FOURIER EXPANSIONS AND INTEGRAL REPRESENTATIONS FOR THE APOSTOL-BERNOULLI ...

16 Pages·2009·0.24 MB·English

MATHEMATICSOFCOMPUTATION
Volume78,Number268,October2009,Pages2193–2208
S0025-5718(09)02230-3
ArticleelectronicallypublishedonJune12,2009
FOURIER EXPANSIONS AND INTEGRAL REPRESENTATIONS
FOR THE APOSTOL-BERNOULLI AND
APOSTOL-EULER POLYNOMIALS
QIU-MINGLUO
Abstract. We investigate Fourier expansions for the Apostol-Bernoulli and
Apostol-EulerpolynomialsusingtheLipschitzsummationformulaandobtain
their integral representations. We give some explicit formulas at rational ar-
guments for these polynomials in terms of the Hurwitz zeta function. We
also derive the integral representations for the classical Bernoulli and Euler
polynomialsandrelatedknownresults.
1. Introduction
Theclassical BernoullipolynomialsandEulerpolynomialsaredefinedbymeans
of the following generating functions (see [1, pp. 804-806] or [18, pp. 25-32])
𝑧𝑒𝑥𝑧 ∑∞ 𝑧𝑛
(1.1) = 𝐵 (𝑥) (∣𝑧∣<2𝜋)
𝑒𝑧 −1 𝑛 𝑛!
𝑛=0
and
2𝑒𝑥𝑧 ∑∞ 𝑧𝑛
(1.2) = 𝐸 (𝑥) (∣𝑧∣<𝜋),
𝑒𝑧+1 𝑛 𝑛!
𝑛=0
( )
respectively. Obviously, 𝐵 := 𝐵 (0), 𝐸 := 2𝑛𝐸 1 are the Bernoulli numbers
𝑛 𝑛 𝑛 𝑛 2
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:
ReceivedbytheeditorJune3,2008and,inrevisedform,September26,2008.
2000 MathematicsSubjectClassification. Primary11B68;Secondary42A16,11M35.
Keywordsandphrases. Lipschitzsummationformula,Fourierexpansion,integralrepresenta-
tion, Apostol-Bernoulli and Apostol-Euler polynomials and numbers, Bernoulli and Euler poly-
nomialsandnumbers,HurwitzZetafunction,Lerch’sfunctionalequation,rationalarguments.
Theauthorexpresseshissinceregratitudetotherefereeforvaluablesuggestionsandcomments.
The author thanks Professor Chi-Wang Shu who helped with the submission of this manuscript
totheWebsubmissionsystemoftheAMS..
The present investigation was supported in part by the PCSIRT Project of the Ministry of
Educationof China underGrant#IRT0621,InnovationProgramof Shanghai MunicipalEduca-
tion Committee of China under Grant #08ZZ24 and Henan Innovation Project For University
ProminentResearchTalents of China underGrant#2007KYCX0021.
⃝c2009AmericanMathematicalSociety
Reverts to public domain 28 years from publication
2193
2194 QIU-MINGLUO
Definition 1.1 (Apostol[2]; seealsoSrivastava[20]). TheApostol-Bernoullipoly-
nomials ℬ (𝑥;𝜆) in 𝑥 are defined by means of the generating function
𝑛
𝑧𝑒𝑥𝑧 ∑∞ 𝑧𝑛
(1.3) = ℬ (𝑥;𝜆)
𝜆𝑒𝑧−1 𝑛 𝑛!
𝑛=0
(∣𝑧∣<2𝜋 when 𝜆=1; ∣𝑧∣<∣log𝜆∣ when 𝜆∕=1)
with, of course,
𝐵 (𝑥)=ℬ (𝑥;1) and ℬ (𝜆):=ℬ (0;𝜆),
𝑛 𝑛 𝑛 𝑛
where ℬ (𝜆) denotes the so-called Apostol-Bernoulli numbers (in fact, it is a func-
𝑛
tion in 𝜆).
Recently, Luo and Srivastava introduced the Apostol-Euler polynomials as fol-
lows:
Definition 1.2 (Luo [14]; see also Luo and Srivastava [13]). The Apostol-Euler
polynomials ℰ (𝑥;𝜆) in 𝑥 are defined by means of the generating function
𝑛
2𝑒𝑥𝑧 ∑∞ 𝑧𝑛
(1.4) = ℰ (𝑥;𝜆) (∣𝑧∣<∣log(−𝜆)∣),
𝜆𝑒𝑧 +1 𝑛 𝑛!
𝑛=0
with, of course,
( )
1
𝐸 (𝑥)=ℰ (𝑥;1) and ℰ (𝜆):=2𝑛ℰ ;𝜆 ,
𝑛 𝑛 𝑛 𝑛 2
where ℰ (𝜆) denote the so-called Apostol-Euler numbers (in fact, it is a function
𝑛
in 𝜆).
Remark 1.3. In Definition 1.1 and Definition 1.2, the original constraints
∣𝑧+log𝜆∣<2𝜋 and ∣𝑧+log𝜆∣<𝜋, respectively, should be replaced by the condi-
tions ∣𝑧∣ < 2𝜋 when 𝜆 = 1; ∣𝑧∣ < ∣log𝜆∣ when 𝜆 ∕= 1 and ∣𝑧∣ < ∣log(−𝜆)∣ for the
referee’sclearanddetailedargumentation. Hence,thecorrespondingconstraintsin
References [13], [14], [15] and [20] should also be such.
TheApostol-BernoulliandApostol-Eulerpolynomialshavebeeninvestigatedby
many people (see, e.g., [2], [4], [5], [9], [13]–[17], [20] and [22]).
D.H.Lehmer[11]gaveanewapproachtoBernoullipolynomials,startingfroma
function equation (Rabbe’s multiplication theorem). H. Haruki and T. M. Rassias
[10]providedthenewintegralrepresentationsfortheBernoulliandEulerpolynomi-
alsaswellasusingasimilarfunctionequation. Recently,D.Cvijovi´c[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 Apostol-
Bernoulli and Apostol-Euler polynomials based on the Lipschitz summation for-
mula, and then provide their integral representations. We obtain some explicit
formulas for the Apostol-Bernoulli and Apostol-Euler polynomials at rational ar-
guments 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 Cvijovi´c or H. Haruki and T. M. Rassias are the
corresponding direct consequences of our formulas.
Thepaperisorganizedasfollows. Inthefirstsectionwerewritethedefinitionsof
Apostol-Bernoulli and Apostol-Euler polynomials. In the second section we derive
FOURIER EXPANSIONS AND INTEGRAL REPRESENTATIONS 2195
FourierexpansionsfortheApostol-BernoulliandApostol-Eulerpolynomials. Inthe
thirdsectionweshowtheirintegralrepresentations. Inthefourthsectionweobtain
theirexplicitformulasatrationalarguments intermsoftheHurwitzzetafunction.
In the fifth section we deduce the corresponding uniform integral representations
for the classical Bernoulli and Euler polynomials and related results of Cvijovi´c
or H. Haruki and T. M. Rassias. In the sixth section we give some applications
and remarks; for example, the classical Euler formula 𝜁(2𝑛) = (−1)𝑛−1(2𝜋)2𝑛𝐵 is
2(2𝑛)! 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) = ,
(𝑛+𝜇)1−𝛼 (−2𝜋𝑖)𝛼 (𝜏 +𝑘)𝛼
𝑛+𝜇>0 𝑘∈ℤ
where𝜇∈ℤandℜ(𝛼)>1or𝜇∈ℝ∖ℤandℜ(𝛼)>0; 𝜏 ∈𝐻 isthecomplexupper
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
𝑛! ∑′ 𝑒2𝜋𝑖𝑘𝑥
(2.2) ℬ (𝑥;𝜆)=−𝛿 (𝑥;𝜆)−
𝑛 𝑛 𝜆𝑥 (2𝜋𝑖𝑘−log𝜆)𝑛
[ [( ) ]
𝑛!𝑖𝑛 ∑∞ exp −2𝜋𝑘𝑥+ 𝑛𝜋 𝑖
(2.3) =−𝛿 (𝑥;𝜆)− 2
𝑛 𝜆𝑥 (2𝜋𝑖𝑘+log𝜆)𝑛
𝑘=1 [( ) ]]
∑∞ exp 2𝜋𝑘𝑥− 𝑛𝜋 𝑖
+ 2 ,
(2𝜋𝑖𝑘−log𝜆)𝑛
𝑘=1
∑
′
where the symbol denotes the standard convention of a sum over the integers
that omits 0; 𝛿 (𝑥;𝜆)=0 or (−1)𝑛𝑛! according as 𝜆=1 or 𝜆∕=1, respectively.
𝑛 𝜆𝑥log𝑛𝜆
Proof. For 0≤𝑥≤1, by (1.3) and the generalized binomial theorem, we have
∑∞ (2𝜋𝑖𝜏)𝑘−1 𝑒2𝜋𝑖𝜏𝑥 ∑∞
(2.4) ℬ (𝑥;𝜆) = =− 𝜆𝑘𝑒2𝜋𝑖(𝑘+𝑥)𝜏
𝑘 𝑘! 𝜆𝑒2𝜋𝑖𝜏 −1
( 𝑘=0 𝑘=0 )
∣log𝜆∣ log∣𝜆∣
∣𝜏∣<1 when 𝜆=1; ∣𝜏∣< when 𝜆∕=1; ℑ𝜏 > .
2𝜋 2𝜋
We differentiate both sides of (2.4) with respect to the variable 𝜏, by 𝑛−1 times
and noting that ℬ (𝑥;𝜆)=𝛿 (see [13, p. 301]). Then we get
0 1,𝜆
∑∞ (2𝜋𝑖)𝑘−1𝜏𝑘−𝑛 (−1)𝑛−1(𝑛−1)!
(2.5) ℬ (𝑥;𝜆) + 𝛿
𝑘 𝑘(𝑘−𝑛)! 2𝜋𝑖𝜏𝑛 1,𝜆
𝑘=𝑛
∑∞
=−(2𝜋𝑖)𝑛−1 𝜆𝑘(𝑘+𝑥)𝑛−1𝑒2𝜋𝑖(𝑘+𝑥)𝜏,
𝑘=0
where 𝛿 is the Kronecker symbol.
1,𝜆
2196 QIU-MINGLUO
On the other hand, letting 𝛼=𝑛,𝜇(cid:13)→𝑥,𝜏 (cid:13)→𝜏 + log𝜆 in (2.1), we find that
2𝜋𝑖
∑ 𝑒−2𝜋𝑖𝑘𝑥 ∑∞
(2.6) (−1)𝑛(𝑛−1)! = 𝜆𝑘+𝑥(𝑘+𝑥)𝑛−1𝑒2𝜋𝑖(𝑘+𝑥)𝜏.
[2𝜋𝑖(𝜏 +𝑘)+log𝜆]𝑛
𝑘∈ℤ 𝑘=0
Combining (2.5) and (2.6), we obtain
∑∞ (2𝜋𝑖)𝑘−1𝜏𝑘−𝑛 (−1)𝑛−1(𝑛−1)!
𝜆𝑥 ℬ (𝑥;𝜆) +𝜆𝑥 𝛿
𝑘 𝑘(𝑘−𝑛)! 2𝜋𝑖𝜏𝑛 1,𝜆
𝑘=𝑛
∑ 𝑒−2𝜋𝑖𝑘𝑥
=(−1)𝑛−1(𝑛−1)!(2𝜋𝑖)𝑛−1 .
[2𝜋𝑖(𝜏 +𝑘)+log𝜆]𝑛
𝑘∈ℤ
Separating this 𝑘 =0 term in the above sum on the right side yields that
∑∞ (2𝜋𝑖)𝑘−1𝜏𝑘−𝑛
(2.7) 𝜆𝑥 ℬ (𝑥;𝜆) =(−1)𝑛−1(𝑛−1)!(2𝜋𝑖)𝑛−1
𝑘 𝑘(𝑘−𝑛)!
𝑘=𝑛
∑′ 𝑒−2𝜋𝑖𝑘𝑥 (−1)𝑛−1(𝑛−1)!(2𝜋𝑖)𝑛−1
× + (1−𝛿 ).
[2𝜋𝑖(𝜏 +𝑘)+log𝜆]𝑛 (2𝜋𝑖𝜏 +log𝜆)𝑛 1,𝜆
Letting 𝜏 →0 in (2.7) we are led at once to the assertion (2.2) of Theorem 2.1.
Noting that 𝑖𝑛 = 𝑒𝑛𝜋𝑖, (−1)𝑛 = 𝑒−𝑛𝜋𝑖 and via a simple calculation, then the
2
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⋅𝑛! ∑ 𝑒(2𝑘−1)𝜋𝑖𝑥
(2.8) ℰ (𝑥;𝜆)=
𝑛 𝜆𝑥 [(2𝑘−1)𝜋𝑖−log𝜆]𝑛+1
𝑘∈ℤ[ [( ) ]
2⋅𝑛!𝑖𝑛+1 ∑∞ exp 𝑛+1𝜋−(2𝑘+1)𝜋𝑥 𝑖
(2.9) = 2
𝜆𝑥 [(2𝑘+1)𝜋𝑖+log𝜆]𝑛+1
𝑘=0 [( ) ]]
∑∞ exp −𝑛+1𝜋+(2𝑘+1)𝜋𝑥 𝑖
+ 2 .
[(2𝑘+1)𝜋𝑖−log𝜆]𝑛+1
𝑘=0
By Theorem 2.1 and Theorem 2.2, we can deduce respectively the Fourier ex-
pansions 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
𝑛! ∑′ 𝑒2𝜋𝑖𝑘𝑥
(2.10) 𝐵 (𝑥)=−
𝑛 (2𝜋𝑖)𝑛 𝑘𝑛
( )
2⋅𝑛! ∑∞ cos 2𝜋𝑘𝑥− 𝑛𝜋
(2.11) =− 2 .
(2𝜋)𝑛 𝑘𝑛
𝑘=1
FOURIER EXPANSIONS AND INTEGRAL REPRESENTATIONS 2197
Corollary 2.4. For 𝑛=0, 0<𝑥<1 and 𝑛>0, 0≤𝑥≤1, we have
2⋅𝑛! ∑ 𝑒(2𝑘−1)𝜋𝑖𝑥
(2.12) 𝐸 (𝑥)=
𝑛 (𝜋𝑖)𝑛+1 (2𝑘−1)𝑛+1
𝑘∈ℤ
[ ]
4⋅𝑛! ∑∞ sin (2𝑘+1)𝜋𝑥− 𝑛𝜋
(2.13) = 2 .
𝜋𝑛+1 (2𝑘+1)𝑛+1
𝑘=0
Remark 2.5. Replacing𝜏 by𝜏+log𝜆+1 in(2.1)andapplyingℰ (𝑥;𝜆)= 2 (see,
2𝜋𝑖 2 0 𝜆+1
for details, [13]–[15]) when we prove the assertion (2.8) of Theorem 2.2.
Remark 2.6. We define the 𝑛-th Apostol-Bernoulli function as
(2.14) ℬˆ (𝑥;𝜆):=ℬ (𝑥;𝜆) (0≤𝑥<1), ℬˆ (𝑥+1;𝜆)=𝜆−1ℬˆ (𝑥;𝜆),
𝑛 𝑛 𝑛 𝑛
which is also called the quasi-periodicity Apostol-Bernoulli polynomials. For any
𝑥∈ℝ,𝑟 ∈ℤ, we have
(2.15) ℬˆ (𝑥;𝜆)=𝜆−[𝑥]ℬ ({𝑥};𝜆), ℬˆ (𝑥+𝑟;𝜆)=𝜆−𝑟ℬˆ (𝑥;𝜆).
𝑛 𝑛 𝑛 𝑛
Herethenotation{𝑥}denotesthefractionalpartof𝑥,andthenotation[𝑥]denotes
the greatest integer not exceeding 𝑥.
Clearly, the Apostol-Bernoulli polynomials ℬ (𝑥;𝜆) (0 ≤ 𝑥 < 1) are the quasi-
𝑛
periodicity functions in 𝑥 with period 1. One of the special cases of the quasi-
periodicity Apostol-Bernoulli polynomials is just Carlitz’s periodic Bernoulli func-
tion [3, p. 661] for 𝜆=1.
Remark 2.7. We define the 𝑛-th Apostol-Euler function as
(2.16) ℰˆ (𝑥;𝜆):=ℰ (𝑥;𝜆) (0≤𝑥<1), ℰˆ (𝑥+1;𝜆)=−𝜆−1ℰˆ (𝑥;𝜆),
𝑛 𝑛 𝑛 𝑛
which is called the quasi-periodicity Apostol-Euler polynomials. For any 𝑥∈ℝ,𝑟 ∈
ℤ, we have
(2.17) ℰˆ (𝑥;𝜆)=(−1)[𝑥]𝜆−[𝑥]ℰ ({𝑥};𝜆), ℰˆ (𝑥+𝑟;𝜆)=(−1)𝑟𝜆−𝑟ℰˆ (𝑥;𝜆).
𝑛 𝑛 𝑛 𝑛
Obviously, the Apostol-Euler polynomials ℰ (𝑥;𝜆) (0 ≤ 𝑥 < 1) are the quasi-
𝑛
periodicity functions in 𝑥 with period 1. One of the special cases of the quasi-
periodicity Apostol-Euler polynomials is just Carlitz’s periodic Euler function
[3, p. 661] for 𝜆=1.
2198 QIU-MINGLUO
Remark 2.8. We employ a useful relationship [15, p. 636, Eq. (38)]
[ ( )]
2 𝑥
(2.18) ℰ (𝑥;𝜆)= ℬ (𝑥;𝜆)−2𝑛+1ℬ ;𝜆2
𝑛 𝑛+1 𝑛+1 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
log𝜆,i.e.,log𝜆=log∣𝜆∣+𝑖arg𝜆(−𝜋 <arg𝜆≤𝜋)when𝜆∕=1;Wechooselog1=0
when 𝜆=1.
3. Integral representations for the Apostol-Bernoulli and
Apostol-Euler polynomials
InthissectionwegivetheintegralrepresentationsfortheApostol-Bernoulliand
Apostol-EulerpolynomialswiththeirFourierexpansions. Forconvenience, wetake
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) ℬ (𝑥;𝑒2𝜋𝑖𝜉)=−Δ (𝑥;𝜉)
𝑛 𝑛
∫
∞ 𝑈(𝑛;𝑥,𝑡)cosh(2𝜋𝜉𝑡)+𝑖𝑉(𝑛;𝑥,𝑡)sinh(2𝜋𝜉𝑡)
−𝑛𝑒−2𝜋𝑖𝑥𝜉 𝑡𝑛−1d𝑡,
cosh2𝜋𝑡−cos2𝜋𝑥
0
(−1)𝑛𝑛!
where Δ (𝑥;𝜉)=0 or according as 𝜉 =0 or 𝜉 ∕=0, respectively, and
𝑛 𝑒2𝜋𝑖𝑥𝜉(2𝜋𝑖𝜉)𝑛
[ ( ) ( ) ]
𝑛𝜋 𝑛𝜋
𝑈(𝑛;𝑥,𝑡)= cos 2𝜋𝑥− −cos 𝑒−2𝜋𝑡 ,
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 (cid:13)→−𝑘 yields
𝑛!𝑒−2𝜋𝑖𝑥𝜉 ∑′ 𝑒−2𝜋𝑖𝑘𝑥
ℬ (𝑥;𝑒2𝜋𝑖𝜉)=−Δ (𝑥;𝜉)− .
𝑛 𝑛 (−2𝜋𝑖)𝑛 (𝑘+𝜉)𝑛
Using the known integral formula
∫
∞ 𝑛!
(3.2) 𝑡𝑛𝑒−𝑎𝑡d𝑡= (𝑛=0,1,...; ℜ(𝑎)>0),
𝑎𝑛+1
0
( )
and noting that −1 𝑛 =𝑒𝑛𝜋𝑖 and (−1)𝑛 =𝑒−𝑛𝜋𝑖, then we have
𝑖 2
FOURIER EXPANSIONS AND INTEGRAL REPRESENTATIONS 2199
{
∫
𝑛𝑒−2𝜋𝑖𝑥𝜉 ∑∞ ∞
ℬ (𝑥;𝑒2𝜋𝑖𝜉) =−Δ (𝑥;𝜉)− 𝑒−2𝜋𝑖𝑘𝑥 𝑡𝑛−1𝑒−(𝑘+𝜉)𝑡d𝑡
𝑛 𝑛 (−2𝜋𝑖)𝑛
𝑘=1 0 }
∫
∑∞ ∞
+(−1)𝑛 𝑒2𝜋𝑖𝑘𝑥 𝑡𝑛−1𝑒−(𝑘−𝜉)𝑡d𝑡
{ 𝑘=1 0
∫
𝑛𝑒−2𝜋𝑖𝑥𝜉 ∞ ∑∞
=−Δ (𝑥;𝜉)− 𝑒−𝜉𝑡𝑡𝑛−1 𝑒−(2𝜋𝑖𝑥+𝑡)𝑘d𝑡
𝑛 (−2𝜋𝑖)𝑛
0 𝑘=1 }
∫
∞ ∑∞
+(−1)𝑛 𝑒𝜉𝑡𝑡𝑛−1 𝑒(2𝜋𝑖𝑥−𝑡)𝑘d𝑡
{ 0 𝑘=1
∫
𝑛𝑒−2𝜋𝑖𝑥𝜉 ∞ 𝑒−2𝜋𝑖𝑥
=−Δ (𝑥;𝜉)− 𝑒−𝜉𝑡𝑡𝑛−1d𝑡
𝑛 (−2𝜋𝑖)𝑛 𝑒𝑡−𝑒−2𝜋𝑖𝑥
0
}
∫
∞ 𝑒2𝜋𝑖𝑥
+(−1)𝑛 𝑒𝜉𝑡𝑡𝑛−1d𝑡
𝑒𝑡−𝑒2𝜋𝑖𝑥
0
{
∫
𝑛𝑒−2𝜋𝑖𝑥𝜉 ∞ 𝑒𝑛𝜋𝑖(𝑒−2𝜋𝑖𝑥−𝑒−𝑡)
=−Δ (𝑥;𝜉)− 2 𝑒−𝜉𝑡𝑡𝑛−1d𝑡
𝑛 2(2𝜋)𝑛 cosh𝑡−cos2𝜋𝑥
0
}
∫
∞ 𝑒−𝑛𝜋𝑖(𝑒2𝜋𝑖𝑥−𝑒−𝑡)
+ 2 𝑒𝜉𝑡𝑡𝑛−1d𝑡 .
cosh𝑡−cos2𝜋𝑥
0
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 poly-
nomials by a similar method.
Theorem 3.2. For 𝑛=1,2,..., 0≤ℜ(𝑥)≤1, ∣𝜉∣< 1, 𝜉 ∈ℝ, we have
2
(3.3) ℰ (𝑥;𝑒2𝜋𝑖𝜉)=2𝑒−2𝜋𝑖𝑥𝜉
𝑛
∫
∞ 𝑋(𝑛;𝑥,𝑡)cosh(2𝜋𝜉𝑡)+𝑖𝑌(𝑛;𝑥,𝑡)sinh(2𝜋𝜉𝑡)
× 𝑡𝑛d𝑡,
cosh2𝜋𝑡−cos2𝜋𝑥
0
where
[ ( ) ( )]
𝑛𝜋 𝑛𝜋
𝑋(𝑛;𝑥,𝑡)= 𝑒−𝜋𝑡sin 𝜋𝑥+ +𝑒𝜋𝑡sin 𝜋𝑥− ,
2 2
[ ( ) ( )]
𝑛𝜋 𝑛𝜋
𝑌(𝑛;𝑥,𝑡)= 𝑒−𝜋𝑡cos 𝜋𝑥+ −𝑒𝜋𝑡cos 𝜋𝑥− .
2 2
On the other hand, we can also arrive at the following different integral repre-
sentations 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
2𝑛𝑒−2𝜋𝑖𝑥𝜉
(3.4) ℬ (𝑥;𝑒2𝜋𝑖𝜉)=−Δ (𝑥;𝜉)+
𝑛 𝑛 (−2𝜋)𝑛
∫
1 𝑈′(𝑛;𝑥,𝑡)cosh(𝜉log𝑡)−𝑖𝑉′(𝑛;𝑥,𝑡)sinh(𝜉log𝑡)
× (log𝑡)𝑛−1d𝑡,
𝑡2−2𝑡cos2𝜋𝑥+1
0
2200 QIU-MINGLUO
(−1)𝑛𝑛!
where Δ (𝑥;𝜉)=0 or according as 𝜉 =0 or 𝜉 ∕=0, respectively, and
𝑛 𝑒2𝜋𝑖𝑥𝜉(2𝜋𝑖𝜉)𝑛
[ ( ) ( )]
𝑛𝜋 𝑛𝜋
𝑈′(𝑛;𝑥,𝑡)= cos 2𝜋𝑥− −𝑡cos ,
2 2
[ ( ) ( )]
𝑛𝜋 𝑛𝜋
𝑉′(𝑛;𝑥,𝑡)= sin 2𝜋𝑥− +𝑡sin .
2 2
Proof. First we substitute cosh2𝜋𝑡= 𝑒2𝜋𝑡+𝑒−2𝜋𝑡 into (3.1). Then we see that
2
(3.5) ℬ (𝑥;𝑒2𝜋𝑖𝜉)=−Δ (𝑥;𝜉)−2𝑛𝑒−2𝜋𝑖𝑥𝜉
𝑛 𝑛
∫
∞ 𝑈(𝑛;𝑥,𝑡)cosh(2𝜋𝜉𝑡)+𝑖𝑉(𝑛;𝑥,𝑡)sinh(2𝜋𝜉𝑡)
× 𝑡𝑛−1d𝑡.
𝑒2𝜋𝑡+𝑒−2𝜋𝑡−2cos2𝜋𝑥
0
Thenmakingthetransformation𝑢=𝑒−2𝜋𝑡 in(3.5), weeasilyobtainformula(3.4).
This completes the proof. □
Similarly, we obtain
Theorem 3.4. For 𝑛=1,2,..., 0≤ℜ(𝑥)≤1, ∣𝜉∣< 1, 𝜉 ∈ℝ, we have
2
4𝑒−2𝜋𝑖𝑥𝜉
(3.6) ℰ (𝑥;𝑒2𝜋𝑖𝜉)=(−1)𝑛
𝑛 𝜋𝑛+1
∫
1 𝑋′(𝑛;𝑥,𝑡)cosh(2𝜉log𝑡)−𝑖 𝑌′(𝑛;𝑥,𝑡)sinh(2𝜉log𝑡)
× (log𝑡)𝑛d𝑡,
𝑡4−2𝑡2cos2𝜋𝑥+1
0
where
[ ( ) ( )]
𝑛𝜋 𝑛𝜋
𝑋′(𝑛;𝑥,𝑡)= 𝑡2sin 𝜋𝑥+ +sin 𝜋𝑥− ,
2 2
[ ( ) ( )]
𝑛𝜋 𝑛𝜋
𝑌′(𝑛;𝑥,𝑡)= 𝑡2cos 𝜋𝑥+ −cos 𝜋𝑥− .
2 2
Remark 3.5. For any integers ℓ, we see easily that ℬ (𝑥;𝑒2𝜋𝑖(ℓ+𝜉)) = ℬ (𝑥;𝑒2𝜋𝑖𝜉),
𝑛 𝑛
ℰ (𝑥;𝑒2𝜋𝑖(ℓ+𝜉)) = ℰ (𝑥;𝑒2𝜋𝑖𝜉). Therefore, the Apostol-Bernoulli polynomials
𝑛 𝑛
ℬ (𝑥;𝑒2𝜋𝑖𝜉)andtheApostol-Eulerpolynomialsℰ (𝑥;𝑒2𝜋𝑖𝜉)aretheperiodicityfunc-
𝑛 𝑛
tions 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. WecanalsoproveTheorem2.1andTheorem2.2byTheorem3.1and
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 Cvijovi´c and Klinowski are the corresponding special cases of our for-
mulas.
The Hurwitz-Lerch zeta function Φ(𝑧,𝑠,𝑎) defined by (cf., e.g., [21, p. 121,
et seq.])
∑∞ 𝑧𝑛
(4.1) Φ(𝑧,𝑠,𝑎):=
(𝑛+𝑎)𝑠
( 𝑛=0 )
𝑎∈ℂ∖ℤ−; 𝑠∈ℂ when ∣𝑧∣<1; ℜ(𝑠)>1 when ∣𝑧∣=1
0
FOURIER EXPANSIONS AND INTEGRAL REPRESENTATIONS 2201
contains, as its special cases, not only the Riemann and Hurwitz zeta functions
( )
1 1 ∑∞ 1
(4.2) 𝜁(𝑠):=Φ(1,𝑠,1)=𝜁(𝑠,1)= 𝜁 𝑠, =
2𝑠−1 2 𝑛𝑠
𝑛=1
and
∑∞ 1 ( )
(4.3) 𝜁(𝑠,𝑎):=Φ(1,𝑠,𝑎)= ℜ(𝑠)>1; 𝑎∈/ ℤ−
(𝑛+𝑎)𝑠 0
𝑛=0
and the Lerch zeta function (or periodic zeta function)
∑∞ 𝑒2𝑛𝜋𝑖𝜉 ( )
(4.4) 𝑙 (𝜉):= =𝑒2𝜋𝑖𝜉 Φ 𝑒2𝜋𝑖𝜉,𝑠,1
𝑠 𝑛𝑠
𝑛=1
(𝜉 ∈ℝ; ℜ(𝑠)>1),
but also such other functions as the polylogarithmic function
∑∞ 𝑧𝑛
(4.5) Li (𝑧):= =𝑧 Φ(𝑧,𝑠,1)
𝑠 𝑛𝑠
( 𝑛=1 )
𝑠∈ℂ when ∣𝑧∣<1; ℜ(𝑠)>1 when ∣𝑧∣=1
and the Lipschitz-Lerch zeta function (cf. [21, p. 122, Eq. 2.5 (11)])
∑∞ 𝑒2𝑛𝜋𝑖𝜉 ( )
(4.6) 𝜙(𝜉,𝑎,𝑠):= =Φ 𝑒2𝜋𝑖𝜉,𝑠,𝑎 =:𝐿(𝜉,𝑠,𝑎)
(𝑛+𝑎)𝑠
( 𝑛=0 )
𝑎∈ℂ∖ℤ−; ℜ(𝑠)>0 when 𝜉 ∈ℝ∖ℤ; ℜ(𝑠)>1 when 𝜉 ∈ℤ ,
0
which was first studied by Rudolf Lipschitz (1832-1903) and Matya´ˇs Lerch (1860-
1922) in connection with Dirichlet’s famous theorem on primes in arithmetic pro-
gressions.
Recently, H. M. Srivastava made use of Apostol’s formula (see [2, p. 164])
( )
ℬ 𝑎;𝑒2𝜋𝑖𝜉
(4.7) 𝜙(𝜉,𝑎,1−𝑛)=Φ(𝑒2𝜋𝑖𝜉,1−𝑛,𝑎)=− 𝑛 (𝑛∈ℕ)
𝑛
and Lerch’s functional equation (see [2, p. 161, (1.4)])
(4.8)
{ [( ) ]
Γ(𝑠) 1
𝜙(𝜉,𝑎,1−𝑠)= exp 𝑠−2𝑎𝜉 𝜋𝑖 𝜙(−𝑎,𝜉,𝑠)
(2𝜋)𝑠 2
[( ) ] }
1
+exp − 𝑠+2𝑎(1−𝜉) 𝜋𝑖 𝜙(𝑎,1−𝜉,𝑠)
2
(𝑠∈ℂ; 0<𝜉 <1)
2202 QIU-MINGLUO
to derive the following formula of Apostol-Bernoulli polynomials at rational argu-
ments (see [20, p. 84, Eq. (4.6)]):
(4.9)
( ) { ( ) [( ) ]
𝑝 𝑛! ∑𝑞 𝜉+𝑗−1 𝑛 2(𝜉+𝑗−1)𝑝
ℬ ;𝑒2𝜋𝑖𝜉 =− 𝜁 𝑛, exp − 𝜋𝑖
𝑛 𝑞 (2𝑞𝜋)𝑛 𝑞 2 𝑞
𝑗=1
( ) [( ) ]}
∑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 Apostol-
Bernoulli 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)
( ) ( ) { ( ) [( ) ]
𝑝 𝑝 𝑛! ∑𝑞 𝑗+𝜉 𝑛 2(𝑗+𝜉)𝑝
ℬ ;𝑒2𝜋𝑖𝜉 =−Δ ;𝜉 − 𝜁 𝑛, exp − 𝜋𝑖
𝑛 𝑞 𝑛 𝑞 (2𝜋𝑞)𝑛 𝑞 2 𝑞
𝑗=1
( ) [( ) ]}
∑𝑞 𝑗−𝜉 𝑛 2(𝑗−𝜉)𝑝
+ 𝜁 𝑛, exp − + 𝜋𝑖
𝑞 2 𝑞
𝑗=1
(−1)𝑛𝑛!
holdstrueintermsoftheHurwitzzetafunction,whereΔ (𝑥;𝜉)=0or
𝑛 𝑒2𝜋𝑖𝑥𝜉(2𝜋𝑖𝜉)𝑛
according as 𝜉 =0 or 𝜉 ∕=0, respectively.
Proof. We employ formula (2.3),
[ [( ) ] [( ) ]]
𝑛!𝑖𝑛 ∑∞ exp −2𝜋𝑘𝑥+ 𝑛𝜋 𝑖 ∑∞ exp 2𝜋𝑘𝑥− 𝑛𝜋 𝑖
ℬ (𝑥;𝜆)=−𝛿 (𝑥;𝜆)− 2 + 2 ,
𝑛 𝑛 𝜆𝑥 (2𝜋𝑖𝑘+log𝜆)𝑛 (2𝜋𝑖𝑘−log𝜆)𝑛
𝑘=1 𝑘=1
so that, in view of the definition (4.1) and the elementary series identity
∑∞ ∑ℓ ∑∞
(4.12) 𝑓(𝑘)= 𝑓(ℓ𝑘+𝑗) (ℓ∈ℕ),
𝑘=1 𝑗=1𝑘=0
we find the formula:
(4.13)
𝑛!𝑖𝑛𝜆−𝑥
ℬ (𝑥;𝜆)=−𝛿 (𝑥;𝜆)−
𝑛 𝑛 (2𝜋𝑖ℓ)𝑛
⎡
∑ℓ ( 2𝜋𝑖𝑗−log𝜆) [( 𝑛𝜋) ]
×⎣ Φ 𝑒2𝜋𝑖ℓ𝑥,𝑛, exp 2𝜋𝑗𝑥− 𝑖
2𝜋𝑖ℓ 2
𝑗=1
⎤
∑ℓ ( 2𝜋𝑖𝑗+log𝜆) [ ( 𝑛𝜋) ]
+ Φ 𝑒−2𝜋𝑖ℓ𝑥,𝑛, exp − 2𝜋𝑗𝑥+ 𝑖 ⎦.
2𝜋𝑖ℓ 2
𝑗=1
Setting 𝜆 = exp(2𝜋𝑖𝜉), 𝑥 = 𝑝, ℓ = 𝑞 in (4.13), we then obtain the assertion of
𝑞
Theorem 4.1. This completes the proof. □