) (4.7)

2𝜋𝑖𝜉

𝜙(𝜉, 𝑎, 1 − 𝑛) = Φ(𝑒

( ) ℬ𝑛 𝑎; 𝑒2𝜋𝑖𝜉 , 1 − 𝑛, 𝑎) = − 𝑛

(𝑛 ∈ ℕ)

and Lerch’s functional equation (see [2, p. 161, (1.4)]) (4.8)

Γ(𝑠) 𝜙(𝜉, 𝑎, 1 − 𝑠) = (2𝜋)𝑠

) ] { [( 1 𝑠 − 2𝑎𝜉 𝜋𝑖 𝜙 (−𝑎, 𝜉, 𝑠) exp 2 ) ] } [( 1 + exp − 𝑠 + 2𝑎(1 − 𝜉) 𝜋𝑖 𝜙(𝑎, 1 − 𝜉, 𝑠) 2 (𝑠 ∈ ℂ; 0 < 𝜉 < 1)

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to derive the following formula of Apostol-Bernoulli polynomials at rational arguments (see [20, p. 84, Eq. (4.6)]): (4.9) { 𝑞 ( ( ) ) [( ) ] ∑ 𝑝 2𝜋𝑖𝜉 𝑛 2(𝜉 + 𝑗 − 1)𝑝 𝜉+𝑗−1 𝑛! ℬ𝑛 ;𝑒 − 𝜁 𝑛, =− exp 𝜋𝑖 𝑞 (2𝑞𝜋)𝑛 𝑗=1 𝑞 2 𝑞 ( ) [( ) ]} 𝑞 ∑ 𝑗−𝜉 𝑛 2(𝑗 − 𝜉)𝑝 𝜁 𝑛, + exp − + 𝜋𝑖 𝑞 2 𝑞 𝑗=1 (4.10)

(𝑛 ∈ ℕ ∖ {1} ; 𝑞 ∈ ℕ; 𝑝 ∈ ℤ; 𝜉 ∈ ℝ).

Below we obtain similar formulas by using Fourier expansions for the ApostolBernoulli polynomials and Apostol-Euler polynomials, respectively. Theorem 4.1. For 𝑛 ∈ ℕ ∖ {1} , 𝑞 ∈ ℕ, 𝑝 ∈ ℤ, 𝜉 ∈ ℝ, ∣𝜉∣ < 1, the following formula of Apostol-Bernoulli polynomials at rational arguments (4.11) { 𝑞 ( ( ) ( ) ) [( ) ] ∑ 𝑝 2𝜋𝑖𝜉 𝑝 𝑛 2(𝑗 + 𝜉)𝑝 𝑗+𝜉 𝑛! ℬ𝑛 𝜁 𝑛, = − Δ𝑛 ;𝑒 ;𝜉 − exp − 𝜋𝑖 𝑞 𝑞 (2𝜋𝑞)𝑛 𝑗=1 𝑞 2 𝑞 ( ) [( ) ]} 𝑞 ∑ 𝑗−𝜉 𝑛 2(𝑗 − 𝜉)𝑝 𝜁 𝑛, + exp − + 𝜋𝑖 𝑞 2 𝑞 𝑗=1 holds true in terms of the Hurwitz zeta function, where Δ𝑛 (𝑥; 𝜉) = 0 or according as 𝜉 = 0 or 𝜉 ∕= 0, respectively.

(−1)𝑛 𝑛! 𝑒2𝜋𝑖𝑥𝜉 (2𝜋𝑖𝜉)𝑛

Proof. We employ formula (2.3), [∞ )] ) ]] [( [( ∞ ∑ exp 2𝜋𝑘𝑥 − 𝑛𝜋 𝑛!𝑖𝑛 ∑ exp −2𝜋𝑘𝑥 + 𝑛𝜋 2 𝑖 2 𝑖 ℬ𝑛 (𝑥; 𝜆) = −𝛿𝑛 (𝑥; 𝜆) − 𝑥 + , 𝜆 (2𝜋𝑖𝑘 + log 𝜆)𝑛 (2𝜋𝑖𝑘 − log 𝜆)𝑛 𝑘=1

𝑘=1

so that, in view of the definition (4.1) and the elementary series identity ∞ ∑

(4.12)

𝑓 (𝑘) =

𝑘=1

ℓ ∑ ∞ ∑

𝑓 (ℓ𝑘 + 𝑗)

(ℓ ∈ ℕ) ,

𝑗=1 𝑘=0

we find the formula: (4.13)

𝑛!𝑖𝑛 𝜆−𝑥 ℬ𝑛 (𝑥; 𝜆) = − 𝛿𝑛 (𝑥; 𝜆) − (2𝜋𝑖ℓ)𝑛 ⎡ ( ) ℓ [( ∑ 2𝜋𝑖𝑗 − log 𝜆 𝑛𝜋 ) ] 2𝜋𝑖ℓ𝑥 ⎣ × Φ 𝑒 , 𝑛, exp 2𝜋𝑗𝑥 − 𝑖 2𝜋𝑖ℓ 2 𝑗=1

⎤ ( ) ) ] [ ( 2𝜋𝑖𝑗 + log 𝜆 𝑛𝜋 𝑖 ⎦. Φ 𝑒−2𝜋𝑖ℓ𝑥 , 𝑛, + exp − 2𝜋𝑗𝑥 + 2𝜋𝑖ℓ 2 𝑗=1 ℓ ∑

Setting 𝜆 = exp(2𝜋𝑖𝜉), 𝑥 = 𝑝𝑞 , ℓ = 𝑞 in (4.13), we then obtain the assertion of Theorem 4.1. This completes the proof. □


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If we make use of the equivalent of (2.9) as [ ∞ [( ) ] 2 ⋅ 𝑛!𝑖𝑛+1 ∑ exp 𝑛+1 2 𝜋 − (2𝑘 − 1)𝜋𝑥 𝑖 ℰ𝑛 (𝑥; 𝜆) = 𝑛+1 𝜆𝑥 [(2𝑘 − 1)𝜋𝑖 + log 𝜆] 𝑘=1 (4.14) [( ) ]] ∞ ∑ exp − 𝑛+1 2 𝜋 + (2𝑘 − 1)𝜋𝑥 𝑖 + 𝑛+1 [(2𝑘 − 1)𝜋𝑖 − log 𝜆] 𝑘=1 and the elementary series identity (4.12), by an analogous method, we provide that Theorem 4.2. For 𝑛, 𝑞 ∈ ℕ, 𝑝 ∈ ℤ, 𝜉 ∈ ℝ, ∣𝜉∣ < 1, the following formula of Apostol-Euler polynomials at rational arguments ( ) 𝑝 2𝜋𝑖𝜉 2 ⋅ 𝑛! ;𝑒 ℰ𝑛 = 𝑞 (2𝑞𝜋)𝑛+1 { 𝑞 ( ) [( ) ] ∑ 𝑛 + 1 (2𝑗 + 2𝜉 − 1)𝑝 2𝑗 + 2𝜉 − 1 − × 𝜁 𝑛 + 1, exp 𝜋𝑖 (4.15) 2𝑞 2 𝑞 𝑗=1 ( ) [( ) ]} 𝑞 ∑ 2𝑗 − 2𝜉 − 1 𝑛 + 1 (2𝑗 − 2𝜉 − 1)𝑝 + 𝜁 𝑛 + 1, + exp − 𝜋𝑖 2𝑞 2 𝑞 𝑗=1 holds true in terms of the Hurwitz zeta function. Upon the special cases of (4.11) and (4.15), for 𝜉 = 0, are respectively the following known results given earlier by Cvijović and Klinowski. Corollary 4.3 ([6, p. 1529, Theorem A]). For 𝑛 ∈ ℕ ∖ {1} , 𝑞 ∈ ℕ, 𝑝 ∈ ℤ, the following formula for the classical Bernoulli polynomials ( ( ) ) ( ) 𝑞 𝑝 2𝑗𝑝𝜋 𝑛𝜋 𝑗 2 ⋅ 𝑛! ∑ − 𝜁 𝑛, 𝐵𝑛 =− cos 𝑞 (2𝑞𝜋)𝑛 𝑗=1 𝑞 𝑞 2 holds true. Corollary 4.4 ([6, p. 1529, Theorem B]). For 𝑛, 𝑞 ∈ ℕ, 𝑝 ∈ ℤ, the following formula for the classical Euler polynomials ( ( ) ) ( ) 𝑞 𝑝 (2𝑗 − 1)𝑝𝜋 𝑛𝜋 2𝑗 − 1 4 ⋅ 𝑛! ∑ − 𝜁 𝑛 + 1, 𝐸𝑛 = sin 𝑞 (2𝑞𝜋)𝑛+1 𝑗=1 2𝑞 𝑞 2 holds true. Remark 4.5. We may also prove formula (4.15) by applying the relationship (2.18) and Theorem 4.1. Remark 4.6. In view of Remark 3.5, we say that 𝜉 is any real number in Theorem 4.1 and Theorem 4.2. Remark 4.7. Obviously, Srivastava’s formula (4.9) is an equivalent with our formula (4.11).

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5. Integral representations for the Bernoulli and Euler polynomials In this section we will see that Theorem 5.1 and Theorem 5.2 below involve the results of Cvijović or H. Haruki and T. M. Rassias. By (3.1) and (3.3) for 𝜉 = 0, it follows that we give the uniform integral representations for the classical Bernoulli and Euler polynomials, respectively. Theorem 5.1. For 𝑛 = 1, 2, . . . , 0 ≤ ℜ(𝑥) ≤ 1, we have ) ( ( ) ∫ ∞ − 𝑒−2𝜋𝑡 cos 𝑛𝜋 cos 2𝜋𝑥 − 𝑛𝜋 2 2 𝑡𝑛−1 d𝑡. (5.1) 𝐵𝑛 (𝑥) = −𝑛 cosh 2𝜋𝑡 − cos 2𝜋𝑥 0 Theorem 5.2. For 𝑛 = 1, 2, . . . , 0 ≤ ℜ(𝑥) ≤ 1, we have ) ( ( ∫ ∞ 𝜋𝑡 + 𝑒−𝜋𝑡 sin 𝜋𝑥 + 𝑒 sin 𝜋𝑥 − 𝑛𝜋 2 (5.2) 𝐸𝑛 (𝑥) = 2 cosh 2𝜋𝑡 − cos 2𝜋𝑥 0

𝑛𝜋 2

) 𝑡𝑛 d𝑡.

Remark 5.3. Theorem 5.1 and Theorem 5.2 above show the uniform integral representations for the classical Bernoulli and Euler polynomials which were never found in the classical literature, for example [1], [8] and [18], etc. So these uniform formulas are interesting in this subject. By (3.4) and (3.6) for 𝜉 = 0, we easily find the following additional integral representations for the classical Bernoulli and Euler polynomials, respectively. Theorem 5.4. For 𝑛 = 1, 2, . . . , 0 ≤ ℜ(𝑥) ≤ 1, we have ) ( ) ( ∫ 1 − 𝑡 cos 𝑛𝜋 cos 2𝜋𝑥 − 𝑛𝜋 𝑛 2𝑛 2 2 (log 𝑡)𝑛−1 d𝑡. (5.3) 𝐵𝑛 (𝑥) = (−1) (2𝜋)𝑛 0 𝑡2 − 2𝑡 cos 2𝜋𝑥 + 1 Theorem 5.5. For 𝑛 = 1, 2, . . . , 0 ≤ ℜ(𝑥) ≤ 1, we have ) ( ( ∫ 1 + 𝑡2 sin 𝜋𝑥 + sin 𝜋𝑥 − 𝑛𝜋 𝑛 4 2 (5.4) 𝐸𝑛 (𝑥) = (−1) 𝑛+1 𝜋 𝑡4 − 2𝑡2 cos 2𝜋𝑥 + 1 0

𝑛𝜋 2

) (log 𝑡)𝑛 d𝑡.

We see easily that Theorem 5.4 and Theorem 5.5 imply the main results of Cvijović [7, p. 170, Theorem 1] or H. Haruki and T. M. Rassias [10, p. 82, Theorem (ii)(iv)], i.e., (5.5) (5.6) (5.7) (5.8)

2(2𝑛) 𝐵2𝑛 (𝑥) = (−1) (2𝜋)2𝑛

∫

1

cos 2𝜋𝑥 − 𝑡 (log 𝑡)2𝑛−1 d𝑡, 2 − 2𝑡 cos 2𝜋𝑥 + 1 𝑡 0 ∫ 2(2𝑛 − 1) 1 sin 2𝜋𝑥 𝑛 𝐵2𝑛−1 (𝑥) = (−1) (log 𝑡)2𝑛−2 d𝑡, (2𝜋)2𝑛−1 0 𝑡2 − 2𝑡 cos 2𝜋𝑥 + 1 ∫ 1 4 (𝑡2 + 1) sin 𝜋𝑥 𝐸2𝑛 (𝑥) = (−1)𝑛 2𝑛+1 (log 𝑡)2𝑛 d𝑡, 4 2 𝜋 0 𝑡 − 2𝑡 cos 2𝜋𝑥 + 1 ∫ 1 4 (𝑡2 − 1) cos 𝜋𝑥 (log 𝑡)2𝑛−1 d𝑡. 𝐸2𝑛−1 (𝑥) = (−1)𝑛 2𝑛 4 2 𝜋 0 𝑡 − 2𝑡 cos 2𝜋𝑥 + 1 𝑛

On the other hand, Theorem 5.1 and Theorem 5.2 also imply the classical integral representations for the Bernoulli polynomials and Euler polynomials, respectively


(see [18, pp. 27, 31]): (5.9) (5.10) (5.11) (5.12)

∫

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∞

cos 2𝜋𝑥 − 𝑒−2𝜋𝑡 2𝑛−1 𝑡 d𝑡, cosh 2𝜋𝑡 − cos 2𝜋𝑥 0 ∫ ∞ sin 2𝜋𝑥 𝑡2𝑛−2 d𝑡, 𝐵2𝑛−1 (𝑥) = (−1)𝑛 (2𝑛 − 1) cosh 2𝜋𝑡 − cos 2𝜋𝑥 0 ∫ ∞ sin 𝜋𝑥 cosh 𝜋𝑡 𝑛 𝑡2𝑛 d𝑡, 𝐸2𝑛 (𝑥) = 4(−1) cosh 2𝜋𝑡 − cos 2𝜋𝑥 0 ∫ ∞ cos 𝜋𝑥 sinh 𝜋𝑡 𝑛 𝑡2𝑛−1 d𝑡. 𝐸2𝑛−1 (𝑥) = 4(−1) cosh 2𝜋𝑡 − cos 2𝜋𝑥 0

𝐵2𝑛 (𝑥) = (−1)𝑛+1 (2𝑛)

Remark 5.6. If we make an appropriate transformation 𝑢 = 𝑒−2𝜋𝑡 in (5.9) and (5.10), and make a suitable transformation 𝑢 = 𝑒−𝜋𝑡 in (5.11) and (5.12), respectively, then we can directly obtain (5.5)–(5.8); i.e., the main results of Cvijović or H. Haruki and T. M. Rassias are only a very simple transmogrification for the corresponding classical cases (5.9)–(5.12). Therefore, in view of this reason, we say that (5.5)–(5.8) are not new integral representations for the classical Bernoulli polynomials and Euler polynomials. 6. Further observations and consequences By formula (2.3) of Theorem 2.1 for 𝑥 = 0, we obtain the relationship between the Apostol-Bernoulli numbers and the Hurwitz zeta function as follows: ( [ ) ( )] (−1)𝑛−1 𝑛! log 𝜆 log 𝜆 𝑛 ℬ𝑛 (𝜆) = 𝜁 𝑛, 1 − (−1) (6.1) + 𝜁 𝑛, . (2𝜋𝑖)𝑛 2𝜋𝑖 2𝜋𝑖 Letting 𝜆 = 1, 𝑛 → 2𝑛 in (6.1), we at once produce the following famous Euler formula (see, e.g, [1, p. 807, 23.2.16]): (6.2)

𝜁(2𝑛) =

(−1)𝑛−1 (2𝜋)2𝑛 𝐵2𝑛 . 2(2𝑛)!

On the other hand, we define zeta functions as (6.3)

𝛽(𝑛; 𝜉) =

∞ ∑ 𝑘=0

(−1)𝑘 , (2𝑘 + 2𝜉 + 1)𝑛

𝛽(𝑛) =

∞ ∑ 𝑘=0

(−1)𝑘 (2𝑘 + 1)𝑛

(𝜉 ∈ ℝ).

Setting 𝜆 = 𝑒2𝜋𝑖𝜉 in (2.9) of Theorem 2.2, we have [ ∞ [( ) ] ∑ exp 𝑛+1 𝜋 − (2𝑘 + 1)𝜋𝑥 𝑖 2 ⋅ 𝑛! 2𝜋𝑖𝜉 2 ) = 𝑛+1 2𝜋𝑖𝜉𝑥 ℰ𝑛 (𝑥; 𝑒 𝜋 𝑒 (2𝑘 + 2𝜉 + 1)𝑛+1 𝑘=0 (6.4) [( ) ]] ∞ ∑ exp − 𝑛+1 𝜋 + (2𝑘 + 1)𝜋𝑥 𝑖 2 + . (2𝑘 − 2𝜉 + 1)𝑛+1 𝑘=0 ( ) Taking 𝑥 = 12 in (6.4) and noting that ℰ𝑛 (𝜆) = 2𝑛 ℰ𝑛 12 ; 𝜆 and (6.3), we readily obtain the following relationship between the Apostol-Euler numbers ℰ𝑛 (𝑒2𝜋𝑖𝜉 ) and the zeta function 𝛽(𝑛; 𝜉): (6.5)

ℰ𝑛 (𝑒2𝜋𝑖𝜉 ) =

2𝑛+1 𝑖𝑛 ⋅ 𝑛! [𝛽(𝑛 + 1; 𝜉) + (−1)𝑛 𝛽(𝑛 + 1; −𝜉)] 𝜋 𝑛+1 𝑒𝜋𝑖𝜉

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or (6.6)

𝛽(2𝑛 + 1; 𝜉) + 𝛽(2𝑛 + 1; −𝜉) =

(−1)𝑛 𝑒𝜋𝑖𝜉 ( 𝜋 )2𝑛+1 ℰ2𝑛 (𝑒2𝜋𝑖𝜉 ). (2𝑛)! 2

Further putting 𝜉 = 0 in (6.6), we arrive directly at the following well-known formula (see [1, p. 807, 23.2.22]): (6.7)

𝛽(2𝑛 + 1) =

∞ ∑ 𝑘=0

(−1)𝑘 (−1)𝑛 ( 𝜋 )2𝑛+1 = 𝐸2𝑛 . 2𝑛+1 (2𝑘 + 1) 2(2𝑛)! 2

We can also obtain the formulas (6.2) and (6.7) by (4.11) and (4.15), respectively. By Theorem 3.1, Theorem 3.2, Theorem 3.3 and Theorem 3.4, respectively, we easily find the following integral representations for the Apostol-Bernoulli and Apostol-Euler numbers: (6.8) ℬ𝑛 (𝑒2𝜋𝑖𝜉 ) = −Δ𝑛 (0; 𝜉) ( ) ( ) ∫ ∞ −2𝜋𝑡 −2𝜋𝑡 cos 𝑛𝜋 ) cosh(2𝜋𝜉𝑡) + 𝑖 sin 𝑛𝜋 ) sinh(2𝜋𝜉𝑡) 𝑛−1 2 (1 − 𝑒 2 (1 + 𝑒 𝑡 d𝑡, −𝑛 cosh 2𝜋𝑡 − 1 0 (6.9) 2𝑛 ℬ𝑛 (𝑒2𝜋𝑖𝜉 ) = −Δ𝑛 (0; 𝜉) + (−2𝜋)𝑛 ( 𝑛𝜋 ) ( ) ∫ 1 cos 𝑛𝜋 2 (1 − 𝑡) cosh(𝜉 log 𝑡) − 𝑖 sin 2 (1 + 𝑡) sinh(𝜉 log 𝑡) (log 𝑡)𝑛−1 d𝑡, × 𝑡2 − 2𝑡 + 1 0 (6.10) ℰ𝑛 (𝑒2𝜋𝑖𝜉 ) = 2𝑛+2 𝑒−𝜋𝑖𝜉 ( 𝑛𝜋 ) ( ) ∫ ∞ cos 𝑛𝜋 2 cosh 𝜋𝑡 cosh(2𝜋𝜉𝑡) + 𝑖 sin 2 cosh 𝜋𝑡 sinh(2𝜋𝜉𝑡) 𝑛 × 𝑡 d𝑡, cosh 2𝜋𝑡 + 1 0 (6.11) 2𝑛+2 𝑒−𝜋𝑖𝜉 ℰ𝑛 (𝑒2𝜋𝑖𝜉 ) = (−1)𝑛 𝑛+1 ( 𝑛𝜋 ) ( 𝜋) ∫ 1 cos 𝑛𝜋 2 cosh(2𝜉 log 𝑡) − 𝑖 sin 2 sinh(2𝜉 log 𝑡) (log 𝑡)𝑛 d𝑡. × 2+1 𝑡 0 Further setting 𝜉 = 0 in (6.8)–(6.11), respectively, we deduce the integral representations for the classical Bernoulli numbers and Euler numbers as follows (see, e.g., [18, pp. 28-32]): ( 𝑛𝜋 ) ∫ ∞ 𝑡𝑛−1 𝑒−𝜋𝑡 csch(𝜋𝑡) d𝑡 𝐵𝑛 = −𝑛 cos 2 0 ( ) ∫ 1 3𝑛𝜋 (log 𝑡)𝑛−1 2𝑛 d𝑡, = cos 2 (2𝜋)𝑛 0 1−𝑡 ∫ ( 𝑛𝜋 ) ∞ 𝐸𝑛 = 2𝑛+1 cos 𝑡𝑛 sech(𝜋𝑡) d𝑡 2 0 ( ) ∫ 3𝑛𝜋 2𝑛+2 1 (log 𝑡)𝑛 = cos d𝑡. 2 𝜋 𝑛+1 0 1 + 𝑡2


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Recently, Garg et al. [9] gave an extension of Apostol’s formula (4.7) as (6.12)

ℬ𝑛 (𝑎; 𝜆) = −𝑛Φ(𝜆, 1 − 𝑛, 𝑎)

(𝑛 ∈ ℕ, 𝜆 ∈ ℂ, ∣𝜆∣ ≤ 1, 𝑎 ∈ ℂ ∖ ℤ− 0 ).

By (1.4) and the binomial theorem, we have ∞ ∑ 𝑛=0

(6.13)

ℰ𝑛 (𝑎; 𝜆)

∞ ∑ 2𝑒𝑎𝑧 𝑧𝑛 = 𝑧 =2 (−𝜆)𝑘 𝑒(𝑘+𝑎)𝑧 𝑛! 𝜆𝑒 + 1 𝑘=0 [ ∞ ] ∞ 𝑛 ∑ ∑ 𝑘 𝑛 𝑧 = (−𝜆) (𝑘 + 𝑎) 2 𝑛! 𝑛=0 𝑘=0 [ ] ∞ ∞ ∑ ∑ (−𝜆)𝑘 𝑧𝑛 = . 2 (𝑘 + 𝑎)−𝑛 𝑛! 𝑛=0 𝑘=0

Hence, we also obtain the following interesting relationship between the ApostolEuler polynomials and the Hurwitz-Lerch zeta function: (6.14)

ℰ𝑛 (𝑎; 𝜆) = 2Φ(−𝜆, −𝑛, 𝑎)

(𝑛 ∈ ℕ, 𝜆 ∈ ℂ, ∣𝜆∣ ≤ 1, 𝑎 ∈ ℂ ∖ ℤ− 0 ).

We can prove Theorem 2.1 and Theorem 2.2, respectively, by applying the relationships (6.12) and (6.14) in conjunction with Lerch’s functional equation (4.8). The same as with the elementary series (4.12), we may also prove Theorem 4.1 and Theorem 4.2, respectively. References 1. M. Abramowitz, I. A. Stegun (Editors), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, Fourth Printing, Washington, D.C., 1965. MR757537 (85j:00005a) 2. T. M. Apostol, On the Lerch zeta function, Pacific J. Math. 1 (1951), 161–167. MR0043843 (13:328b) 3. L. Carlitz, Multiplication formulas for products of Bernoulli and Euler polynomials, Pacific J. Math., 9 (1959), 661–666. MR0108601 (21:7317) 4. M. Cenkci, M. Can, Some results on 𝑞-analogue of the Lerch zeta function, Adv. Stud. Contemp. Math., 12 (2006), 213–223. MR2213080 (2007c:11098) 5. J. Choi, P. J. Anderson, H. M. Srivastava, Some 𝑞-extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order 𝑛, and the multiple Hurwitz zeta function, Appl. Math. Comput. (2007), 199 (2008), 723–737. MR2420600 6. D. Cvijovi´ c, J. Klinowski, New formulae for the Bernoulli and Euler polynomials at rational arguments, Proc. Amer. Math. Soc., 123 (1995), 1527–1535. MR1283544 (95g:11085) 7. D. Cvijovi´ c, The Haruki-Rassias and related integral representations of the Bernoulli and Euler polynomials, J. Math. Anal. Appl., 337 (2008), 169–173. MR2356063 (2008i:33028) 8. A. Erd´ elyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher Transcendental Functions, Volumes I–III, McGraw-Hill, New York, 1953–1955. MR0698779 (84h:33001a); MR0698780 (84h:330016); MR0066496 (16:586c) 9. M. Garg, K. Jain, H. M. Srivastava, Some relationships between the generalized ApostolBernoulli polynomials and Hurwitz-Lerch zeta functions, Integral Transforms Spec. Funct., 17 (2006), no. 11, 803–815. MR2263956 (2007h:11028) 10. H. Haruki, T. M. Rassias, New integral representations for Bernoulli and Euler polynomials, J. Math. Anal. Appl., 175 (1993), 81–90. MR1216746 (94e:39016) 11. D. H. Lehmer, A new approach to Bernoulli polynomials, American Math. Monthly, 95 (1988), 905–911. MR979133 (90c:11014) 12. R. Lipschitz, Untersuchung der Eigenschaften einer Gattung von unendlichen Reihen, J. Reine und Angew. Math. CV (1889), 127–156. 13. Q.-M. Luo, H. M. Srivastava, Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials, J. Math. Anal. Appl., 308 (2005), 290–302. MR2142419 (2006e:33012)

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14. Q.-M. Luo, Apostol-Euler polynomials of higher order and Gaussian hypergeometric functions, Taiwanese J. Math. 10 (2006), 917–925. MR2229631 (2007c:33005) 15. Q.-M. Luo, H. M. Srivastava, Some relationships between the Apostol-Bernoulli and ApostolEuler polynomials, Comput. Math. Appl., 51 (2006), 631–642. MR2207447 (2006k:42050) 16. Q.-M. Luo, An explicit relationship between the generalized Apostol-Bernoulli and ApostolEuler polynomials associated with 𝜆−Stirling numbers of the second kind, Houston J. Math., accepted in press. 17. Q.-M. Luo, The multiplication formulas for the Apostol-Bernoulli and Apostol-Euler polynomials of higher order, Integral Transforms Spec. Funct., accepted in press. 18. W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Third Enlarged Edition, Springer-Verlag, New York, 1966. MR0232968 (38:1291) 19. P. C. Pasles, W. A. Pribitkin, A generalization of the Lipschitz summation formula and some applications, Proc. Amer. Math. Soc., 129 (2001), 3177–3184. MR1844990 (2002e:11059) 20. H. M. Srivastava, Some formulas for the Bernoulli and Euler polynomials at rational arguments, Math. Proc. Cambridge Philos. Soc. 129 (2000), 77–84. MR1757780 (2001f:11033) 21. H. M. Srivastava, J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, Boston and London, 2001. MR1849375 (2003a:11107) 22. W. Wang, C. Jia, T. Wang, Some results on Apostol-Bernoulli and Apostol-Euler polynomials, Comput. Math. Appl., 55 (2008), 1322–1332. MR2394371 Department of Mathematics, East China Normal University, Shanghai 200241, People’s Republic of China –and– Department of Mathematics, Jiaozuo University, Henan Jiaozuo 454003, People’s Republic of China E-mail address: [email protected] E-mail address: [email protected]