Some properties of the generalized Apostol-type polynomials

13 Pages·2013·0.23 MB·English

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RESEARCH OpenAccess
Some properties of the generalized
Apostol-type polynomials
Da-QianLu1andQiu-MingLuo2*
*Correspondence:
[email protected] Abstract
2DepartmentofMathematics,
Inthispaper,westudysomepropertiesofthegeneralizedApostol-typepolynomials
ChongqingNormalUniversity,
ChongqingHigherEducationMega (see(LuoandSrivastavainAppl.Math.Comput.217:5702-5728,2011)),includingthe
Center,HuxiCampus,Chongqing, recurrencerelations,thedifferentialequationsandsomeotherconnectedproblems,
401331,People’sRepublicofChina
whichextendsomeknownresults.Wealsodeducesomepropertiesofthe
Fulllistofauthorinformationis
availableattheendofthearticle generalizedApostol-Eulerpolynomials,thegeneralizedApostol-Bernoulli
polynomials,andApostol-Genocchipolynomialsofhighorder.
MSC: Primary11B68;secondary33C65
Keywords: generalizedApostoltypepolynomials;recurrencerelations;differential
equations;connectedproblems;quasi-monomial
1 Introduction,definitionsandmotivation
TheclassicalBernoullipolynomialsB (x),theclassicalEulerpolynomialsE (x) andthe
n n
classicalGenocchipolynomialsG (x),togetherwiththeirfamiliargeneralizationsB(α)(x),
n n
E(α)(x)andG(α)(x)of(realorcomplex)orderα,areusuallydefinedbymeansofthefol-
n n
lowinggeneratingfunctions(see,fordetails,[],pp.-and[],p.etseq.;seealso
[]andthereferencescitedtherein):
(cid:2) (cid:3)
z α (cid:4)∞ zn (cid:5) (cid:6)
·exz= B(α)(x) |z|<π , (.)
ez– n n!
n=
(cid:2) (cid:3)
 α (cid:4)∞ zn (cid:5) (cid:6)
·exz= E(α)(x) |z|<π (.)
ez+ n n!
n=
and
(cid:2) (cid:3)
z α (cid:4)∞ zn (cid:5) (cid:6)
·exz= G(α)(x) |z|<π . (.)
ez+ n n!
n=
Sothat,obviously,theclassicalBernoullipolynomialsB (x),theclassicalEulerpolynomi-
n
alsE (x)andtheclassicalGenocchipolynomialsG (x)aregiven,respectively,by
n n
B (x):=B()(x), E (x):=E()(x) and G (x):=G()(x) (n∈N ). (.)
n n n n n n 
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FortheclassicalBernoullinumbers B ,theclassicalEulernumbers E andtheclassical
n n
GenocchinumbersG ofordern,wehave
n
B :=B ()=B()(), E :=E ()=E()() and G :=G ()=G()(), (.)
n n n n n n n n n
respectively.
Some interesting analogues of the classical Bernoulli polynomials and numbers were
first investigated by Apostol (see [], p., Eq. (.)) and (more recently) by Srivastava
(see[],pp.-).WebeginbyrecallinghereApostol’sdefinitionsasfollows.
Definition.(Apostol[];seealsoSrivastava[]) TheApostol-Bernoullipolynomials
B (x;λ)(λ∈C)aredefinedbymeansofthefollowinggeneratingfunction:
n
(cid:4)∞
zexz zn
= B (x;λ)
λez– n n!
n=
(cid:5) (cid:6)
|z|<π whenλ=;|z|<|logλ|whenλ(cid:4)= (.)
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∈N aredefinedbymeansofthefollowinggeneratingfunction:

(cid:2) (cid:3)
z α (cid:4)∞ zn
·exz= B(α)(x;λ)
λez– n n!
n=
(cid:5) (cid:6)
|z|<π whenλ=;|z|<|logλ|whenλ(cid:4)= (.)
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-EulerpolynomialsE(α)(x;λ)(λ∈C)oforderα∈N
n 
aredefinedbymeansofthefollowinggeneratingfunction:
(cid:2) (cid:3)
 α·exz=(cid:4)∞ E(α)(x;λ)zn (cid:5)|z|<(cid:7)(cid:7)log(–λ)(cid:7)(cid:7)(cid:6) (.)
λez+ n n!
n=
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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-Genocchi
polynomialsof(realorcomplex)orderα,whicharedefinedasfollows:
Definition. TheApostol-GenocchipolynomialsG(α)(x;λ)(λ∈C)oforderα∈N are
n 
definedbymeansofthefollowinggeneratingfunction:
(cid:2) (cid:3)
z α·exz=(cid:4)∞ G(α)(x;λ)zn (cid:5)|z|<(cid:7)(cid:7)log(–λ)(cid:7)(cid:7)(cid:6) (.)
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-Genocchi numbers of order α and the Apostol-Genocchi polynomials, respec-
tively.
Recently, Luo and Srivastava [] introduced a unification (and generalization) of the
above-mentionedthreefamiliesofthegeneralizedApostoltypepolynomials.
Definition . (Luo and Srivastava []) The generalized Apostol type polynomials
F(α)(x;λ;u,v)(α∈N ,λ,u,v∈C)oforderαaredefinedbymeansofthefollowinggener-
n 
atingfunction:
(cid:2) (cid:3)
uzv α·exz=(cid:4)∞ F(α)(x;λ;u,v)zn (cid:5)|z|<(cid:7)(cid:7)log(–λ)(cid:7)(cid:7)(cid:6), (.)
λez+ n n!
n=
where
F(α)(λ;u,v):=F(α)(;λ;u,v) (.)
n n
denotetheso-calledApostoltypenumbersoforderα.
Sothat,bycomparingDefinition.withDefinitions.,.and.,wehave
B(α)(x;λ)=(–)αF(α)(x;–λ;,), (.)
n n
E(α)(x;λ)=F(α)(x;λ;,), (.)
n n
G(α)(x;λ)=F(α)(x;λ;,). (.)
n n
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Apolynomialp (x)(n∈N,x∈C)issaidtobeaquasi-monomial[],whenevertwoop-
n
ˆ ˆ
eratorsM,P,calledmultiplicativeandderivative(orlowering)operatorsrespectively,can
bedefinedinsuchawaythat
ˆ
Pp (x)=np (x), (.)
n n–
ˆ
Mp (x)=p (x), (.)
n n+
whichcanbecombinedtogettheidentity
ˆ ˆ
MPp (x)=np (x). (.)
n n
The Appell polynomials [] can be defined by considering the following generating
function:
(cid:4)∞
R (x)
A(t)ext= n tn, (.)
n!
n=
where
(cid:4)∞ (cid:5) (cid:6)
R
A(t)= ktk A()(cid:4)= (.)
k!
k=
isanalyticfunctionatt=.
From[],weknowthatthemultiplicativeandderivativeoperatorsofR (x)are
n
Mˆ =(x+α )+(cid:4)n– αn–k Dn–k, (.)
 (n–k)! x
k=
ˆ
P=D , (.)
x
where
A(cid:5)(t) (cid:4)∞ tn
= α . (.)
n
A(t) n!
n=
Byusing(.),wehavethefollowinglemma.
Lemma.([]) TheAppellpolynomialsR (x)definedby(.)satisfythedifferential
n
equation:
α α α
n– y(n)+ n– y(n–)+···+ y(cid:5)(cid:5)+(x+α )y(cid:5)–ny=, (.)

(n–)! (n–)! !
wherethenumericalcoefficientsα ,k=,,...,n–aredefinedin(.),andarelinked
k
tothevaluesR bythefollowingrelations:
k
(cid:2) (cid:3)
(cid:4)k
k
R = R α .
k+ h k–h
h
h=
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LetP bethevectorspaceofpolynomialswithcoefficientsinC.Apolynomialsequence
{Pn}n≥beapolynomialset.{Pn}n≥iscalledaσ-Appellpolynomialsetoftransferpower
seriesAisgeneratedby
(cid:4)∞
P (x)
G(x,t)=A(t)G (x,t)= n tn, (.)

n!
n=
whereG (x,t)isasolutionofthesystem:

σG (x,t)=tG (x,t),
 
G (x,)=.

In[],theauthorsinvestigatedtheconnectioncoefficientsbetweentwopolynomials.
Andthereisaresultaboutconnectioncoefficientsbetweentwoσ-Appellpolynomialsets.
Lemma.([]) Letσ ∈(cid:6)(–).Let{Pn}n≥and{Qn}n≥betwoσ-Appellpolynomialsets
oftransferpowerseries,respectively,A andA .Then
 
(cid:4)n
n!
Q (x)= α P (x), (.)
n n–m m
m!
m=
where
(cid:4)∞
A (t)
 = α tk.
k
A (t)

k=
Inrecentyears,severalauthorsobtainedmanyinterestingresultsinvolvingtherelated
BernoullipolynomialsandEulerpolynomials[,–].Andin[],theauthorsstudied
someseriesidentitiesinvolvingthegeneralizedApostoltypeandrelatedpolynomials.
Inthispaper,westudysomeotherpropertiesofthegeneralizedApostoltypepolynomi-
alsF(α)(x;λ;u,v),includingtherecurrencerelations,thedifferentialequationsandsome
n
connectionproblems,whichextendsomeknownresults.Asspecial,weobtainsomeprop-
erties of the generalized Apostol-Euler polynomials, the generalized Apostol-Bernoulli
polynomialsandApostol-Genocchipolynomialsofhighorder.
2 Recursionformulasanddifferentialequations
Fromthegeneratingfunction(.),wehave
∂
F(α)(x;λ;u,v)=nF(α)(x;λ;u,v). (.)
∂x n n–
ArecurrencerelationforthegeneralizedApostoltypepolynomialsisgivenbythefol-
lowingtheorem.
Theorem. Foranyintegraln≥,λ∈Candα∈N,thefollowingrecurrencerelation
forthegeneralizedApostoltypepolynomialsF(α)(x;λ;u,v)holdstrue:
n
(cid:2) (cid:3)
αv αλ n!
– F(α)(x;λ;u,v)= · F(α+)(x+;λ;u,v)–xF(α)(x;λ;u,v). (.)
n+ n+ u (n+v)! n+v n
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Proof Differentiating both sides of (.) with respect to t, and using some elementary
algebraandtheidentityprincipleofpowerseries,recursion(.)easilyfollows. (cid:2)
Bysettingλ:=–λ,u=andv=inTheorem.,andthenmultiplying(–)α onboth
sidesoftheresult,wehave:
Corollary. Foranyintegraln≥,λ∈Candα∈N,thefollowingrecurrencerelation
forthegeneralizedApostol-BernoullipolynomialsB(α)(x;λ)holdstrue:
n
(cid:8) (cid:9)
α–(n+) B(α)(x;λ)=αλB(α+)(x+;λ)–xB(α)(x;λ). (.)
n+ n+ n
Bysettingu=andv=inTheorem.,wehavethefollowingcorollary.
Corollary. Foranyintegraln≥,λ∈Candα∈N,thefollowingrecurrencerelation
forthegeneralizedApostol-EulerpolynomialsE(α)(x;λ)holdstrue:
n
αλ
E(α)(x;λ)=xE(α)(x;λ)– E(α+)(x+;λ). (.)
n+ n  n
Bysettingu=andv=inTheorem.,wehavethefollowingcorollary.
Corollary. Foranyintegraln≥,λ∈Candα∈N,thefollowingrecurrencerelation
forthegeneralizedApostol-GenocchipolynomialsG(α)(x;λ)holdstrue:
n
(cid:8) (cid:9)
 α–(n+) G(α)(x;λ)=αλG(α+)(x+;λ)–(n+)xG(α)(x;λ). (.)
n+ n+ n
From (.) and (.), we know that the generalized Appostol type polynomials
F(α)(x;λ;u,v)isAppellpolynomialswith
n
(cid:2) (cid:3)
utv α
A(t)= . (.)
λet+
FromtheEq.()of[],weknowthatG (;λ)=.Sofrom(.)and(.),wecanobtain

thatifv=,wehave
A(cid:5)(t) λα(cid:4)∞ G (;λ) tn
= n+ · . (.)
A(t)  n+ n!
n=
Byusing(.)and(.),wecanobtainthemultiplicativeandderivativeoperatorsofthe
generalizedAppostoltypepolynomialsF(α)(x;λ;u,v)
n
(cid:2) (cid:3)
Mˆ = x+ λαG (;λ) + λα(cid:4)n– Gn–k+(;λ)Dn–k, (.)
   (n–k+)! x
k=
ˆ
P=D . (.)
x
From(.),wecanobtain
∂p n!
F(α)(x;λ;u,v)= F(α)(x;λ;u,v). (.)
∂xp n (n–p)! n–p
Thenbyusing(.),(.)and(.),weobtainthefollowingresult.
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Theorem. Foranyintegraln≥,λ∈Candα∈N,thefollowingrecurrencerelation
forthegeneralizedApostoltypepolynomialsF(α)(x;λ;u,)holdstrue:
n
(cid:2) (cid:3)
λα
F(α)(x;λ;u,)= x+ G (;λ) F(α)(x;λ;u,)
n+   n
(cid:2) (cid:3)
λα(cid:4)n– n G (;λ)
+ n–k+ F(α)(x;λ;u,). (.)
 k n–k+ n–k
k=
Bysettingu=inTheorem.,wehavethefollowingcorollary.
Corollary. Foranyintegraln≥,λ∈Candα∈N,thefollowingrecurrencerelation
forthegeneralizedApostol-EulerpolynomialsE(α)(x;λ)holdstrue:
n
(cid:2) (cid:3) (cid:2) (cid:3)
λα λα(cid:4)n– n G (;λ)
E(α)(x;λ)= x+ G (;λ) E(α)(x;λ)+ n–k+ E(α)(x;λ). (.)
n+   n  k n–k+ n–k
k=
Furthermore,applyingLemma.toF(α)(x;λ;u,),wehavethefollowingtheorem.
n
Theorem. ThegeneralizedApostoltypepolynomialsF(α)(x;λ;u,)satisfythediffer-
n
entialequation:
λαG (;λ) λαG (;λ)
n y(n)+ n– y(n–)+···
 n!  (n–)!
(cid:2) (cid:3)
λαG (;λ) λα
+  y(cid:5)(cid:5)+ x+ G (;λ) y(cid:5)–ny=. (.)

  
Specially,bysettingu=inTheorem.,thenwehavethefollowingcorollary.
Corollary. ThegeneralizedApostol-EulerpolynomialsE(α)(x;λ)satisfythedifferential
n
equation:
λαG (;λ) λαG (;λ)
n y(n)+ n– y(n–)+···
 n!  (n–)!
(cid:2) (cid:3)
λαG (;λ) λα
+  y(cid:5)(cid:5)+ x+ G (;λ) y(cid:5)–ny=. (.)

  
3 Connectionproblems
From(.)and(.),weknowthatthegeneralizedApostoltypepolynomialsF(α)(x;λ;
n
u,v)areaD -Appellpolynomialset,whereD denotesthederivativeoperator.
x x
FromTable  in [],weknow thatthederivative operatorsofmonomials xn andthe
Gould-Hopperpolynomialsgm(x,h)[]areallD .AndtheirtransferpowerseriesA(t)
n x
areandehtm,respectively.
ApplyingLemma.toP (x)=xnandQ (x)=F(α)(x;λ;u,v),wehavethefollowingthe-
n n n
orem.
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Theorem.
(cid:2) (cid:3)
(cid:4)n
n
F(α)(x;λ;u,v)= F(α) (λ;u,v)xm, (.)
n m n–m
m=
whereF(α)(λ;u,v)istheso-calledApostoltypenumbersoforderαdefinedby(.).
n
Bysettingλ:=–λ,u=andv=inTheorem.,andthenmultiplying(–)α onboth
sidesoftheresult,wehavethefollowingcorollary.
Corollary.
(cid:2) (cid:3)
(cid:4)n
n
B(α)(x;λ)= B(α) (λ)xm, (.)
n m n–m
m=
whichisjustEq.(.)of[].
Bysettingu=andv=inTheorem.,wehavethefollowingcorollary.
Corollary.
(cid:2) (cid:3)
(cid:4)n
n
E(α)(x;λ)= E(α) (λ)xm. (.)
n m n–m
m=
Bysettingu=andv=inTheorem.,wehavethefollowingcorollary.
Corollary.
(cid:2) (cid:3)
(cid:4)n
n
G(α)(x;λ)= G(α) (λ)xm, (.)
n m n–m
m=
whichisjustEq.()of[].
ApplyingLemma.toP (x)=F (x;λ;u,v)andQ (x)=F(α)(x;λ;u,v),wehavethefol-
n n n n
lowingtheorem.
Theorem.
(cid:2) (cid:3)
(cid:4)n
n
F(α)(x;λ;u,v)= F(α–)(λ;u,v)F (x;λ;u,v), (.)
n m n–m m
m=
whereF(α)(λ;u,v)istheso-calledApostoltypenumbersoforderαdefinedby(.).
n
Bysettingλ:=–λ,u=andv=inTheorem.,andthenmultiplying(–)α onboth
sidesoftheresult,wehavethefollowingcorollary.
Corollary.
(cid:2) (cid:3)
(cid:4)n
n
B(α)(x;λ)= B(α–)(λ)B (x;λ), (.)
n m n–m m
m=
whichisjustEq.(.)of[].
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Bysettingu=andv=inTheorem.,wehavethefollowingcorollary.
Corollary.
(cid:2) (cid:3)
(cid:4)n
n
E(α)(x;λ)= E(α–)(λ)E (x;λ). (.)
n m n–m m
m=
Bysettingu=andv=inTheorem.,wehavethefollowingcorollary.
Corollary.
(cid:2) (cid:3)
(cid:4)n
n
G(α)(x;λ)= G(α–)(λ)G (x;λ). (.)
n m n–m m
m=
ApplyingLemma.toP (x)=gm(x,h)andQ (x)=F(α)(x;λ;u,v),wehavethefollowing
n n n n
theorem.
Theorem.
(cid:10) (cid:11)
(cid:4)n n! [(n(cid:4)–r)/m] hk
F(α)(x;λ;u,v)= (–)k F(α) (λ;u,v) gm(x,h). (.)
n r! k!(n–r–mk)! n–r–mk r
r= k=
Bysettingλ:=–λ,u=andv=inTheorem.,andthenmultiplying(–)α onboth
sidesoftheresult,wehavethefollowingcorollary.
Corollary.
(cid:10) (cid:11)
(cid:4)n n! [(n(cid:4)–r)/m] hk
B(α)(x;λ)= (–)k B(α) (λ) gm(x,h), (.)
n r! k!(n–r–mk)! n–r–mk r
r= k=
whichisjustEq.(.)of[].
Bysettingu=andv=inTheorem.,wehavethefollowingcorollary.
Corollary.
(cid:10) (cid:11)
(cid:4)n n! [(n(cid:4)–r)/m] hk
E(α)(x;λ)= (–)k E(α) (λ) gm(x,h). (.)
n r! k!(n–r–mk)! n–r–mk r
r= k=
Bysettingu=andv=inTheorem.,wehavethefollowingcorollary.
Corollary.
(cid:10) (cid:11)
(cid:4)n n! [(n(cid:4)–r)/m] hk
G(α)(x;λ)= (–)k G(α) (λ) gm(x,h). (.)
n r! k!(n–r–mk)! n–r–mk r
r= k=
Whenvα=,applyingLemma.toP (x)=E(α–)(x;λ)andQ (x)=F(α)(x;λ;u,v),we
n n n n
havethefollowingtheorem.
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Theorem. Ifvα=,thenwehave
(cid:2) (cid:3)
(cid:4)n
n
F(α)(x;λ;u,v)= (u–)αG (λ)E(α–)(x;λ). (.)
n m n–m m
m=
Bysettingλ:=–λ,u=andv=inTheorem.,andthenmultiplying(–)α onboth
sidesoftheresult,wehavethefollowingcorollary.
Corollary.
(cid:2) (cid:3)
(cid:4)n
 n
B (x;λ)=– G (–λ)xm. (.)
n n–m
 m
m=
Bysettingu=andv=inTheorem.,wehavethefollowingcorollary.
Corollary.
(cid:2) (cid:3)
(cid:4)n
 n
G (x;λ)=– G (λ)xm, (.)
n n–m
 m
m=
whichisjustthecaseofα=in(.).
When v= or α=,applyingLemma.to P (x)=G(α–)(x;λ) and Q (x)=F(α)(x;λ;
n n n n
u,v),wecanobtainthefollowingtheorem.
Theorem. Ifv=orα=,wehave
(cid:2) (cid:3)
(cid:4)n
n
F(α)(x;λ;u,v)= (u–)αG (λ)G(α–)(x;λ). (.)
n m n–m m
m=
Bysettingλ:=–λ,u=andv=inTheorem.,andthenmultiplying(–)α onboth
sidesoftheresult,wehavethefollowingcorollary.
Corollary.
(cid:2) (cid:3)(cid:2) (cid:3)
(cid:4)n n  α
B(α)(x;λ)= – G (–λ)G(α–)(x;–λ). (.)
n m  n–m m
m=
Whenα=in(.),itisjust(.).
Bysettingu=andv=inTheorem.,wehavethefollowingcorollary.
Corollary.
(cid:2) (cid:3)
(cid:4)n
n
G(α)(x;λ)= G (λ)G(α–)(x;λ), (.)
n m n–m m
m=
whichisequalto(.).