ÖzarslanAdvancesinDifferenceEquations2013,2013:116 http://www.advancesindifferenceequations.com/content/2013/1/116 RESEARCH OpenAccess Hermite-based unified Apostol-Bernoulli, Euler and Genocchi polynomials MehmetAliÖzarslan* *Correspondence: mehmetali.ozarslan@emu.edu.tr 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. ÖzarslanAdvancesinDifferenceEquations2013,2013:116 Page2of13 http://www.advancesindifferenceequations.com/content/2013/1/116 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 . (.)  ÖzarslanAdvancesinDifferenceEquations2013,2013:116 Page3of13 http://www.advancesindifferenceequations.com/content/2013/1/116 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: ÖzarslanAdvancesinDifferenceEquations2013,2013:116 Page4of13 http://www.advancesindifferenceequations.com/content/2013/1/116 • 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. ÖzarslanAdvancesinDifferenceEquations2013,2013:116 Page5of13 http://www.advancesindifferenceequations.com/content/2013/1/116 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+ ÖzarslanAdvancesinDifferenceEquations2013,2013:116 Page6of13 http://www.advancesindifferenceequations.com/content/2013/1/116 (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= ÖzarslanAdvancesinDifferenceEquations2013,2013:116 Page7of13 http://www.advancesindifferenceequations.com/content/2013/1/116 (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. ÖzarslanAdvancesinDifferenceEquations2013,2013:116 Page8of13 http://www.advancesindifferenceequations.com/content/2013/1/116 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= ÖzarslanAdvancesinDifferenceEquations2013,2013:116 Page9of13 http://www.advancesindifferenceequations.com/content/2013/1/116 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= ÖzarslanAdvancesinDifferenceEquations2013,2013:116 Page10of13 http://www.advancesindifferenceequations.com/content/2013/1/116 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=

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

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