Some unified formulas and representations for the Apostol-type polynomials

16 Pages·2015·1.4 MB·English

LuandLuoAdvancesinDifferenceEquations (2015) 2015:137
DOI10.1186/s13662-015-0480-0
RESEARCH OpenAccess
Some unified formulas and
representations for the Apostol-type
polynomials
Da-QianLu1andQiu-MingLuo2*
*Correspondence:
[email protected] Abstract
2DepartmentofMathematics,
Recently,afamilyoftheApostol-typepolynomialswasintroducedbyLuoand
ChongqingNormalUniversity,
ChongqingHigherEducationMega Srivastava(Appl.Math.Comput.217:5702-5728(2011)).Inthispaper,wefurther
Center,HuxiCampus,Chongqing, investigatetheApostol-typepolynomialsandobtaintheirunifiedmultiplication
401331,People’sRepublicofChina
formulaandexplicitrepresentationsintermsoftheGaussianhypergeometric
Fulllistofauthorinformationis
availableattheendofthearticle functionandthegeneralizedHurwitzzetafunction.Wealsoshowsomespecialcases,
whichincludethecorrespondingresultsofLuo,Garg,Srivastava,Ozden,andÖzarslan
etc.
MSC: Primary11B68;secondary11M35;11B73;33C05
Keywords: (generalized)Apostol-typepolynomials;Apostol-Eulerpolynomials;
Apostol-Bernoullipolynomials;Genocchipolynomialsofhighorder;multiplication
formula;Gaussianhypergeometricfunction;(generalized)Hurwitzzetafunction
1 Introduction,definitions,andmotivation
Throughout this paper, we always make use of the following notations: N={,,,...}
denotesthesetofnaturalnumbers,N ={,,,,...}denotesthesetofnonnegativein-

tegers,Z–={,–,–,–,...}denotesthesetofnonpositiveintegers,Zdenotesthesetof

integers,Rdenotesthesetofrealnumbers,andCdenotesthesetofcomplexnumbers.
The symbol (a) denotes the shifted factorial (or the Pochhammer symbol), defined,
k
a∈C,by
⎧
⎨
(cid:2)(a+k) , k=,
(a) = = (.)
k (cid:2)(a) ⎩a(a+)···(a+k–), k∈N.
Thesymbol{n} denotesthefallingfactorial,defined,a∈C,by
k
⎧
⎨
, k=,
{a} = (.)
k ⎩
a(a–)···(a–k+)= (cid:2)(a+) , k∈N,
(cid:2)(a–k+)
where(cid:2)(x)istheusualgammafunction.
TheclassicalBernoullipolynomialsB (x),EulerpolynomialsE (x),andGenocchipoly-
n n
nomialsG (x),togetherwiththeirfamiliargeneralizationsB(α)(x),E(α)(x),andG(α)(x)of
n n n n
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LuandLuoAdvancesinDifferenceEquations (2015) 2015:137 Page2of16
orderα,areusuallydefinedbymeansofthefollowinggeneratingfunctions(see,forde-
tails,[],pp.-and[]):
(cid:5) (cid:6)
z α (cid:7)∞ zn (cid:8) (cid:9)
exz= B(α)(x) |z|<π , (.)
ez– n n!
n=
(cid:5) (cid:6)
 α (cid:7)∞ zn (cid:8) (cid:9)
exz= E(α)(x) |z|<π (.)
ez+ n n!
n=
and
(cid:5) (cid:6)
z α (cid:7)∞ zn (cid:8) (cid:9)
exz= G(α)(x) |z|<π . (.)
ez+ n n!
n=
Thus,theBernoullipolynomialsB (x),EulerpolynomialsE (x),andGenocchipolynomi-
n n
alsG (x)aregiven,respectively,by
n
B (x):=B()(x), E (x):=E()(x) and G (x):=G()(x) (n∈N ). (.)
n n n n n n 
TheBernoullinumbersB ,EulernumbersE ,andGenocchinumbersG are,respectively,
n n n
B :=B ()=B()(), E :=E ()=E()() and G :=G ()=G()(). (.)
n n n n n n n n n
SomeinterestinganalogsoftheclassicalBernoullipolynomialsandnumberswerefirst
investigatedbyApostol(see[],p.,Eq.(.))and(morerecently)bySrivastava(see[],
pp.-).WebeginbyrecallinghereApostol’sdefinitionsasfollows.
Definition.(Apostol[];seealsoSrivastava[]) TheApostol-Bernoullipolynomials
B (x;λ)(λ∈C)aredefinedbymeansofthefollowinggeneratingfunction:
n
zexz (cid:7)∞ zn (cid:8) (cid:9)
= B (x;λ) |z|<π whenλ=;|z|<|logλ|whenλ(cid:4)= (.)
λez– n n!
n=
with,ofcourse,
B (x)=B (x;) and B (λ):=B (;λ), (.)
n n n n
whereB (λ)denotestheso-calledApostol-Bernoullinumbers.
n
Recently,LuoandSrivastava[]furtherextendedtheApostol-Bernoullipolynomialsas
theso-calledApostol-Bernoullipolynomialsoforderα.
Definition.(LuoandSrivastava[]) TheApostol-BernoullipolynomialsB(α)(x;λ)(λ∈
n
C)oforderα(α∈N)aredefinedbymeansofthefollowinggeneratingfunction:
(cid:5) (cid:6)
z α (cid:7)∞ zn
·exz= B(α)(x;λ)
λez– n n!
n=
(cid:8) (cid:9)
|z|<π whenλ=;|z|<|logλ|whenλ(cid:4)= (.)
LuandLuoAdvancesinDifferenceEquations (2015) 2015:137 Page3of16
with,ofcourse,
B(α)(x)=B(α)(x;) and B(α)(λ):=B(α)(;λ), (.)
n n n n
whereB(α)(λ)denotestheso-calledApostol-Bernoullinumbersoforderα.
n
Ontheotherhand,Luo[]gaveananalogousextensionofthegeneralizedEulerpoly-
nomialsastheso-calledApostol-Eulerpolynomialsoforderα.
Definition.(Luo[]) TheApostol-Eulerpolynomials E(α)(x;λ) oforder α (α,λ∈C)
n
aredefinedbymeansofthefollowinggeneratingfunction:
(cid:5) (cid:6)
 α·exz=(cid:7)∞ E(α)(x;λ)zn (cid:8)|z|<(cid:10)(cid:10)log(–λ)(cid:10)(cid:10)(cid:9) (.)
λez+ n n!
n=
with,ofcourse,
E(α)(x)=E(α)(x;) and E(α)(λ):=E(α)(;λ), (.)
n n n n
whereE(α)(λ)denotestheso-calledApostol-Eulernumbersoforderα.
n
OnthesubjectoftheGenocchipolynomialsG (x) andtheirvariousextensions,are-
n
markablylargenumberofinvestigationshaveappearedintheliterature(see,forexample,
[–]).Moreover,Luo(see[])introducedandinvestigatedtheApostol-Genocchipoly-
nomialsof(realorcomplex)orderα,whicharedefinedasfollows.
Definition. TheApostol-GenocchipolynomialsG(α)(x;λ)(λ∈C)oforderα (α∈N)
n
aredefinedbymeansofthefollowinggeneratingfunction:
(cid:5) (cid:6)
z α·exz=(cid:7)∞ G(α)(x;λ)zn (cid:8)|z|<(cid:10)(cid:10)log(–λ)(cid:10)(cid:10)(cid:9) (.)
λez+ n n!
n=
with,ofcourse,
G(α)(x)=G(α)(x;), G(α)(λ):=G(α)(;λ),
n n n n
(.)
G (x;λ):=G()(x;λ) and G (λ):=G()(λ),
n n n n
where G (λ), G(α)(λ), and G (x;λ) denote the so-called Apostol-Genocchi numbers, the
n n n
Apostol-Genocchinumbersoforderα,andtheApostol-Genocchipolynomials,respec-
tively.
Ozdenetal.[]investigatedthefollowingunification(andgeneralization)ofthegen-
eratingfunctionsofthethreefamiliesofApostol-typepolynomials:
(cid:7)∞
–κzκ zn
exz= Y (x;κ,a,b)
βbez–ab n,β n!
n=
(cid:8) (cid:10) (cid:10) (cid:9)
|z|<π whenβ=a;|z|<(cid:10)blog(β/a)(cid:10)whenβ(cid:4)=a;κ,β∈C;a,b∈C\{} . (.)
LuandLuoAdvancesinDifferenceEquations (2015) 2015:137 Page4of16
In[]Özarslanfurthergaveanextensionoftheabovedefinition(.)asfollows:
(cid:5) (cid:6)
–κzκ α (cid:7)∞ zn
exz= Y(α)(x;κ,a,b)
βbez–ab n,β n!
n=
(cid:8) (cid:10) (cid:10)
α∈N;|z|<π whenβ=a;|z|<(cid:10)blog(β/a)(cid:10)
(cid:9)
whenβ(cid:4)=a;κ,β∈C;a,b∈C\{} (.)
andgavesomeidentitiesforY(α)(x;κ,a,b).
n,β
Recently,LuoandSrivastava[]furtherextendedtheApostol-typepolynomialsasfol-
lows.
Definition . (Luo and Srivastava []) The generalized Apostol-type polynomials
F(α)(x;λ;μ;ν) of order α (α,λ,μ;ν ∈C) are defined by means of the following generat-
n
ingfunction:
(cid:5) (cid:6)
μzν αexz=(cid:7)∞ F(α)(x;λ;μ;ν)zn (cid:8)|z|<(cid:10)(cid:10)log(–λ)(cid:10)(cid:10)(cid:9). (.)
λez+ n n!
n=
BycomparingDefinition.withDefinitions.,.and.,wereadilyfindthat
B(α)(x;λ)=(–)αF(α)(x;–λ;;) (α∈N), (.)
n n
E(α)(x;λ)=F(α)(x;λ;;) (α∈C) (.)
n n
and
G(α)(x;λ)=F(α)(x;λ;;) (α∈N). (.)
n n
Furthermore,ifwecomparethegeneratingfunctions(.),(.)and(.),wereadily
seethat
(cid:5) (cid:5) (cid:6) (cid:6)
 β b
Y (x;κ,a,b)=– F() x;– ;–κ;κ , (.)
n,β ab n a
(cid:5) (cid:5) (cid:6) (cid:6)
 β b
Y(α)(x;κ,a,b)=(–)α F(α) x;– ;–κ;κ . (.)
n,β abα n a
Moreinvestigationsofthissubjectcanbefoundin[,,–].
TheaimofthispaperistogivethemultiplicationformulafortheApostol-typepolyno-
mialsF(α)(x;λ;μ;ν)andobtainanexplicitrepresentationofF(α)(x;λ;μ;ν)intermsofthe
n n
Gausshypergeometricfunction F (a,b;c;z).Westudysomerelationsbetweenthefam-
 
ilyofApostol-typepolynomialsF(α)(x;λ;μ;ν)andthefamilyofHurwitzzetafunctions
n
(cid:9) (z,s,a).Somespecialcasesalsoareshown.
μ
2 MultiplicationformulafortheApostol-typepolynomials
InthissectionwegiveaunifiedmultiplicationformulafortheApostol-typepolynomials
F(α)(x;λ;μ;ν). We will see that some well-known results are the corresponding special
n
casesofourresult.
Firstweneedthefollowinglemmas.
LuandLuoAdvancesinDifferenceEquations (2015) 2015:137 Page5of16
Lemma.(Multinomialidentity[],p.,TheoremB) Ifx ,x ,...,x arecommuting
  m
elementsofaring(⇐⇒ xx =xx,≤i<j≤m),thenwehaveforallintegersn≥:
i j j i
(cid:5) (cid:6)
(cid:7)
n
(x +x +···+x )n= x ax a···x am, (.)
  m   m
a ,a ,...,a
aa+,aa+,.·..·,·a+mam≥=n   m
thelastsummationtakesplaceoverallpositiveorzerointegersa ≥suchthata +a +
i  
···+a =n,where
m
(cid:5) (cid:6)
n n!
:= ,
a ,a ,...,a a !a !···a !
  m   m
arecalledmultinomialcoefficientsdefinedby[],p.,DefinitionB.
Lemma.(Generalizedmultinomialidentity[],p.,Eq.(m)) If x ,x ,...,x are
  m
commutingelementsofaring(⇐⇒ xx =xx,≤i<j≤m),thenwehaveforallrealor
i j j i
complexvariableα:
(cid:5) (cid:6)
(cid:7)
α
(+x +x +···+x )α= x vx v···x vm, (.)
  m   m
v ,v ,...,v
v,v,...,vm≥   m
thelastsummationtakesplaceoverallpositiveorzerointegersv ≥,where
i
(cid:5) (cid:6)
α := {α}v+v+···+vm = α(α–)(α–)···(α–v–v–···–vm+)
v ,v ,...,v v !v !···v ! v !v !···v !
  m   m   m
arecalledgeneralizedmultinomialcoefficientsdefinedby[],p.,Eq.(C(cid:9)(cid:9)).
Theorem. (Multiplicationformula) For μ,ν,r∈N and ν ≤, n,l∈N , α,λ∈C,we

have
(cid:5) (cid:6)
(cid:7)
α
F(α)(rx;λ;μ;ν)=rn–να
n v ,v ,...,v
v,v,...,vr–≥   r–
(cid:5) (cid:6)
m
×(–λ)mF(α) x+ ;λr;μ;ν , rodd, (.)
n r
(cid:5) (cid:6)
(cid:7)
(–)lμlrn–νl l
F(l)(rx;λ;μ;ν)=
n (n+) v ,v ,...,v
(–ν)l ≤v,v,...,vr–≤l   r–
v+v+···+vr–=l
(cid:5) (cid:6)
m
×(–λ)mB(l) x+ ;λr , reven, (.)
n+(–ν)l r
wherem=v +v +···+(r–)v .
  r–
Proof Itisnotdifficulttoshowthat
 –λez+λez+···+(–λ)r–e(r–)z
=– . (.)
λez+ (–λ)rerz–
LuandLuoAdvancesinDifferenceEquations (2015) 2015:137 Page6of16
Whenrisodd,by(.)and(.)weget
(cid:7)∞
zn
F(α)(rx;λ;μ;ν)
n n!
n=
(cid:5) (cid:6) (cid:5) (cid:6)
 μ(rz)ν α λrerz+ α
= erxz
rνα λrerz+ λez+
(cid:5) (cid:6) (cid:11) (cid:12)
 μ(rz)ν α (cid:7)r–(cid:8) (cid:9) α
= –λez k erxz
rνα λrerz+
k=
(cid:5) (cid:6) (cid:5) (cid:6)
=  (cid:7) α (–λ)m μ(rz)ν αe(x+mr)rz
rνα v ,v ,...,v λrerz+
v,v,...,vr–≥   r–
(cid:13) (cid:5) (cid:6) (cid:5) (cid:6)(cid:14)
(cid:7)∞ (cid:7)
α m zn
= rn–να (–λ)mF(α) x+ ;λr;μ;ν . (.)
v ,v ,...,v n r n!
n= v,v,...,vr–≥   r–
Comparingthecoefficientsof zn onbothsidesof (.),weobtaintheassertion(.)of
n!
Theorem..
Whenriseven,wecansimilarlyprovetheassertion(.)ofTheorem..Theproofis
complete. (cid:2)
Itfollowsthatwecandeducethewell-knownformulasfromTheorem..
Letting λ(cid:10)−→–λ, taking μ= and ν = in (.) and (.) and noting (.), we can
obtainthefollowingmainresultofLuo(see[],p.,Theorem.).
Corollary . For r,α ∈N, n∈N , λ∈C, the following multiplication formula for the

Apostol-Bernoullipolynomialsofhigherorderholdstrue:
(cid:5) (cid:6) (cid:5) (cid:6)
(cid:7)
α m
B(α)(rx;λ)=rn–α λmB(α) x+ ;λr , (.)
n v ,v ,...,v n r
v,v,...,vr–≥   r–
wherem=v +v +···+(r–)v .
  r–
Takingμ=andν=in(.)and(.),andnoting(.),wecanobtainthefollowing
mainresultofLuo(see[],p.,Theorem.).
Corollary. Forr∈N,n,l∈N ,α,λ∈C,thefollowingmultiplicationformulaforthe

Apostol-Eulerpolynomialsofhigherorderholdstrue:
(cid:5) (cid:6) (cid:5) (cid:6)
(cid:7)
α m
E(α)(rx;λ)=rn (–λ)mE(α) x+ ;λr , rodd, (.)
n v ,v ,...,v n r
v,v,...,vr–≥   r–
(cid:5) (cid:6)
(cid:7)
(–)lrn l
E(l)(rx;λ)=
n (n+) v ,v ,...,v
l ≤v,v,...,vr–≤l   r–
v+v+···+vr–=l
(cid:5) (cid:6)
m
×(–λ)mB(l) x+ ;λr , reven, (.)
n+l r
wherem=v +v +···+(r–)v .
  r–
LuandLuoAdvancesinDifferenceEquations (2015) 2015:137 Page7of16
Takingμ=ν=in(.)and(.),andnoting(.),wecanobtainthefollowingmain
result(see[],p.,Corollary.).
Corollary. Forα,r∈N,n,l∈N ,λ∈C,thefollowingmultiplicationformulaforthe

Apostol-Genocchipolynomialsofhigherorderholdstrue:
(cid:5) (cid:6) (cid:5) (cid:6)
(cid:7)
α m
G(α)(rx;λ)=rn–α (–λ)mG(α) x+ ;λr , rodd, (.)
n v ,v ,...,v n r
v,v,...,vr–≥   r–
(cid:5) (cid:6)
(cid:7)
l
G(l)(rx;λ)=(–)lrn–l
n v ,v ,...,v
≤v,v,...,vr–≤l   r–
v+v+···+vr–=l
(cid:5) (cid:6)
m
×(–λ)mB(l) x+ ;λr , reven, (.)
n r
wherem=v +v +···+(r–)v .
  r–
Takingλ=–(β)b,μ=–κ,ν=κin(.),andnoting(.),wecanobtainthefollowing
a
multiplication formulasfor the polynomials Y(α)(x;κ,a,b) and Y (x;κ,a,b) defined by
n,β n,β
(.)and(.),respectively.
Corollary. Forκ,μ,ν,m,n,l,r∈N ,α,λ∈C,wehave

Y(α)(rx;κ,a,b)
n,β
(cid:5) (cid:6)(cid:5) (cid:6) (cid:5) (cid:6)
(cid:7) α β bm m
=rn–κα a(r–)bαY(α) x+ ;κ;a;br (.)
v ,v ,...,v a n,β r
v,v,...,vr–≥   r–
(cid:5) (cid:6)(cid:5) (cid:6) (cid:5) (cid:6)
(cid:7) α β bm m
=rn–κα a(r–)bαY(α) x+ ;κ;ar;b , (.)
v ,v ,...,v a n,βr r
v,v,...,vr–≥   r–
wherem=v +v +···+(r–)v .
  r–
Settingα=l=in(.)and(.),respectively,wehave(see[],p.,Theorem)
thefollowing.
Corollary. Forκ,μ,ν,n,r∈N ,λ∈C,wehave

(cid:5) (cid:6) (cid:5) (cid:6)
(cid:7)r– β bj j
Y (rx;κ,a,b)=rn–κ a(r–)bY x+ ;κ;a;br (.)
n,β n,β
a r
j=
(cid:5) (cid:6) (cid:5) (cid:6)
(cid:7)r– β bj j
=rn–κ a(r–)bYn,βr x+ ;κ;ar;b . (.)
a r
j=
Remark. In[],p.,Theorem.,oneofthemainresultofÖzarslanisnotright,
thecorrectformshouldbe(.)and(.)ofCorollary..
Remark . In fact, setting λ=–(β)b, μ=–κ, ν =κ in (.) and noting (.), we
a
deducethemultiplicationformulaswhicharerightonlywhenr isodd.Inthesameway
LuandLuoAdvancesinDifferenceEquations (2015) 2015:137 Page8of16
astheproofof[],p.,Theorem.,wecanobtainthemultiplicationformulas(.)
and(.)ofCorollary..
3 AunifiedrepresentationinconjunctionwiththeGausshypergeometric
function
In this section we obtain a unified representation of the Apostol-type polynomials
F(l)(x;λ;μ;ν)withtheGaussianhypergeometricfunctions.
n
Theorem. Forμ,ν,n,l∈N ,λ∈C,wehave

F(l)(x;λ;μ;ν)
n
(cid:5) (cid:6) (cid:5) (cid:6)(cid:5) (cid:6)
n (cid:7)n–νl l+k– n–νl (–λ)k
=μl(νl)!
νl k k (λ+)l+k
k=
(cid:5) (cid:6) (cid:5) (cid:6)
(cid:7)k
k m
× (–)m mk(x+m)n–νl–k F –n+νl+k,k;k+; , (.)
 
m m+x
m=
whereF(a,b;c;z)denotesGaussianhypergeometricfunctionsdefinedby(see[],p.,Eq.
())
(cid:7)∞
(a) (b) zn
F(a,b;c;z):= n n , |z|<. (.)
(c) n!
n
n=
Proof Lettingα=l∈Nin(.),wehave
(cid:5) (cid:6)
(cid:7)∞ zn μzν l
F(l)(x;λ;μ;ν) = exz. (.)
n n! λez+
n=
Differentiatingbothsidesof (.)withrespecttothevariablezyields
(cid:13)(cid:5) (cid:6) (cid:14)
μzν l
F(l)(x;λ;μ;ν)=Dn exz
n z λez+
z=
(cid:5) (cid:6)
(cid:7)n (cid:15) (cid:8) (cid:9) (cid:16)
n
=μl xn–sDs zνl λez+ –l
s z z=
s=
(cid:5) (cid:6) (cid:5) (cid:6)
(cid:7)n (cid:15)(cid:8) (cid:9) (cid:16)
n s
=μl xn–s(νl)! Ds–νl λez+ –l
s νl z z=
s=νl
(cid:5) (cid:6) (cid:5) (cid:6)
(cid:7)n (cid:15)(cid:8) (cid:8) (cid:9)(cid:9) (cid:16)
n s
=μl xn–s(νl)! Ds–νl λ++λ ez– –l ,
s νl z z=
s=νl
whereD = d isthedifferentialoperator.
z dz
Applyingthegeneralizedbinomialtheorem
(cid:5) (cid:6) (cid:5) (cid:10) (cid:10) (cid:6)
(cid:7)∞ (cid:10) (cid:10)
(a+b)–α= α+l– a–α–l(–b)l α∈C,(cid:10)(cid:10)b(cid:10)(cid:10)<
l a
l=
LuandLuoAdvancesinDifferenceEquations (2015) 2015:137 Page9of16
and the generating function of the Stirling numbers of the second kind S(n,k) (see, for
details,[],p.,TheoremA),
(cid:7)∞
(ez–)k zn
= S(n,k) ,
k! n!
n=
wefindthat
F(l)(x;λ;μ;ν)
n
(cid:5) (cid:6) (cid:5) (cid:6) (cid:5) (cid:6)
(cid:7)n (cid:7)∞ (cid:15)(cid:8) (cid:9) (cid:16)
n s l+k–
=μl xn–s(νl)! (λ+)–l–k(–λ)kDs–νl ez– k
s νl k z z=
s=νl k=
(cid:5) (cid:6) (cid:5) (cid:6) (cid:5) (cid:6)
(cid:7)n (cid:7)s–νl
n s l+k–
=μl xn–s(νl)! (–λ)k(λ+)–l–kk!S(s–νl,k).
s νl k
s=νl k=
Noting(see[],p.,Eq.())
(cid:5) (cid:6)
(cid:7)k
 k
S(n,k)= (–)k–j jn
k! j
j=
andthewell-knowncombinatorialidentity
(cid:5) (cid:6)(cid:5) (cid:6) (cid:5) (cid:6)(cid:5) (cid:6)
n k n n–s
= ,
k s s n–k
wereadilyobtain
F(l)(x;λ;μ;ν)
n
(cid:5) (cid:6) (cid:5) (cid:6) (cid:5) (cid:6)
(cid:7)n (cid:7)s–νl
n s l+k–
=μl xn–s(νl)!
s νl k
s=νl k=
(cid:5) (cid:6)
(cid:7)k
k
×(–λ)k(λ+)–l–k (–)k–m ms–νl
m
m=
(cid:5) (cid:6) (cid:5) (cid:6)(cid:5) (cid:6) (cid:5) (cid:6)
n (cid:7)n–νl (cid:7)n n–νl l+k– (–λ)kxn–s (cid:7)k k
=μl(νl)! (–)k–m ms–νl
νl n–s k (λ+)l+k m
k=s=k+νl m=
(cid:5) (cid:6) (cid:5) (cid:6)(cid:5) (cid:6)
n (cid:7)n–νln(cid:7)–k–νl n–νl l+k–
=μl(νl)!
νl n–s–νl–k k
k= s=
(cid:5) (cid:6)
(–λ)kxn–s–k–νl (cid:7)k k
× (–)k–m ms+k
(λ+)l+k m
m=
(cid:5) (cid:6) (cid:5) (cid:6) (cid:5) (cid:6)
n (cid:7)n–νl l+k– (–λ)kxn–k–νl (cid:7)k k
=μl(νl)! (–)k–m mk
νl k (λ+)l+k m
k= m=
(cid:5) (cid:6)(cid:5) (cid:6)
n(cid:7)–k–νl n–νl m s
× .
n–s–νl–k x
s=
LuandLuoAdvancesinDifferenceEquations (2015) 2015:137 Page10of16
(cid:8) (cid:9)
Notingthat(inviewof n =whenk>nork<)
k
(cid:5) (cid:6) (cid:5) (cid:6)
(cid:7)n (cid:7)∞
n n
= ,
k k
k= k=
andcombiningthedefinitionoftheGaussianhypergeometricfunction
(cid:7)∞
(a) (b) zn
n n
F (a,b;c;z):= ,
 
(c) n!
n
n=
weobtain
(cid:5) (cid:6) (cid:5) (cid:6)(cid:5) (cid:6)
n (cid:7)n–νl l+k– n–νl
F(l)(x;λ;μ;ν)=μl(νl)!
n νl k k
k=
(cid:5) (cid:6)
(–λ)kxn–k–νl (cid:7)k k
× (–)m mk
(λ+)l+k m
m=
(cid:5) (cid:6)
m
× F –n+νl+k,;k+;– . (.)
 
x
ApplyingthePfaff-Kummerhypergeometrictransformation[],p.,Eq.(..),
(cid:5) (cid:6)
(cid:8) (cid:10) (cid:10) (cid:9)
F (a,b;c;z)=(–z)–a F a,c–b;c; z c∈/ Z–:(cid:10)arg(–z)(cid:10)≤π–(cid:10)(<(cid:10)<π) ,
    z– 
to(.),wearriveatthedesiredequation,(.).Thiscompletesourproof. (cid:2)
Belowweshowsomespecialcasesof (.).
Lettingλ(cid:10)−→–λ,takingμ=andν=in(.)andnoting(.),weeasilyobtainthe
followingexplicitformulafortheApostol-Bernoullipolynomials:
(cid:5) (cid:6) (cid:5) (cid:6)(cid:5) (cid:6)
n (cid:7)n–l l+k– n–l λk
B(l)(x;λ)=l!
n l k k (λ–)k
k=
(cid:5) (cid:6) (cid:5) (cid:6)
(cid:7)k
k m
× (–)m mk(x+m)n–l–k F –n+l+k,k;k+; , (.)
 
m m+x
m=
with n,l∈N , λ∈C\{}, which is just the main result of Luo and Srivastava (see [],

p.,Theorem).
Takingμ=andν =in(.)andnoting(.),wecanobtainthefollowingexplicit
formulafortheApostol-Eulerpolynomials:
(cid:5) (cid:6)(cid:5) (cid:6)
(cid:7)n l+k– n (–λ)k
E(l)(x;λ)=l
n k k (λ+)l+k
k=
(cid:5) (cid:6) (cid:5) (cid:6)
(cid:7)k
k m
× (–)m mk(x+m)n–k F –n+k,k;k+; , (.)
 
m m+x
m=
withn,l∈N ,λ∈C\{–},whichisjustthemainresultofLuo(see[],p.,Theorem).
