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AraciandzerAdvancesinDierenceEquations(2015)2015:272DOI10.
1186/s13662-015-0610-8RESEARCHOpenAccessExtendedq-Dedekind-typeDaehee-Changheesumsassociatedwithextendedq-EulerpolynomialsSerkanAraci1*andzenzer2*Correspondence:mtsrkn@hotmail.
com1DepartmentofEconomics,FacultyofEconomics,AdministrativeandSocialSciences,HasanKalyoncuUniversity,Gaziantep,27410,TurkeyFulllistofauthorinformationisavailableattheendofthearticleAbstractInthepresentpaper,weaimtospecifyap-adiccontinuousfunctionforanoddprimeinsideap-adicq-analogoftheextendedDedekind-typesumsofhigherorderaccordingtoextendedq-Eulerpolynomials(orweightedq-Eulerpolynomials)whichisderivedfromafermionicp-adicq-deformedintegralonZp.
MSC:11S80;11B68Keywords:Dedekindsums;q-Dedekind-typesums;p-adicq-integral;extendedq-Eulernumbersandpolynomials1IntroductionLetpbechosenasaxedoddprimenumber.
InthispaperZp,Qp,CandCpwill,respec-tively,denotetheringofp-adicrationalintegers,theeldofp-adicrationalnumbers,thecomplexnumbers,andthecompletionofanalgebraicclosureofQp.
LetvpbeanormalizedexponentialvaluationofCpby|p|p=p–vp(p)=p.
Whenonetalksofaq-extension,qisvariouslyconsideredasanindeterminate,acom-plexnumberq∈Corap-adicnumberq∈Cp.
Ifq∈C,weassumethat|q|<.
Ifq∈Cp,weassumethat|–q|p<(see,fordetails,[–]).
ThefollowingmeasureisdenedbyKim:foranypositiveintegernand≤aExtendedq-Eulerpolynomials(alsoknownasweightedq-Eulerpolynomials)arede-nedbyE(α)n,q(x)=Zp–qα(x+ξ)–qαndμq(ξ)()2015Araciandzer.
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AraciandzerAdvancesinDierenceEquations(2015)2015:272Page2of5forn∈Z+:={,,,,.
.
.
}.
Wenotethatlimq→E(α)n,q(x)=En(x),whereEn(x)arenthEulerpolynomials,whicharedenedbytherule∞n=En(x)tnn!
=etxet+,|t|<π(fordetails,see[]).
Inthecasex=in(),thenwehaveE(α)n,q():=E(α)n,q,whicharecalledextendedq-Eulernumbers(orweightedq-Eulernumbers).
Extendedq-Eulernumbersandpolynomialshavethefollowingexplicitformulas:E(α)n,q=+q(–qα)nnl=nl(–)l+qαl+,()E(α)n,q(x)=+q(–qα)nnl=nl(–)lqαlx+qαl+,()E(α)n,q(x)=nl=nlqαlxE(α)l,q–qαx–qαn–l.
()Moreover,ford∈Nwithd≡(mod),E(α)n,q(x)=+q+qd–qαd–qαnd–a=(–)aE(α)n,qx+ad;()see[].
Foranypositiveintegerh,kandm,Dedekind-typeDCsumsaregivenbyKimin[,],and[]asfollows:Sm(h,k)=k–M=(–)M–MkEmhMk,whereEm(x)aremthperiodicEulerfunctions.
Kim[]derivedsomeinterestingpropertiesforDedekind-typeDCsumsandconsid-eredap-adiccontinuousfunctionforanoddprimenumbertocontainap-adicq-analogofthehigherorderDedekind-typeDCsumskmSm+(h,k).
Simsek[]gaveaq-analogofDedekind-typesumsandderivedinterestingproperties.
Furthermore,Aracietal.
stud-iedDedekind-typesumsinaccordancewithmodiedq-Eulerpolynomialswithweightα[],modiedq-Genocchipolynomialswithweightα[],andweightedq-Genocchipolynomials[].
Recently,weightedq-BernoullinumbersandpolynomialswererstdenedbyKimin[].
Next,manymathematicians,byutilizingKim'spaper[],haveintroducedvariousgeneralizationofsomeknownspecialpolynomialssuchasBernoullipolynomials,Eulerpolynomials,Genocchipolynomials,andsoon,whicharecalledweightedq-Bernoulli,weightedq-Euler,andweightedq-Genocchipolynomialsin[,,–].
AraciandzerAdvancesinDierenceEquations(2015)2015:272Page3of5Bythesamemotivationoftheaboveknowledge,wegiveaweightedp-adicq-analogofthehigherorderDedekind-typeDCsumskmSm+(h,k)whicharederivedfromafermionicp-adicq-deformedintegralonZp.
2Extendedq-Dedekind-typesumsassociatedwithextendedq-EulerpolynomialsLetwbetheTeichmüllercharacter(modp).
Forx∈Zp:=Zp/pZp,setx:q=w–(x)–qx–q.
LetaandNbepositiveintegerswith(p,a)=andp|N.
WenowconsiderC(α)qs,a,N:qN=w–(a)a:qαs∞j=sjqαaj–qαN–qαajE(α)j,qN.
Inparticular,ifm+≡(modp–),thenC(α)qm,a,N:qN=–qαa–qαmmj=mjqαajE(α)j,qN–qαN–qαaj=–qαN–qαmZp–qαN(ξ+aN)–qαNmdμqN(ξ).
Thus,C(α)q(m,a,N:qN)isacontinuousp-adicextensionof–qαN–qαmE(α)m,qNaN.
Let[·]betheGausssymbolandlet{x}=x–[x].
Thus,wearenowreadytointroducetheq-analogofthehigherorderDedekind-typeDCsumsJ(α)m,q(h,k:ql)bytheruleJ(α)m,qh,k:ql=k–M=(–)M––qαM–qαkZp–qα(lξ+l{hMk})–qαlmdμql(ξ).
Ifm+≡(modp–),–qαk–qαm+k–M=(–)M––qαM–qαkZp–qαk(ξ+hMk)–qαkmdμqk(ξ)=k–M=(–)M––qαM–qα–qαk–qαmZp–qαk(ξ+hMk)–qαkmdμqk(ξ),wherep|k,(hM,p)=foreachM.
By(),weeasilystatethefollowing:–qαk–qαm+J(α)m,qh,k:qk=k–M=–qαM–qα–qαk–qαm(–)M–AraciandzerAdvancesinDierenceEquations(2015)2015:272Page4of5*Zp–qαk(ξ+hMk)–qαkmdμqk(ξ)=k–M=(–)M––qαM–qαC(α)qm,(hM)k:qk,()where(hM)kdenotestheintegerxsuchthat≤xItisnotdiculttoindicatethefollowing:Zp–qα(x+ξ)–qαkdμq(ξ)=–qαm–qαk+q+qmm–i=(–)iZp–qαm(ξ+x+im)–qαmkdμqm(ξ).
()Onaccountof()and(),weeasilyseethat–qαN–qαmZp–qαN(ξ+aN)–qαNmdμqN(ξ)=+qN+qNpp–i=(–)i–qαNp–qαmZp–qαpN(ξ+a+iNpN)–qαpNmdμqpN(ξ).
()Becauseof(),(),and(),wedevelopthep-adicintegrationasfollows:C(α)qs,a,N:qN=+qN+qNp≤i≤p–a+iN=(modp)(–)iC(α)qs,(a+iN)pN,pN:qpN.
So,C(α)qm,a,N:qN=–qαN–qαmZp–qαN(ξ+aN)–qαNmdμqN(ξ)––qαNp–qαmZp–qαpN(ξ+a+iNpN)–qαpNmdμqpN(ξ),where(p–a)Ndenotestheintegerxwith≤xTherefore,wehavek–M=(–)M––qαM–qαC(α)qm,hM,k:qk=–qαk–qαm+J(α)m,qh,k:qk––qαk–qαm+*–qαkp–qαkJ(α)m,qp–h,k:qpk,wherepkandphmforeachM.
Thus,wegivethefollowingdenition,whichseemsinterestingforfurtherstudyingthetheoryofDedekindsums.
AraciandzerAdvancesinDierenceEquations(2015)2015:272Page5of5DenitionLeth,kbepositiveintegerwith(h,k)=,pk.
Fors∈Zp,wedeneap-adicDedekind-typeDCsumsasfollows:J(α)p,qs:h,k:qk=k–M=(–)M––qαM–qαC(α)qm,hM,k:qk.
Asaresultoftheabovedenition,westatethefollowingtheorem.
Theorem.
Form+≡(modp–)and(p–a)Ndenotestheintegerxwith≤xInthespecialcaseα=,ourapplicationsintheoryofDedekindsumsresembleKim'sresultsin[].
Theseresultsseemtobeinterestingforfurtherstudiesasin[,]and[].
CompetinginterestsTheauthorsdeclarethattheyhavenocompetinginterests.
Authors'contributionsAllauthorscontributedequallytothiswork.
Allauthorsreadandapprovedtherevisedmanuscript.
Authordetails1DepartmentofEconomics,FacultyofEconomics,AdministrativeandSocialSciences,HasanKalyoncuUniversity,Gaziantep,27410,Turkey.
2DepartmentofMathematics,FacultyofScienceandArts,KrklareliUniversity,Krklareli,39000,Turkey.
AcknowledgementsTheauthorsthankthereviewersfortheirhelpfulcommentsandsuggestions,whichhaveimprovedthequalityofthepaper.
Received:22June2015Accepted:15August2015References1.
Araci,S,Acikgoz,M,Park,KH:Anoteontheq-analogueofKim'sp-adicloggamma-typefunctionsassociatedwithq-extensionofGenocchiandEulernumberswithweightα.
Bull.
KoreanMath.
Soc.
50(2),583-588(2013)2.
Araci,S,Erdal,D,Seo,JJ:Astudyonthefermionicp-adicq-integralrepresentationonZpassociatedwithweightedq-Bernsteinandq-Genocchipolynomials.
Abstr.
Appl.
Anal.
2011,ArticleID649248(2011)3.
Araci,S,Acikgoz,M,Seo,JJ:Explicitformulasinvolvingq-Eulernumbersandpolynomials.
Abstr.
Appl.
Anal.
2012,ArticleID298531(2012).
doi:10.
1155/2012/2985314.
Araci,S,Acikgoz,M,Esi,A:Anoteontheq-Dedekind-typeDaehee-Changheesumswithweightαarisingfrommodiedq-Genocchipolynomialswithweightα.
J.
AssamAcad.
Math.
5,47-54(2012)5.
Kim,T:Anoteonp-adicq-Dedekindsums.
C.
R.
Acad.
BulgareSci.
54,37-42(2001)6.
Kim,T:Noteonq-Dedekind-typesumsrelatedtoq-Eulerpolynomials.
Glasg.
Math.
J.
54,121-125(2012)7.
Kim,T:NoteonDedekindtypeDCsums.
Adv.
Stud.
Contemp.
Math.
18,249-260(2009)8.
Kim,T:Themodiedq-Eulernumbersandpolynomials.
Adv.
Stud.
Contemp.
Math.
16,161-170(2008)9.
Kim,T:q-Volkenbornintegration.
Russ.
J.
Math.
Phys.
9,288-299(2002)10.
Kim,T:Onaq-analogueofthep-adicloggammafunctionsandrelatedintegrals.
J.
NumberTheory76,320-329(1999)11.
Kim,T:Ontheweightedq-Bernoullinumbersandpolynomials.
Adv.
Stud.
Contemp.
Math.
21(2),207-215(2011)12.
Rim,SH,Jeong,J:Anoteonthemodiedq-Eulernumbersandpolynomialswithweightα.
Int.
Math.
Forum6(65),3245-3250(2011)13.
Ryoo,CS:Anoteontheweightedq-Eulernumbersandpolynomials.
Adv.
Stud.
Contemp.
Math.
21,47-54(2011)14.
Seo,JJ,Araci,S,Acikgoz,M:q-Dedekind-typeDaehee-Changheesumswithweightαassociatedwithmodiedq-Eulerpolynomialswithweightα.
J.
ChungcheongMath.
Soc.
27(1),1-8(2014)15.
Simsek,Y:q-Dedekindtypesumsrelatedtoq-zetafunctionandbasicL-series.
J.
Math.
Anal.
Appl.
318,333-351(2006)16.
Sen,E,Acikgoz,M,Araci,S:Anoteonthemodiedq-Dedekindsums.
NotesNumberTheoryDiscreteMath.
19(3),60-65(2013)