Recurrence formulae for Apostol Bernoulli and Apostol Euler polynomials

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R E S E A R C H

Open Access

Recurrence formulae for Apostol-Bernoulli

and Apostol-Euler polynomials

Yuan He

*

_{and Chunping Wang}

*_{Correspondence:}

hyyhe@yahoo.com.cn Faculty of Science, Kunming University of Science and Technology, Kunming, Yunnan 650500, People’s Republic of China

Abstract

In this paper, using generating functions and combinatorial techniques, we extend Agoh and Dilcher’s quadratic recurrence formula for Bernoulli numbers in (Agoh and Dilcher in J. Number Theory 124:105-122, 2007) to Apostol-Bernoulli and

Apostol-Euler polynomials and numbers.

MSC: 11B68; 05A19

Keywords: Apostol-Bernoulli polynomials and numbers; Apostol-Euler polynomials and numbers; recurrence formula

1 Introduction

The classical Bernoulli polynomialsBn(x) and Euler polynomialsEn(x) have played

impor-tant roles in many branches of mathematics such as number theory, combinatorics, special functions and analysis, and they are usually deﬁned by means of the following generating functions:

text et_{– }=

∞

n= Bn(x)

tn n!

|t|< π, e

xt

et_{+ }=

∞

n= En(x)

tn n!

|t|<π. (.)

In particular,Bn=Bn() andEn= nEn(/) are called the classical Bernoulli numbers and

Euler numbers, respectively. Numerous interesting properties for these polynomials and numbers have been explored; see, for example, [–].

Recently, using some relationships involving Bernoulli numbers, Agoh and Dilcher [] extended Euler’s well-known quadratic recurrence formula on the classical Bernoulli num-bers

(B+B)n= n

i=

n i

BiBn–i= –nBn–– (n– )Bn (n≥) (.)

to obtain an explicit expression for (Bk+Bm)nwith arbitrary non-negative integersk,m, nandkandmnot both zero as follows:

(Bk+Bm)n= n

i=

n i

Bk+iBm+n–i

= –k!m!(n+δ(k,m)(m+k+ )) (m+k+ )! Bm+n+k

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+

m+k

i=

(–)i Bm+k+–i

m+k+  –i

(–)k

k+ 

i

k+  –i k+  n–

im k+ 

+ (–)m

m+ 

i

m+  –i m+  n–

ik m+ 

Bn+i–, (.)

whereδ(k,m) =  whenk=  orm= , andδ(k,m) =  otherwise. As further applications, Agoh and Dilcher [] derived some new types of recurrence formulae on the classical Bernoulli numbers and showed that the values ofBndepend only onBn,Bn+, . . . ,Bn

for a positive integernand, similarly, forBn+andBn+.

In this paper, using generating functions and combinatorial techniques, we extend the above mentioned Agoh and Dilcher quadratic recurrence formula for Bernoulli numbers to Apostol-Bernoulli and Apostol-Euler polynomials and numbers. These results also lead to some known ones related to the formulae on products of the classical Bernoulli and Euler polynomials and numbers stated in Nielsen’s classical book [].

2 Preliminaries and known results

We ﬁrst recall the Apostol-Bernoulli polynomials which were introduced by Apostol [] (see also Srivastava [] for a systematic study) in order to evaluate the value of the Hurwitz-Lerch zeta function. For simplicity, we here start with the Apostol-Bernoulli polynomials B(α)

n (x;λ) of (real or complex) orderαgiven by Luo and Srivastava []

t

λet_{– } α

ext=

∞

n=

B(α) n (x;λ)

tn n!

|t+logλ|< π; α:= . (.)

Especially the caseα=  in (.) gives the Apostol-Bernoulli polynomials which are de-noted byBn(x;λ) =B()n (x;λ). Moreover,Bn(λ) =Bn(;λ) are called the Apostol-Bernoulli

numbers. Further, Luo [] introduced the Apostol-Euler polynomialsEn(α)(x;λ) of orderα:



λet_{+ } α

ext=

∞

n=

E(α) n (x;λ)

tn n!

|t+logλ|<π; α:= . (.)

The Apostol-Euler polynomialsEn(x;λ) and Apostol-Euler numbersEn(λ) are given by

En(x;λ) =En()(x;λ) andEn(λ) = nEn(/;λ), respectively. Obviously,Bn(x;λ) andEn(x;λ)

reduce toBn(x) andEn(x) whenλ= . Several interesting properties for Apostol-Bernoulli

and Apostol-Euler polynomials and numbers have been presented in [–]. Next we give some basic properties for Apostol-Bernoulli and Apostol-Euler polynomials of orderαas stated in [, ].

Proposition . Diﬀerential relations of the Apostol-Bernoulli and Apostol-Euler polyno-mials of orderα:for non-negative integers k and n with≤k≤n,

∂k ∂xk B

(α) n (x;λ)

=n!B

(α) n–k(x;λ)

(n–k)! ,

∂k ∂xk E

(α) n (x;λ)

=n!E

(α) n–k(x;λ)

(n–k)! . (.)

Proposition . Diﬀerence equations of the Apostol-Bernoulli and Apostol-Euler polyno-mials of orderα:for a positive integer n,

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and

λE_n(α)(x+ ;λ) +E(α)

n (x;λ) = En(α–)(x;λ)

E()

n (x;λ) =xn

. (.)

Proposition . Addition theorems of the Apostol-Bernoulli and Apostol-Euler polynomi-als of orderα:for a suitable parameterβand a non-negative integer n,

B(α+β)

n (x+y;λ) = n

i=

n i

B(α) i (x;λ)B

(β) n–i(y;λ)

B()

n (x;λ) =xn

, (.)

and

E(α+β)

n (x+y;λ) = n

i=

n i

E(α) i (x;λ)E

(β) n–i(y;λ)

E()

n (x;λ) =xn

. (.)

Proposition . Complementary addition theorems of the Bernoulli and Apostol-Euler polynomials of orderα:for a non-negative integer n,

B(α)

n (α–x;λ) =

(–)n

λα B

(α) n

x;λ–, E_n(α)(α–x;λ) =(–)

n

λα E

(α) n

x;λ–. (.)

Settingα= ,β=  andα=λ= ,β=  in the formulae (.) to (.), we immediately get the corresponding formulae for the Apostol-Bernoulli and Apostol-Euler polynomials, and the classical Bernoulli and Euler polynomials, respectively. It is worth noting that the casesα=λ=  in Proposition . are also called the symmetric distributions of the clas-sical Bernoulli and Euler polynomials. The above propositions will be very useful to in-vestigate the quadratic recurrence formulae for the Apostol-Bernoulli and Apostol-Euler polynomials in the next two sections.

3 Recurrence formulae for Apostol-Bernoulli polynomials

In what follows, we shall always denote byδ,λthe Kronecker symbol which is deﬁned by

δ,λ=  or  according toλ=  orλ= , and also denoteB–n(x;λ) =E–n(x;λ) =  for any

pos-itive integern. Before stating the quadratic recurrence formula for the Apostol-Bernoulli polynomials, we begin with a summation formula concerning the quadratic recurrence of the Apostol-Bernoulli polynomials.

Theorem . Let k,m,n be non-negative integers.Then

n

i=

n i

Bi(x;μ) +Bm+n–i(y;λμ) k

= –δ,λ

n+k

m+ Bm+n+k(x+y;λμ)

+Bm(y;λ)Bn+k(x+y;μ) + n+k

m+ Bm+(y;λ)Bn+k–(x+y;μ)

+ (n+k)

m

i=

m i

(–)m+iBn+k+i–(x+y;λμ)

Bm+–i(x;λ–)

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Proof Multiplying both sides of the identity

Makingk-times derivative for the above identity (.) with respect tov, with the help of the Leibniz rule, we obtain

k

On the other hand, by Taylor’s theorem, we have

(u+v)ex(u+v)

Applying (.), (.) and (.) to (.), in view of the Cauchy product and the complemen-tary addition theorem of the Apostol-Bernoulli polynomials, we derive

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+k

∞

m=

∞

n= _m

i=

m i

(–)m+iBn+k+i–(x+y;λμ)

Bm+–i(x;λ–) m+  –i

um m!

vn n!

–kδ,λ

∞

m=

∞

n=

Bm+n+k–(x+y;λμ) um–

m!

vn

n!. (.)

Comparing the coeﬃcients of um_vn+_/_m_!(_n_{+ )! and} _um_/_m_{! in (.), we conclude our}

proof.

As a special case of Theorem ., we have the following

Corollary . Let m and n be positive integers.Then



m m

i=

m i

(–)m–iBm–i

y–x;λ–Bn+i–(y;λμ) –



mBm(x;λ)Bn–(y;μ)

= –

n n

i=

n i

Bn–i(y–x;μ)Bm+i–(x;λμ) +



nBm–(x;λ)Bn(y;μ). (.)

Proof Settingk=  and substitutingyforxandxforyin Theorem ., we obtain that for non-negative integersmandn, we have

n

i=

n i

Bn–i(y;μ)Bm+i(x;λμ)

=Bm(x;λ)Bn(x+y;μ) + n

m+ Bm+(x;λ)Bn–(x+y;μ)

+ n

m+ 

m+

i=

m+ 

i

(–)m+iBm+–i

y;λ–Bn+i–(x+y;λμ). (.)

Hence, replacingybyy–xin the above gives the desired result.

Remark . Settingλ=μ= ,x=tandy=  –tin Corollary ., byBn( –x) = (–)nBn(x)

for a non-negative integern, we immediately get the generalization of Woodcock’s identity on the classical Bernoulli numbers, see [, ],



m m

i=

m i

(–)i_B

m–i(t)Bn+i–(t) –



mBm(t)Bn–(t)

=

n n

i=

n i

(–)iBn–i(t)Bm+i–(t) –



nBn(t)Bm–(t) (m,n≥). (.)

Applying the diﬀerence equation and symmetric distribution of the classical Bernoulli polynomials, one can get (–)n_B

n(–x) =Bn(x) +nxn–for a positive integern.

Combin-ing with Corollary ., we have another symmetric expression on the classical Bernoulli numbers due to Agoh, see [, ],



m m

i=

m i

Bm–iBn+i–+



n n

i=

n i

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Using Theorem ., we shall give the following quadratic recurrence formula for Apostol-Bernoulli polynomials.

Theorem . Let k,m,n be non-negative integers.Then

Bk(x;μ) +Bm(y;λμ) n

= –δ,λ

k!m!(n+δ(k,m)(m+k+ ))

(m+k+ )! Bm+n+k(x+y;λμ)

+

m+k

i=

(–)m+i

n

m

i

–k

m i– 

Bn+i–(x+y;λμ)Bm+k+–i

(x;λ–)

m+k+  –i

+

m+k

i=

(–)k+i

n

k

i

–m

k i– 

Bn+i–(x+y;μ)Bm+k+–i

(y;λ)

m+k+  –i, (.)

whereδ(k,m) = –when k=m= ,δ(k,m) = when k= ,m≥or m= ,k≥,and

δ(k,m) = otherwise.

Proof Settingk=  and substitutingm+kformin Theorem ., we have

B(x;μ) +Bm+k(y;λμ) n

= –δ,λ

nBm+n+k(x+y;λμ)

m+k+  +Bm+k(y;λ)Bn(x+y;μ)

+nBm+k+(y;λ)Bn–(x+y;μ)

m+k+  +n

m+k

i=

m+k i

(–)m+k+i

×Bn+i–(x+y;λμ)Bm+k+–i

(x;λ–)

m+k+  –i , (.)

which is just the caseδ(k,m) = – or  in Theorem .. Next, consider the caseδ(k,m) = . We shall use induction onkin Theorem . to prove Theorem .. Clearly, the casek=  in Theorem . gives

B(x;μ) +Bm(y;λμ) n

+B(x;μ) +Bm+(y;λμ) n

= –δ,λ

n+ 

m+ Bm+n+(x+y;λμ)

+Bm(y;λ)Bn+(x+y;μ) + n+ 

m+ Bm+(y;λ)Bn(x+y;μ)

+ (n+ )

m

i=

m i

(–)m+iBn+i(x+y;λμ)Bm+–i

(x;λ–)

m+  –i . (.)

Noting that for any non-negative integerk,

m

i=

m i

(–)m+iBn+k+i–(x+y;λμ)Bm+–i

(x;λ–₎ m+  –i

=

m+k

i=

m i–k

(–)m+k+iBn+i–(x+y;λμ)Bm+k+–i

(x;λ–₎

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Hence, substituting (.) and (.) to (.), we get the casek=  of Theorem .. Now assume that Theorem . holds for all positive integers being less thank. From (.) and (.), we obtain

Since (.) holds for all positive integers being less thank, we have

k–

It follows from (.) that

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k–

By substituting the above identities (.)-(.) to (.), we get

k–

Thus, putting (.) and (.) to (.) concludes the induction step. This completes the

proof of Theorem ..

Obviously, the casesλ=μ=  andx=y=  in Theorem . lead to the Agoh-Dilcher quadratic recurrence formula (.). We now use Theorem . to give the following formula on products of the Apostol-Bernoulli polynomials.

Corollary . Let m and n be positive integers.Then

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Proof Takingn=  and then substitutingmfork,nform,λforμandμforλin Theo-rem ., we obtain that for positive integersmandn,

Bm(x;λ)Bn(y;λμ) =n m

i=

m i

(–)iBm–i(x+y;λ)Bn+i

(y;μ)

n+i

+m n

i=

n i

(–)iBn–i(x+y;λμ)

Bm+i(x;μ–) m+i

–δ,μm!n!

(m+n)!Bm+n(x+y;λμ). (.)

Substituting  –yforyandμ–_for_λμ_{in (.) and using the complementary addition}

theorem for Apostol-Bernoulli polynomials, we get our result.

Remark . The casesλ=μ=  andx=yin Corollary . together with the factB= –/

andBn+=  for a positive integern(see,e.g., []) will yield that for positive integersm

andn, see [, ],

Bm(x)Bn(x) = [m+n ]

i=

n

m

i

+m

n

i

Bi

Bm+n–i(x) m+n– i+ (–)

m+ m!n!

(m+n)!Bm+n, (.)

where [x] is the maximum integer less than or equal to the real numberx.

4 Recurrence formulae for mixed Apostol-Bernoulli and Apostol-Euler polynomials

In this section, we shall use the above methods to give some quadratic recurrence formu-lae for mixed Apostol-Bernoulli and Apostol-Euler polynomials. As in the proof of Theo-rem ., we need the following summation formula concerning the quadratic recurrence of mixed Apostol-Bernoulli and Apostol-Euler polynomials.

Theorem . Let k,m,n be non-negative integers.Then

n

i=

n i

Ei(x;μ) +Bm+n–i(y;λμ) k

=

m

i=

m i

(–)m+iEm–i

x;λ–Bn+k+i(x+y;λμ)

– 

 mEm–(y;λ)En+k(x+y;μ) + (n+k)Em(y;λ)En+k–(x+y;μ)

. (.)



λeu_{+ }



μev_{+ }=

λeu

λeu_{+ }–



μev_{+ }



λμeu+v_{– } (.)

by (u+v)exv+y(u+v)_{, we have}

exv

μev_{+ }

(u+v)ey(u+v)

λμeu+v_{– } =λ

e(–x)u

λeu_{+ }

(u+v)e(x+y)(u+v)

λμeu+v_{– } – u+v

 eyu

λeu_{+ }

e(x+y)v

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Makingk-times derivative for the above identity (.) with respect tov, with the help of the Leibniz rule, we get

k

Applying (.) and (.) to (.), in view of the Cauchy product and the complementary addition theorem for the Apostol-Euler polynomials, we obtain

∞

we complete the proof of Theorem ..

We now give a special case of Theorem .. We have the following

Corollary . Let m and n be positive integers.Then

m

Proof Settingk=  and substitutingyforxandxforyin Theorem ., we obtain that for positive integersmandn,

n

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In particular, settingλ=μ= ,x=tandy=  –tin Corollary ., by the factEn( – x) = (–)n_E

n(x) for a non-negative integern, we obtain the following symmetric identity

involving the Bernoulli and Euler polynomials, see [].

Corollary . Let m and n be positive integers.Then

m

i=

m i

(–)iEm–i(t)Bn+i(t) – m

Em–(t)En(t)

=

n

i=

n i

(–)iEn–i(t)Bm+i(t) – n

En–(t)Em(t). (.)

Now we apply Theorem . to give the following recurrence formula for mixed Apostol-Bernoulli and Apostol-Euler polynomials.

Theorem . Let k,m,n be non-negative integers.Then

Ek(x;μ) +Bm(y;λμ) n

=

m+k

i=

m i

(–)m+iBn+i(x+y;λμ)Em+k–i

x;λ–

–  

m+k

i=

(–)k+i

n

k i

–m

k i– 

En–+i(x+y;μ)Em+k–i(y;λ). (.)

Proof The proof is similar to that of Theorem ., and therefore we leave out some of the more obvious details. Clearly, the casek=  in Theorem . is complete. Next, consider the casek≥ in Theorem .. Assume that Theorem . holds for all positive integers being less thank. In light of (.), we have

= –

+

m+k

i=

m i–k

(–)m+k+iBn+i(x+y;λμ)Em+k–i

x;λ–

–

k–

j=

k j

Ej(x;μ) +Bm+k–j(y;λμ) n

. (.)

It follows from (.) and (.) that

= –

+

m+k

i=

(–)m+iBn+i(x+y;λμ)Em+k–i

x;λ–

k–

j=

(–)k––j

k

j

m+k–j i

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– 

Thus, applying (.), (.) and (.) to (.), we conclude the induction step. This

completes the proof of Theorem ..

Corollary . Let m be a non-negative integer and n be a positive integer.Then

Em(x;λ)Bn(y;μ) =

tion theorems of the Apostol-Bernoulli and Apostol-Euler polynomials, the desired result

follows immediately.

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Theorem . Let k,m,n be non-negative integers.Then



Makingk-times derivative for the above identity (.) with respect tov, with the help of the Leibniz rule, we derive

k

Note that from Taylor’s theorem, we have

ex(u+v)

Applying (.), (.) and (.) to (.), in view of the Cauchy product and the comple-mentary addition theorem for the Apostol-Bernoulli polynomials, we obtain

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Remark . Settingk=  in Theorem ., one can easily reobtain Corollary ..

Theorem . Let k,m,n be non-negative integers.Then

Ek(x;μ) +Em(y;λμ) n

= δ,λ

k!m!

(m+k+ )!Em+n+k+(x+y;λμ)

– 

m+k

i=

(–)m+i

m

i

En+i(x+y;λμ)

Bm+k+–i(x;λ–) m+k+  –i

– 

m+k

i=

(–)k+i

k

i

En+i(x+y;μ)Bm+k+–i

(y;λ)

m+k+  –i. (.)

Proof Clearly, the casek=  in Theorem . leads to the casek=  in Theorem .. Now consider the casek≥ in Theorem .. Assume that Theorem . holds for all positive integers being less thank. By (.) we have

= 

m+  δ,λEm+n+k+(x+y;λμ) –Bm+(y;λ)En+k(x+y;μ)

– 

m+k

i=

m i–k

(–)m+k+iEn+i(x+y;λμ)Bm+k+–i

(x;λ–₎ m+k+  –i

–

k–

j=

k j

Ej(x;μ) +Em+k–j(y;λμ) n

. (.)

It follows from (.) and (.) that

= 

m+  Em+n+k+(x+y;λμ) –Bm+(y;λ)En+k(x+y;μ)

– 

m+k

i=

m i–k

(–)m+k+iEn+i(x+y;λμ)

Bm+k+–i(x;λ–) m+k+  –i

– δ,λ

Em+n+k+(x+y;λμ) k+m+ 

k–

j= _k

j _m₊_k j

– 

m+k

i=

(–)m+iEn+i(x+y;λμ)

×Bm+k+–i(x;λ–) m+k+  –i

k–

j=

(–)k––j

k

j

m+k–j i

– 

m+k

i=

(–)k+iEn+i(x+y;μ)Bm+k+–i

(y;λ)

m+k+  –i k–

j=

(–)k––j

k

j

j i

. (.)

Thus, applying (.), (.) and (.) to (.), we conclude the induction step. This

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Corollary . Let m and n be non-negative integers.Then

Em(x;λ)En(y;μ) =  m

i=

m i

(–)m–iEm–i

y–x;λ–Bn++i(y;λμ)

n+  +i

– 

n

i=

n i

En–i(y–x;μ)Bm++i

(x;λμ)

m+  +i

– δ,λμ

(–)m_m_!_n_!

(m+n+ )!Em+n+

y–x;λ–. (.)

Proof Takingn=  in Theorem ., and substitutingmfork,nform,λforμandμforλ, we get

Em(x;λ)En(y;λμ) = δ,μ

m!n!

(m+n+ )!Em+n+(x+y;λμ)

– 

m

i=

m i

(–)iEm–i(x+y;λμ)Bn++i

(x;μ–₎ n+  +i

– 

n

i=

n i

(–)iEn–i(x+y;λ)Bm++i

(y;μ)

m+  +i . (.)

Hence, substituting  –yforyandμ–_for_λμ_{in (.), in view of the complementary}

addition theorems of the Apostol-Bernoulli and Apostol-Euler polynomials, the desired

result follows immediately.

Remark . The casesλ=μ=  andx=yin Corollaries . and . will yield the corre-sponding formulae forBm(x)En(x) andEm(x)En(x) presented in []. We leave them to the

interested readers for an exercise. For diﬀerent proofs of Corollaries ., . and ., we refer to [].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and signiﬁcantly in writing this paper.

Acknowledgements

The authors express their gratitude to the referees for careful reading and helpful comments on the previous version of this work. This work is supported by the National Natural Science Foundation of China (Grant No. 11071194).

Received: 29 August 2012 Accepted: 21 November 2012 Published: 10 December 2012 References

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Nat. Soc.2012, Article ID 927953 (2012)

doi:10.1186/1687-1847-2012-209

Cite this article as:He and Wang:Recurrence formulae for Apostol-Bernoulli and Apostol-Euler polynomials.