Hermite-based unified Apostol-Bernoulli, Euler and Genocchi polynomials

13 Pages·2013·0.3 MB·English

ÖzarslanAdvancesinDifferenceEquations2013,2013:116
http://www.advancesindifferenceequations.com/content/2013/1/116
RESEARCH OpenAccess
Hermite-based unified Apostol-Bernoulli,
Euler and Genocchi polynomials
MehmetAliÖzarslan*
*Correspondence:
[email protected] Abstract
DepartmentofMathematics,
Inthispaper,weintroduceaunifiedfamilyofHermite-basedApostol-Bernoulli,Euler
FacultyofArtsofSciences,Eastern
MediterraneanUniversity, andGenocchipolynomials.Weobtainsomesymmetryidentitiesbetweenthese
Gazimagusa,NorthCyprus, polynomialsandthegeneralizedsumofintegerpowers.Wegiveexplicitclosed-form
Mersin10,Turkey
formulaeforthisunifiedfamily.Furthermore,weproveafiniteseriesrelationbetween
thisunificationand3d-Hermitepolynomials.
MSC: Primary11B68;secondary33C05
Keywords: Hermite-basedApostol-Bernoullipolynomials;Hermite-based
Apostol-Eulerpolynomials;Hermite-basedApostol-Genocchipolynomials;
generalizedsumofintegerpowers;generalizedsumofalternativeintegerpowers
1 Introduction
Recently,Khanetal.[]introducedtheHermite-basedAppellpolynomialsviathegener-
atingfunction
G(x,y,z;t)=A(t)exp(Mt),
where
∂ ∂
M=x+y +z
∂x ∂x
isthemultiplicativeoperatorofthe-variableHermitepolynomials,whicharedefinedby
(cid:2) (cid:3) (cid:4)∞ tn
exp xt+yt+zt = H()(x,y,z) (.)
n n!
n=
and
(cid:4)∞
A(t)= a tn, a (cid:3)=.
n 
n=
ByusingtheBerrydecouplingidentity,
eA+B=em/e((–m)A/+A)eB, [A,B]=mA/
©2013Özarslan;licenseeSpringer.ThisisanOpenAccessarticledistributedunderthetermsoftheCreativeCommonsAttribu-
tionLicense(http://creativecommons.org/licenses/by/2.0),whichpermitsunrestricteduse,distribution,andreproductioninany
medium,providedtheoriginalworkisproperlycited.
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theyobtainedthegeneratingfunctionoftheHermite-basedAppellpolynomials A (x,
H n
y,z)as
(cid:2) (cid:3) (cid:4)∞ tn
G(x,y,z;t)=A(t)exp xt+yt+zt = A (x,y,z) .
H n
n!
n=
LettingA(t)= t ,theydefinedHermite-Bernoullipolynomials B (x,y,z)by
et– H n
t (cid:2) (cid:3) (cid:4)∞ tn
exp xt+yt+zt = B (x,y,z) , |t|<π.
et– H n n!
n=
ForA(t)=  ,theydefinedHermite-Eulerpolynomials E (x,y,z)by
et+ H n
 (cid:2) (cid:3) (cid:4)∞ tn
exp xt+yt+zt = E (x,y,z) , |t|<π
et+ H n n!
n=
andforA(t)= t ,theydefinedHermite-Genocchipolynomials G (x,y,z)by
et+ H n
t (cid:2) (cid:3) (cid:4)∞ tn
exp xt+yt+zt = G (x,y,z) , |t|<π.
et+ H n n!
n=
Recently,theauthorconsideredthefollowingunificationoftheApostol-Bernoulli,Euler
andGenocchipolynomials
(cid:5) (cid:6)
–ktk α (cid:4)∞ tn
f(α)(x;t;k,β):= ext= P(α)(x;k,a,b)
a,b βbet–ab n,β n!
n=
(cid:2) (cid:3)
k∈N ;a,b∈R\{};α,β∈C

andobtainedtheexplicitrepresentationofthisunifiedfamily,intermsofGaussianhyper-
geometricfunction.Somesymmetryidentitiesandmultiplicationformulaarealsogiven
in[].NotethatthefamilyofpolynomialsP()(x,y,z;k,a,b)wasinvestigatedin[].
n,β
Weorganizethepaperasfollows.
In Section,we introducetheunification oftheHermite-based generalized Apostol-
Bernoulli,EulerandGenocchipolynomials P(α)(x,y,z;k,a,b) andgivesummationfor-
H n,β
mulas for this unification. In Section , we obtain some symmetry identities for these
polynomials.InSection,wegiveexplicitclosed-formformulaeforthisunifiedfamily.
Furthermore, we prove a finite series relation between this unification and d-Hermite
polynomials.
2 Hermite-basedgeneralizedApostol-Bernoulli,EulerandGenocchi
polynomials
Inthispaper,weconsiderthefollowinggeneralclassofpolynomials:
(cid:5) (cid:6)
–ktk α (cid:4)∞ tn
f(α)(x,y,z;t;k,β):= ext+yt+zt = P(α)(x,y,z;k,a,b)
a,b βbet–ab H n,β n!
n=
(cid:2) (cid:3)
k∈N ;a,b∈R\{};α,β∈C . (.)

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Fortheexistenceoftheexpansion,weneed
(i) |t|<π whenα∈C,k=and(β)b=;|t|<π whenα∈N ,k=,,...and
a 
(β)b=;|t|<|blog(β)|whenα∈N ,k∈Nand(β)b(cid:3)=(or(cid:3)=–);x,y,z∈R,β∈C,
a a  a
a,b∈C/{};α:=;
(ii) |t|<π when(β)b=–;|t|<|blog(β)|when(β)b(cid:3)=–;x,y,z∈R,k=,α,β∈C,
a a a
a,b∈C/{};α:=;
(iii) |t|<π whenα∈N and(β)b=–;x,y,z∈R,k∈N,β∈C,a,b∈C/{};α:=,
 a
wherew=|w|eiθ,–π ≤θ <π andlog(w)=log(|w|)+iθ.
Fork=a=b=andβ=λin(.),wedefinethefollowing.
Definition. Letα∈N ,λbeanarbitrary(realorcomplex)parameterandx,y,z∈R.

TheHermite-basedgeneralizedApostol-Bernoullipolynomialsaredefinedby
(cid:5) (cid:6)
t α (cid:2) (cid:3) (cid:4)∞ tn
exp xt+yt+zt = B(α)(x,y,z;λ)
λet– H n n!
n=
(cid:2) (cid:7) (cid:7)
|t|<π whenα∈Candλ=;|t|<(cid:7)log(λ)(cid:7)
(cid:3)
whenα∈N andλ(cid:3)=;x,y,z∈R;α:= .

Itisclearthat
P(α)(x,y,z;,,)= B(α)(x,y,z;λ).
H n,λ H n
Some special cases of the Hermite-based generalized Apostol-Bernoulli polynomials
(someofwhicharedefinition)arelistedbelow:
• B()(x,y,z;λ):= B (x,y,z;λ)iscalledHermite-basedApostol-Bernoulli
H n H n
polynomials.
• B (x,y,z;)= B (x,y,z)istheHermite-Bernoullipolynomials.
H n H n
• B (x,,;λ):=B (x;λ)istheApostol-Bernoullipolynomials(see[–]).When
H n n
λ=,wehavetheclassicalBernoullipolynomials.
• B (;λ):=B (λ)aretheApostol-Bernoullinumbers.λ=givestheclassicalBernoulli
n n
numbers.
Settingk+=–a=b=andβ=λin(.),wegetthefollowing.
Definition . Let α and λ ((cid:3)= –) be an arbitrary (real or complex) parameter and
x,y,z∈R.TheHermite-basedgeneralizedApostol-Eulerpolynomialsaredefinedby
(cid:5) (cid:6)
 α (cid:2) (cid:3) (cid:4)∞ tn
exp xt+yt+zt = E(α)(x,y,z;λ)
λet+ H n n!
n=
(cid:2) (cid:7) (cid:7) (cid:3)
|t|<π whenλ=;|t|<(cid:7)log(–λ)(cid:7)whenλ(cid:3)=;x,y,z∈R,α∈C;α:= .
Obviously,wehave
P(α)(x,y,z;,–,)= E(α)(x,y,z;λ).
H n,λ H n
SomespecialcasesoftheHermite-basedgeneralizedApostol-Eulerpolynomials(some
ofwhicharedefinition)arelistedbelow:
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• E()(x,y,z;λ):= E (x,y,z;λ)iscalledHermite-basedApostol-Eulerpolynomials.
H n H n
• E (x,y,z;)= E (x,y,z)istheHermite-Eulerpolynomials.
H n H n
• E (x,,;λ):=E (x;λ)istheApostol-Eulerpolynomials(see[]).Forλ=,wehave
H n n
theclassicalEulerpolynomials.
• nE (;λ):=E (λ)aretheApostol-Eulernumbers.Thecaseλ=givestheclassical
n  n
Eulernumbers.
Choosingk=–a=b=andβ=λin(.),wedefinethefollowing.
Definition . Let α and λ ((cid:3)= –) be an arbitrary (real or complex) parameter and
x,y,z ∈ R. The Hermite-based generalized Apostol-Genocchi polynomials are defined
by
(cid:5) (cid:6)
t α (cid:2) (cid:3) (cid:4)∞ tn
exp xt+yt+zt = G(α)(x,y,z;λ)
λet+ H n n!
n=
(cid:2) (cid:7) (cid:7)
|t|<π whenα∈N andλ=;|t|<(cid:7)log(–λ)(cid:7)

(cid:3)
whenα∈N andλ(cid:3)=;x,y,z∈R;α:= .

Itiseasilyseenthat
(cid:5) (cid:6)
–
P(α) x,y,z;, , = Gα(x,y,z;λ).
H n,λ  H n

Some special cases of the Hermite-based generalized Apostol-Genocchi polynomials
(someofwhicharedefinition)arelistedbelow:
• G()(x,y,z;λ):= G (x,y,z;λ)iscalledHermite-basedApostol-Genocchi
H n H n
polynomials.
• G (x,y,z;)= G (x,y,z)istheHermite-Genocchipolynomials.
H n H n
• G (x,,;λ):=G (x;λ)istheApostol-Genocchipolynomials(see[,]).When
H n n
λ=,wehavetheclassicalGenocchipolynomials.
• G (;λ):=G (λ)aretheApostol-Genocchinumbers.λ=givestheclassical
n n
Genocchinumbers.
FinallywedefinetheunifiedHermite-basedApostolpolynomialsby
(cid:4)∞
–ktk tn
f()(x;t;k,β):= ext+yt+zt = P (x,y,z;k,a,b)
a,b βbet–ab H n,β n!
n=
(cid:2) (cid:3)
k∈N ;a,b∈R\{};β∈C .

Thusitisclearthat P (x,y,z;k,a,b)= P()(x,y,z;k,a,b)andthatwehavethefollowing
H n,β H n,β
observationsatonce:
• P (x,y,z;,,)= B (x,y,z;λ)aretheHermite-basedApostol-Bernoulli
H n,λ H n
polynomials.
• P (x,y,z;,–,)= E(x,y,z;λ)aretheHermite-basedApostol-Eulerpolynomials.
H n,λ H
• HPn,λ(x,y,z;,–,)=HGn(x,y,z;λ)aretheHermite-basedApostol-Genocchi
polynomials.
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For the other generalization, we refer [–] and []. Now we give some relations
betweentheabovementionedApostolpolynomials.
Using(.),wegetthefollowingidentityatonce.
Theorem . Let α,k ∈N ; a,b∈R\{}; β ∈C be such that the conditions (i)-(iii) are

satisfied.Then,thefollowingrelation
(cid:5) (cid:6)
(cid:4)n
n
P(α) (x,y,z;k,a,b) P(α)(u,v,w;k,a,b)= P(α)(x+u,y+v,z+w;k,a,b)
r H n–r,β H r,β H n,β
r=
holdstrue.
Corollary. Foreachn∈N,thefollowingrelation
(cid:5) (cid:6)
(cid:4)n
n
B(α)(x,y,z;λ) B(β)(u,v,w;λ)= B(α+β)(x+u,y+v,z+w;λ)
k H n–k H k H n
k=
holdstruefortheHermite-basedgeneralizedApostol-Bernoullipolynomials.
Corollary. Foreachn∈N,thefollowingrelation
(cid:5) (cid:6)
(cid:4)n
n
E(α)(x,y,z;λ) E(β)(u,v,w;λ)= E(α+β)(x+u,y+v,z+w;λ)
k H n–k H k H n
k=
holdstruefortheHermite-basedgeneralizedApostol-Eulerpolynomials.
Corollary. Foreachn∈N,thefollowingrelation
(cid:5) (cid:6)
(cid:4)n
n
G(α)(x,y,z;λ) G(β)(u,v,w;λ)= G(α+β)(x+u,y+v,z+w;λ)
k H n–k H k H n
k=
holdstruefortheHermite-basedgeneralizedApostol-Genocchipolynomials.
Theorem. Foreachn∈N,thefollowingrelation
(cid:5) (cid:6) (cid:5) (cid:6)
(cid:4)n
n x+u y+v z+w
B(α)(x,y,z;λ) E(α)(u,v,w;λ)=nB(α) , , ;λ
k H n–k H k H n   
k=
holdstruebetweentheHermite-basedgeneralizedApostol-BernoulliandEulerpolynomi-
als.
Proof Bydirectcalculations,wehave
(cid:5) (cid:6)
(cid:4)∞
x+u y+v z+w (t)n
B(α) , , ;λ
H n    n!
n=
(cid:5) (cid:6) (cid:8)(cid:5) (cid:6) (cid:5) (cid:6) (cid:5) (cid:6) (cid:9)
t α x+u y+v z+w
= exp t+ (t)+ (t)
λet–   
(cid:5) (cid:6) (cid:5) (cid:6)
t α (cid:2) (cid:3)  α (cid:2) (cid:3)
= exp xt+yt+zt exp ut+vt+wt
λet– λet+
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(cid:4)∞ (cid:4)∞
tn tk
= B(α)(x,y,z;λ) E(α)(u,v,w;λ)
H n n! H k k!
n= k=
(cid:5) (cid:6)
(cid:4)∞ (cid:4)n n tn
= B(α)(x,y,z;λ) E(α)(u,v,w;λ) .
k H n–k H k n!
n= k=
Comparingthecoefficientsof tn onbothsides,wegettheresult. (cid:2)
n!
3 Symmetryidentitiesfortheunifiedfamily
(cid:10)
For each k ∈N , the sum S (n)= n ik is known as the power sum and we have the
 k i=
followinggeneratingrelation:
(cid:4)∞
tk e(n+)t–
S (n) =+et+et+···+ent= .
k k! et–
k=
Foranarbitraryrealorcomplexλ,thegeneralizedsumofintegerpowersS (n,λ)isde-
k
fined,in[],viathefollowinggeneratingrelation:
(cid:4)∞
tk λe(n+)t–
S (n,λ) = .
k k! λet–
k=
ItclearthatS (n,)=S (n).
k k (cid:10)
Foreachk∈N ,thesumM (n)= n (–)kikisknownasthesumofalternativeinteger
 k i=
powers.Thefollowinggeneratingrelationisstraightforward:
(cid:4)∞
tk –(–et)(n+)
M (n) =–et+et–···+(–)nent= .
k k! et+
k=
For an arbitrary real or complex λ, the generalized sum of alternative integer powers
M (n,λ)isdefined,in[],by
k
(cid:4)∞
tk –λ(–et)(n+)
M (n,λ) = .
k k! λet+
k=
ClearlyM (n,)=M (n).Ontheotherhand,ifniseven,then
k k
S (n,–λ)=M (n,λ). (.)
k k
Westartbyobtainingcertainsymmetryidentities,whichincludestheresultsgivenin
[–]and[],wheny=z=.
Theorem. Letc,d,m∈N,n∈N besuchthattheconditions(i)-(iii)aresatisfiedwith

treplacedbyctanddt.Thenwehavethefollowingsymmetryidentity:
(cid:5) (cid:6)
(cid:4)n (cid:2) (cid:3)
n
cn–rdr+k P(m) dx,dy,dz;k,a,b
r H n–r,β
r=
(cid:5) (cid:6) (cid:5) (cid:5) (cid:6) (cid:6)
(cid:4)r r β b (cid:2) (cid:3)
× S c–; P(m–) cX,cY,cZ;k,a,b
l l a H r–l,β
l=
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(cid:5) (cid:6)
(cid:4)n (cid:2) (cid:3)
n
= dn–rcr+k P(m) cx,cy,cz;k,a,b
r H n–r,β
r=
(cid:5) (cid:6) (cid:5) (cid:5) (cid:6) (cid:6)
(cid:4)r r β b (cid:2) (cid:3)
× S d–; P(m–) dX,dY,dZ;k,a,b .
l l a H r–l,β
l=
Proof Let
(–k)(m–)tkm–kecdxt+y(cdt)+z(cdt)(βbecdt–ab)ecdXt+Y(cdt)+Z(cdt)
G(t):= .
(βbect–ab)m(βbedt–ab)m
ExpandingG(t)intoaseries,weget
(cid:5) (cid:6) (cid:5) (cid:6)
 –kcktk m βbecdt–ab
G(t)= ecdxt+y(cdt)+z(cdt)
ckmdk(m–) βbect–ab βbedt–ab
(cid:5) (cid:6)
–kdktk m–
× ecdXt+Y(cdt)+Z(cdt)
βbedt–ab
(cid:11) (cid:12)(cid:11) (cid:5) (cid:5) (cid:6) (cid:6) (cid:12)
 (cid:4)∞ (cid:2) (cid:3)(ct)n (cid:4)∞ β b (dt)l
= P(m) dx,dy,dz;k,a,b S c–;
ckmdk(m–) H n,β n! l a l!
n= l=
(cid:11) (cid:12)
(cid:4)∞ (cid:2) (cid:3)(dt)r
× P(m–) cX,cY,cZ;k,a,b .
H r,β r!
r=
Now,usingCorollaryin[,p.],weget
(cid:11) (cid:5) (cid:6)
(cid:4)∞ (cid:4)n (cid:2) (cid:3)
 n
G(t)= cn–rdr+k P(m) dx,dy,dz;k,a,b
ckmdkm r H n–r,β
n= r=
(cid:5) (cid:6) (cid:5) (cid:5) (cid:6) (cid:6) (cid:12)
(cid:4)r r β b (cid:2) (cid:3) tn
× S c–; P(m–) cX,cY,cZ;k,a,b . (.)
l l a H r–l,β n!
l=
Inasimilarmanner,
(cid:5) (cid:6) (cid:5) (cid:6)
 –kdktk m βbecdt–ab
G(t)= ecdxt+y(cdt)+z(cdt)
dkmck(m–) βbect–ab βbedt–ab
(cid:5) (cid:6)
–kcktk m–
× ecdXt+Y(cdt)+Z(cdt)
βbedt–ab
(cid:11) (cid:5) (cid:6)
(cid:4)∞ (cid:4)n (cid:2) (cid:3)
 n
= dn–rcr+k P(m) cx,cy,cz;k,a,b
ckmdkm r H n–r,β
n= r=
(cid:5) (cid:6) (cid:5) (cid:5) (cid:6) (cid:6) (cid:12)
(cid:4)r r β b (cid:2) (cid:3) tn
× S d–; P(m–) dX,dY,dZ;k,a,b . (.)
l l a H r–l,β n!
l=
From(.)and(.),wegettheresult. (cid:2)
Fork=a=b=andβ=λwegetthefollowingcorollaryatonce.
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Corollary. Forallc,d,m∈N,n∈N ,λ∈C,wehavethefollowingsymmetryidentity

fortheHermitebasedgeneralizedApostol-Bernoullipolynomials:
(cid:5) (cid:6)
(cid:4)n (cid:2) (cid:3)
n
cn–rdr+ B(m) dx,dy,dz,λ
r H n–r
r=
(cid:5) (cid:6)
(cid:4)r (cid:2) (cid:3)
r
× S(c–;λ) B(m–) cX,cY,cZ,λ
l l H r–l
l=
(cid:5) (cid:6)
(cid:4)n (cid:2) (cid:3)
n
= dn–rcr+ B(m) cx,cy,cz,λ
r H n–r
r=
(cid:5) (cid:6)
(cid:4)r (cid:2) (cid:3)
r
× S(d–;λ) B(m–) dX,dY,dZ,λ .
l l H r–l
l=
Fork+=–a=b=andβ=λweget,byconsidering(.)that
Corollary. Forallm∈N,n∈N ,λ∈C,wehaveforeachpairofpositiveevenintegers

candd,orforeachpairofpositiveoddintegerscandd,
(cid:5) (cid:6)
(cid:4)n (cid:2) (cid:3)
n
cn–rdr+ E(m) dx,dy,dz,λ
r H n–r
r=
(cid:5) (cid:6)
(cid:4)r (cid:2) (cid:3)
r
× M(c–;λ) E(m–) cX,cY,cZ,λ
l l H r–l
l=
(cid:5) (cid:6)
(cid:4)n (cid:2) (cid:3)
n
= dn–rcr+ E(m) cx,cy,cz,λ
r H n–r
r=
(cid:5) (cid:6)
(cid:4)r (cid:2) (cid:3)
r
× M(d–;λ) E(m–) dX,dY,dZ,λ .
l l H r–l
l=
Lettingk=–a=b=andβ=λandtakingintoaccount(.)thatwehavethefollow-
ing.
Corollary. Forallm∈N,n∈N ,λ∈C,wehaveforeachpairofpositiveevenintegers

candd,orforeachpairofpositiveoddintegerscandd,that
(cid:5) (cid:6)
(cid:4)n (cid:2) (cid:3)
n
cn–rdr+ G(m) dx,dy,dz,λ
r H n–r
r=
(cid:5) (cid:6)
(cid:4)r (cid:2) (cid:3)
r
× M(c–;λ) G(m–) cX,cY,cZ,λ
l l H r–l
l=
(cid:5) (cid:6)
(cid:4)n (cid:2) (cid:3)
n
= dn–rcr+ G(m) cx,cy,cz,λ
r H n–r
r=
(cid:5) (cid:6)
(cid:4)r (cid:2) (cid:3)
r
× M(d–;λ) G(m–) dX,dY,dZ,λ .
l l H r–l
l=
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4 Closed-formformulaeforHermite-basedgeneralizedApostolpolynomials
Inthissection,takingintoaccounttherelations
(cid:5) (cid:6)
–ktk α (cid:4)∞ tn
f(α)(x,y,z;t;k,β):= ext+yt+zt = P(α)(x,y,z;k,a,b) ,
a,b βbet–ab H n,β n!
n=
(cid:5) (cid:6)
(cid:4)∞
–ktk tn
f()(x,y,z;t;k,β):= ext+yt+zt = P (x,y,z;k,a,b) ,
a,b βbet–ab H n,β n!
n=
weobservethefollowingfact:
(cid:8) (cid:5) (cid:6)(cid:9)
x y z α
f() , , ;t;k,β =f(α)(x,y,z;t;k,β). (.)
a,b α α α a,b
Using(.),westartbyprovingthefollowingclosedformsummationformula:
Theorem. Lettheconditions(i)-(iii)besatisfied.Thefollowingsummationformula:
(cid:5) (cid:6)(cid:8) (cid:5) (cid:6)
(cid:4)n
n x y z
P(α) (x,y,z;k,a,b) P , , ;k,a,b
l H n–l+,β H l,β α α α
l=
(cid:5) (cid:6)(cid:9)
x y z
–α P(α) (x,y,z;k,a,b) P , , ;k,a,b =
H n–l,β H l+,β α α α
holdstrue.
Proof Takinglogarithmsonbothsidesof(.)andthendifferentiatingwithrespecttot,
weget
(cid:5) (cid:6)
∂f(α)(x,y,z;t;k,β) x y z
a,b f() , , ;t;k,β
∂t a,b α α α
∂f()(x, y, z;t;k,β)
=αf(α)(x,y,z;t;k,β) a,b α α α .
a,b ∂t
Insertingthecorrespondinggeneratingrelations,weobtain
(cid:5) (cid:6)
(cid:4)∞ (cid:4)∞
tn– x y z tl
n P(α)(x,y,z;k,a,b) P , , ;k,a,b
H n,β n! H l,β α α α l!
n= l=
(cid:5) (cid:6)
(cid:4)∞ (cid:4)∞
tn x y z tl–
=α P(α)(x,y,z;k,a,b) l P , , ;k,a,b ,
H n,β n! H l,β α α α l!
n= l=
andhence
(cid:5) (cid:6)
(cid:4)∞ (cid:4)∞
tn x y z tl
P(α) (x,y,z;k,a,b) P , , ;k,a,b
H n+,β n! H l,β α α α l!
n= l=
(cid:5) (cid:6)
(cid:4)∞ (cid:4)∞
tn x y z tl
=α P(α)(x,y,z;k,a,b) P , , ;k,a,b .
H n,β n! H l+,β α α α l!
n= l=
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Usingthefactthat(see[,p.,Lemma])
(cid:4)∞ (cid:4)∞ (cid:4)∞ (cid:4)n
A(n,l)= A(n–l,l), (.)
n= l= n= l=
weget
(cid:11) (cid:5) (cid:6) (cid:5) (cid:6)(cid:12)
(cid:4)∞ (cid:4)n n x y z tn
P(α) (x,y,z;k,a,b) P , , ;k,a,b
l H n–l+,β H l,β α α α n!
n= l=
(cid:11) (cid:5) (cid:6) (cid:5) (cid:6)(cid:12)
(cid:4)∞ (cid:4)n n x y z tn
=α P(α) (x,y,z;k,a,b) P , , ;k,a,b .
l H n–l,β H l+,β α α α n!
n= l=
Whencetheresult. (cid:2)
Corollary. Letk=a=b=andβ=λ.Forallm∈N,n∈N ,λ∈C,wehavethefol-

lowingclosedformsummationformulaforthegeneralizedApostol-Bernoullipolynomials:
(cid:5) (cid:6)(cid:8) (cid:5) (cid:6)
(cid:4)n
n x y z
B(α) (x,y,z;λ) B , , ;λ
k H n–k+ H k α α α
k=
(cid:5) (cid:6)(cid:9)
x y z
–α B(α)(x,y,z;λ)B , , ;λ =.
H n–k k+ α α α
Corollary. Letk+=–a=b=andβ=λ.Forallm∈N,n∈N ,λ∈C,wehavethe

followingclosedformsummationformulaforthegeneralizedApostol-Eulerpolynomials:
(cid:5) (cid:6)(cid:8) (cid:5) (cid:6)
(cid:4)n
n x y z
E(α) (x,y,z;λ) E , , ;λ
k H n–k+ H k α α α
k=
(cid:5) (cid:6)(cid:9)
x y z
–α E(α)(x,y,z;λ)E , , ;λ =.
H n–k k+ α α α
Corollary. Letk=–a=b=andβ=λ.Forallm∈N,n∈N ,λ∈C,wehavethe

followingclosedformsummationformulaforthegeneralizedApostol-Genocchipolynomi-
als:
(cid:5) (cid:6)(cid:8) (cid:5) (cid:6)
(cid:4)n
n x y z
G(α) (x,y,z;λ) G , , ;λ
k H n–k+ H k α α α
k=
(cid:5) (cid:6)(cid:9)
x y z
–α G(α)(x,y,z;λ)G , , ;λ =.
H n–k k+ α α α
Theorem. Lettheconditions(i)-(iii)besatisfied.Thenwehavethefollowingrelation
betweenHermitebasedApostolpolynomialsandd-Hermitepolynomials:
P(α) (X,Y,Z;k,a,b)
H n+m,β
(cid:5) (cid:6)(cid:5) (cid:6)
(cid:4)n,m
n m
= H()(X–x,Y –y,Z–z) P(α) (x,y,z;k,a,b).
r l r+l H n+m–r–l
r,l=