R E S E A R C H
Open Access
Some unified formulas and
representations for the Apostol-type
polynomials
Da-Qian Lu
1and Qiu-Ming Luo
2**Correspondence: [email protected] 2Department of Mathematics, Chongqing Normal University, Chongqing Higher Education Mega Center, Huxi Campus, Chongqing, 401331, People’s Republic of China Full list of author information is available at the end of the article
Abstract
Recently, a family of the Apostol-type polynomials was introduced by Luo and Srivastava (Appl. Math. Comput. 217:5702-5728 (2011)). In this paper, we further investigate the Apostol-type polynomials and obtain their unified multiplication formula and explicit representations in terms of the Gaussian hypergeometric function and the generalized Hurwitz zeta function. We also show some special cases, which include the corresponding results of Luo, Garg, Srivastava, Ozden, and Özarslan
etc.
MSC: Primary 11B68; secondary 11M35; 11B73; 33C05
Keywords: (generalized) Apostol-type polynomials; Apostol-Euler polynomials; Apostol-Bernoulli polynomials; Genocchi polynomials of high order; multiplication formula; Gaussian hypergeometric function; (generalized) Hurwitz zeta function
1 Introduction, definitions, and motivation
Throughout this paper, we always make use of the following notations:N={, , , . . .} denotes the set of natural numbers,N={, , , , . . .}denotes the set of nonnegative in-tegers,Z–
={, –, –, –, . . .}denotes the set of nonpositive integers,Zdenotes the set of
integers,Rdenotes the set of real numbers, andCdenotes the set of complex numbers. The symbol (a)k denotes the shifted factorial (or the Pochhammer symbol), defined,
a∈C, by
(a)k=
(a+k)
(a) = ⎧ ⎨ ⎩
, k= ,
a(a+ )· · ·(a+k– ), k∈N. (.)
The symbol{n}kdenotes the falling factorial, defined,a∈C, by
{a}k= ⎧ ⎨ ⎩
, k= ,
a(a– )· · ·(a–k+ ) =(a(a–+)k+), k∈N, (.)
where(x) is the usual gamma function.
The classical Bernoulli polynomialsBn(x), Euler polynomialsEn(x), and Genocchi poly-nomialsGn(x), together with their familiar generalizationsB(nα)(x),En(α)(x), andG(nα)(x) of
orderα, are usually defined by means of the following generating functions (see, for de-tails, [], pp.- and []):
z ez–
α
exz=
∞
n= B(α)
n (x)
zn
n!
|z|< π , (.)
ez+ α
exz=
∞
n= E(nα)(x)z
n
n!
|z|<π (.)
and
z ez+
α
exz=
∞
n= G(α)
n (x)
zn
n!
|z|<π . (.)
Thus, the Bernoulli polynomialsBn(x), Euler polynomialsEn(x), and Genocchi polynomi-alsGn(x) are given, respectively, by
Bn(x) :=B()n (x), En(x) :=E()n (x) and Gn(x) :=Gn()(x) (n∈N). (.)
The Bernoulli numbersBn, Euler numbersEn, and Genocchi numbersGnare, respectively,
Bn:=Bn() =B()n (), En:=En() =En()() and Gn:=Gn() =G()n (). (.)
Some interesting analogs of the classical Bernoulli polynomials and numbers were first investigated by Apostol (see [], p., Eq. (.)) and (more recently) by Srivastava (see [], pp.-). We begin by recalling here Apostol’s definitions as follows.
Definition .(Apostol []; see also Srivastava []) The Apostol-Bernoulli polynomials
Bn(x;λ) (λ∈C) are defined by means of the following generating function:
zexz
λez– =
∞
n= Bn(x;λ)
zn
n!
|z|< πwhenλ= ;|z|<|logλ|whenλ= (.)
with, of course,
Bn(x) =Bn(x; ) and Bn(λ) :=Bn(;λ), (.) whereBn(λ) denotes the so-called Apostol-Bernoulli numbers.
Recently, Luo and Srivastava [] further extended the Apostol-Bernoulli polynomials as the so-called Apostol-Bernoulli polynomials of orderα.
Definition .(Luo and Srivastava []) The Apostol-Bernoulli polynomialsB(nα)(x;λ) (λ∈
C) of orderα(α∈N) are defined by means of the following generating function:
z
λez– α
·exz=
∞
n= B(α)
n (x;λ)
zn
n!
with, of course,
B(nα)(x) =B(nα)(x; ) and Bn(α)(λ) :=B(nα)(;λ), (.) whereB(nα)(λ) denotes the so-called Apostol-Bernoulli numbers of orderα.
On the other hand, Luo [] gave an analogous extension of the generalized Euler poly-nomials as the so-called Apostol-Euler polypoly-nomials of orderα.
Definition .(Luo []) The Apostol-Euler polynomialsEn(α)(x;λ) of orderα(α,λ∈C) are defined by means of the following generating function:
λez+ α
·exz=
∞
n= E(α)
n (x;λ)
zn
n!
|z|<log(–λ) (.)
with, of course,
E(nα)(x) =En(α)(x; ) and En(α)(λ) :=En(α)(;λ), (.) whereEn(α)(λ) denotes the so-called Apostol-Euler numbers of orderα.
On the subject of the Genocchi polynomialsGn(x) and their various extensions, a re-markably large number of investigations have appeared in the literature (see, for example, [–]). Moreover, Luo (see []) introduced and investigated the Apostol-Genocchi poly-nomials of (real or complex) orderα, which are defined as follows.
Definition . The Apostol-Genocchi polynomialsGn(α)(x;λ) (λ∈C) of orderα(α∈N) are defined by means of the following generating function:
z
λez+ α
·exz=
∞
n= G(α)
n (x;λ)
zn
n!
|z|<log(–λ) (.)
with, of course,
G(nα)(x) =Gn(α)(x; ), Gn(α)(λ) :=Gn(α)(;λ),
Gn(x;λ) :=Gn()(x;λ) and Gn(λ) :=Gn()(λ),
(.)
where Gn(λ),Gn(α)(λ), andGn(x;λ) denote the so-called Apostol-Genocchi numbers, the Apostol-Genocchi numbers of orderα, and the Apostol-Genocchi polynomials, respec-tively.
Ozdenet al.[] investigated the following unification (and generalization) of the gen-erating functions of the three families of Apostol-type polynomials:
–κzκ
βbez–abe xz=
∞
n=
Yn,β(x;κ,a,b)
zn
n!
In [] Özarslan further gave an extension of the above definition (.) as follows:
–κzκ
βbez–ab α
exz=
∞
n= Y(α)
n,β(x;κ,a,b)
zn
n!
α∈N;|z|< πwhenβ=a;|z|<blog(β/a)
whenβ=a;κ,β∈C;a,b∈C\ {} (.)
and gave some identities forYn(α,β)(x;κ,a,b).
Recently, Luo and Srivastava [] further extended the Apostol-type polynomials as fol-lows.
Definition . (Luo and Srivastava []) The generalized Apostol-type polynomials
F(α)
n (x;λ;μ;ν) of orderα (α,λ,μ;ν∈C) are defined by means of the following generat-ing function:
μzν
λez+ α
exz=
∞
n= F(α)
n (x;λ;μ;ν)
zn n!
|z|<log(–λ) . (.)
By comparing Definition . with Definitions ., . and ., we readily find that
B(α)
n (x;λ) = (–)
αF(α)
n (x; –λ; ; ) (α∈N), (.)
E(α)
n (x;λ) =F(
α)
n (x;λ; ; ) (α∈C) (.)
and
G(α)
n (x;λ) =F(
α)
n (x;λ; ; ) (α∈N). (.)
Furthermore, if we compare the generating functions (.), (.) and (.), we readily see that
Yn,β(x;κ,a,b) = –
abF
()
n
x; –
β
a
b ; –κ;κ
, (.)
Y(α)
n,β(x;κ,a,b) = (–)
α
abαF
(α)
n
x; –
β
a
b ; –κ;κ
. (.)
More investigations of this subject can be found in [, , –].
The aim of this paper is to give the multiplication formula for the Apostol-type polyno-mialsFn(α)(x;λ;μ;ν) and obtain an explicit representation ofF(
α)
n (x;λ;μ;ν) in terms of the Gauss hypergeometric functionF(a,b;c;z). We study some relations between the fam-ily of Apostol-type polynomialsFn(α)(x;λ;μ;ν) and the family of Hurwitz zeta functions
μ(z,s,a). Some special cases also are shown.
2 Multiplication formula for the Apostol-type polynomials
In this section we give a unified multiplication formula for the Apostol-type polynomials
F(α)
n (x;λ;μ;ν). We will see that some well-known results are the corresponding special cases of our result.
Lemma .(Multinomial identity [], p., Theorem B) If x,x, . . . ,xm are commuting
elements of a ring(⇐⇒ xixj=xjxi, ≤i<j≤m),then we have for all integers n≥:
(x+x+· · ·+xm)n=
a,a,...,am≥
a+a+···+am=n
n a,a, . . . ,am
xaxa· · ·x
mam, (.)
the last summation takes place over all positive or zero integers ai≥such that a+a+
· · ·+am=n,where
n a,a, . . . ,am
:= n!
a!a!· · ·am! ,
are called multinomial coefficients defined by[],p.,Definition B.
Lemma .(Generalized multinomial identity [], p., Eq. (m)) If x,x, . . . ,xmare
commuting elements of a ring(⇐⇒ xixj=xjxi, ≤i<j≤m),then we have for all real or
complex variableα:
( +x+x+· · ·+xm)α=
v,v,...,vm≥
α
v,v, . . . ,vm
xvxv· · ·xmvm, (.)
the last summation takes place over all positive or zero integers vi≥,where
α
v,v, . . . ,vm
:={α}v+v+···+vm
v!v!· · ·vm!
=α(α– )(α– )· · ·(α–v–v–· · ·–vm+ )
v!v!· · ·vm!
are called generalized multinomial coefficients defined by[],p.,Eq. (C ).
Theorem .(Multiplication formula) Forμ,ν,r∈Nandν≤,n,l∈N,α,λ∈C,we have
F(α)
n (rx;λ;μ;ν) =rn–να
v,v,...,vr–≥
α
v,v, . . . ,vr–
×(–λ)mFn(α)
x+m
r;λ
r;μ;ν
, r odd, (.)
F(l)
n (rx;λ;μ;ν) =
(–)lμlrn–νl (n+ )(–ν)l
≤v,v,...,vr–≤l
v+v+···+vr–=l
l v,v, . . . ,vr–
×(–λ)mB(nl+(–) ν)l
x+m
r;λ
r
, r even, (.)
where m=v+ v+· · ·+ (r– )vr–.
Proof It is not difficult to show that
λez+ = –
–λez+λez+· · ·+ (–λ)r–e(r–)z
Whenris odd, by (.) and (.) we get
Comparing the coefficients of znn! on both sides of (.), we obtain the assertion (.) of Theorem ..
Whenris even, we can similarly prove the assertion (.) of Theorem .. The proof is
complete.
It follows that we can deduce the well-known formulas from Theorem ..
Lettingλ−→–λ, taking μ= andν= in (.) and (.) and noting (.), we can obtain the following main result of Luo (see [], p., Theorem .).
Corollary . For r,α∈N,n∈N,λ∈C,the following multiplication formula for the Apostol-Bernoulli polynomials of higher order holds true:
B(α) Apostol-Euler polynomials of higher order holds true:
Takingμ=ν= in (.) and (.), and noting (.), we can obtain the following main result (see [], p., Corollary .).
Corollary . Forα,r∈N,n,l∈N,λ∈C,the following multiplication formula for the Apostol-Genocchi polynomials of higher order holds true:
G(α) (.) and (.), respectively.
Corollary . Forκ,μ,ν,m,n,l,r∈N,α,λ∈C,we have
Remark . In [], p., Theorem ., one of the main result of Özarslan is not right, the correct form should be (.) and (.) of Corollary ..
as the proof of [], p., Theorem ., we can obtain the multiplication formulas (.) and (.) of Corollary ..
3 A unified representation in conjunction with the Gauss hypergeometric function
In this section we obtain a unified representation of the Apostol-type polynomials
F(l)
Differentiating both sides of (.) with respect to the variablezyields
F(l)
whereDz=dzd is the differential operator. Applying the generalized binomial theorem
and the generating function of the Stirling numbers of the second kindS(n,k) (see, for details, [], p., Theorem A),
(ez– )k
and the well-known combinatorial identity
we readily obtain
Noting that (in view ofnk = whenk>nork< )
and combining the definition of the Gaussian hypergeometric function
F(a,b;c;z) :=
Applying the Pfaff-Kummer hypergeometric transformation [], p., Eq. (..),
F(a,b;c;z) = ( –z)–aF
to (.), we arrive at the desired equation, (.). This completes our proof.
Below we show some special cases of (.).
Lettingλ−→–λ, takingμ= andν= in (.) and noting (.), we easily obtain the following explicit formula for the Apostol-Bernoulli polynomials:
B(l) formula for the Apostol-Euler polynomials:
Takingμ=ν= in (.), and noting (.), we can obtain the following explicit repre-sentation of the generalized Apostol-Genocchi polynomials:
G(l)
4 Some explicit relationships between the generalized Apostol-type polynomials and generalized Hurwitz-Lerch zeta function
A general Hurwitz-Lerch zeta function (z,s,a) defined by (cf.,e.g., [], p.,et seq.)
and the Riemann zeta functionζ(s),
ζ(s) := (,s, ) =ζ(s, ) = s– ζ
s,
(s) > ;a∈/Z
–
, (.)
and the Lerch zeta function:
s(ξ) :=
∞
n= enπiξ
ns =e
πiξ
eπiξ,s, ξ∈R;R(s) > , (.)
but also such other functions as the polylogarithm function:
Lis(z) :=
∞
n= zn
ns =z (z,s, )
s∈Cwhen|z|< ;R(s) > when|z|= (.)
and the Lipschitz-Lerch zeta function (cf. [], p., Eq. .()):
φ(ξ,a,s) :=
∞
n= enπiξ (n+a)s=
eπiξ,s,a =:L(ξ,s,a)
a∈C\Z–;R(s) > whenξ∈R\Z;R(s) > whenξ∈Z, (.)
which was first studied by Rudolf Lipschitz (-) and Matyáš Lerch (-) in connection with Dirichlet’s famous theorem on primes in arithmetic progressions.
A family of the Hurwitz-Lerch zeta functions (ρ,σ)
μ,ν (z,s,a) defined by (seee.g.[], p., Eq. ())
(ρ,σ)
μ,ν (z,s,a) :=
∞
n=
(μ)ρn (ν)σn
zn (n+a)s
μ∈C;a,ν∈C\Z–;ρ,σ∈R+;ρ<σwhens,z∈C;
ρ=σands∈Cwhen|z|< ;ρ=σand(s–μ+ν) > when|z|= , (.) contains, asspecialcases, not only the Hurwitz-Lerch zeta function
(σ,σ)
ν,ν (z,s,a) = (,)μ,ν (z,s,a) = (z,s,a) =
∞
n= zn
(n+a)s (.)
and the Lipschitz-Lerch zeta function φ(ξ,a,s) := (eπiξ,s,a), but also the following generalized Hurwitz-Lerch zeta functions introduced and studied earlier by Goyal and Laddha [], p., Eq. (.):
(,)
μ,(z,s,a) = μ(z,s,a) :=
∞
n=
(μ)n
n!
zn
(n+a)s, (.)
Below we give an explicit relationship between the Apostol-type polynomialsFn(α)(x;λ;
μ;ν) and the Hurwitz-Lerch zeta function μ(z,s,a).
Theorem . For n,ν,α∈N; – <λ;nνα;x∈C\Z–
,μ∈C,the relationship F(α)
n (x;λ;μ;ν) =
μα (να)!
n
να
α(–λ,να–n,x) (.)
holds true.
Proof Applying the generalized binomial theorem
( +w)–α=
∞
r=
α+r–
r
(–w)r |w|<
in (.), we have
∞
n= F(α)
n (x;λ;μ;ν)
zn
n! =
μzν
λez+ α
exz
= μαzνα +λez –αexz
= μαzνα
∞
k=
(α)k
k! (–λ) ke(k+x)z
=
∞
n=
μα
∞
k=
(α)k
k! (–λ)
k(k+x)n
zn+να
n!
=
∞
n=να
μα(να)!
n
να
∞
k=
(α)k
k!
(–λ)k (k+x)να–n
zn
n!. (.)
Noting (.), (.) follows.
Below we see that (.) implies some well-known results.
Letλ−→–λ, takingμ= andν= in (.) and noting (.), we can obtain an explicit relation between the Apostol-Bernoulli polynomials and the Hurwitz-Lerch zeta function:
B(l)
n(x;λ) = (–n)l l(λ,l–n,x)
n,l∈N;nl;|λ|< ;x∈C\Z– . (.) The above result is just one of the main results of Garget al.(see [], p.).
Clearly, we have the following relation between the Apostol-Bernoulli polynomials and the Hurwitz-Lerch zeta function:
Bn(x;λ) = –n (λ, –n,x)
n∈N;|λ|;x∈C\Z– , (.) which is just the result of Apostol (see []).
Takingλ= in (.), we obtain the following well-known relationship between the Bernoulli polynomials and Hurwitz zeta function (see [], p., Theorem .):
Takingx= in (.), we obtain the following the well-known relationship between the Bernoulli numbers and the Riemann zeta function (see [], p., Theorem .):
Bn= –nζ( –n) (n∈N). (.)
Takingμ= andν= in (.) and noting (.), we can obtain the following result of Luo (see [], p., Theorem .):
E(α)
n (x;λ) =
α
α(–λ, –n,x). (.)
Further takingα= in (.), we have the following relation between the Apostol-Euler polynomials and the Hurwitz-Lerch zeta function:
En(x;λ) = (–λ, –n,x)
n∈N; – <λ;x∈C\Z– . (.) Takingλ= in (.), we can obtain the following well-known relation between the Euler polynomials and theL-function:
En(x) = L(–n,x), (.)
where theL-function is defined by
L(s,x) :=
∞
n=
(–)n (n+x)s
(s) > ;x∈C\Z– . (.)
From (.), we further obtain the following well-known relation between the Euler num-bers and thel-function:
En= l(–n), (.)
where thel-function is defined by
l(s) :=
∞
n=
(–)n
ns , (s) > . (.)
Taking μ=ν = in (.), and noting (.), we can deduce the following relation between the Apostol-Genocchi polynomials and Hurwitz-Lerch zeta function (see [], p., Corollary .):
G(l)
n (x;λ) ={n}ll l(–λ,l–n,x)
n,l∈N;nl;|λ|;x∈C\Z– (.) and
Gn(x;λ) = n (–λ, –n,x)
Hurwitz zeta functions [, , ]:
Y(α)
n,β(x;κ,a,b) = (–) α(–
κ)α(κα)!
abα
n
κα
α
β
a
b
,κα–n,x
(.)
and
Yn,β(x;κ,a,b) = –
(–κ)(κ)! ab
n
κ
β
a
b ,κ–n,x
. (.)
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors contributed equally in writing this paper, and they read and approved the final manuscript.
Author details
1Department of Mathematics and Information, Taizhou University, Taizhou, Zhejiang 317000, People’s Republic of China. 2Department of Mathematics, Chongqing Normal University, Chongqing Higher Education Mega Center, Huxi Campus, Chongqing, 401331, People’s Republic of China.
Acknowledgements
The authors sincerely thank the referees for their valuable suggestions and comments. The present investigation was supported byNatural Science Foundation Project of Chongqing,Chinaunder Grant CSTC2011JJA00024,Research Project of Science and Technology of Chongqing Education Commission,Chinaunder Grant KJ120625,Fund of Chongqing Normal University,Chinaunder Grants 10XLR017 and 2011XLZ07.
Received: 12 February 2015 Accepted: 20 April 2015
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