naturalspublishingb2t

J.
Ana.
Num.
Theor.
2,No.
2,59-63(2014)59JournalofAnalysis&NumberTheoryAnInternationalJournalhttp://dx.
doi.
org/10.
12785/jant/020206NewGeneralizationofEulerianPolynomialsandtheirApplicationsSerkanAraci1,,MehmetAcikgoz1,andErdoganSen2,1DepartmentofMathematics,FacultyofScienceandArts,UniversityofGaziantep,27310Gaziantep,Turkey2DepartmentofMathematics,FacultyofScienceandLetters,NamkKemalUniversity,59030Tekirdag,TurkeyReceived:14Feb.
2014,Revised:20Apr.
2014,Accepted:22Apr.
2014Publishedonline:1Jul.
2014Abstract:Inthepresentpaper,weintroduceEulerianpolynomialswithparametersaandbandgivethedenitionofthem.
Byusingthedenitionofgeneratingfunctionforourpolynomials,wederivesomenewidentitiesinAnalyticNumbersTheory.
Also,wegiverelationsbetweenEulerianpolynomialswithparametersaandb,Bernsteinpolynomials,Poly-logarithmfunctions,BernoulliandEulernumbers.
Moreover,weseethatourpolynomialsata=1arerelatedtoEuler-Zetafunctionatnegativeinetegers.
Finally,wegetWitt'sformulafornewgeneralizationofEulerianpolynomialswhichweexpressinthispaper.
Keywords:Eulerianpolynomials,Poly-logarithmfunctions,Stirlingnumbersofthesecondkind,Bernsteinpolynomials,Bernoullinumbers,EulernumbersandEuler-Zetafunction,p-adicfermionicintegralonZp.
2010MATHEMATICSSUBJECTCLASSIFICATION.
Primary05A10,11B65;Secondary11B68,11B73.
1IntroductionTheBernoullinumbersandpolynomials,Eulernumbersandpolynomials,Genocchinumbersandpolynomials,Stirlingnumbersofthesecondkind,BernsteinpolynomialsandEulerianpolynomialspossessmanyinterestingpropertiesnotonlyincomplexanalysis,andanalyticnumberstheorybutalsoinmathematicalphysicsrelatedtoknottheoryandζ-function,andp-adicanalysis.
Thesepolynomialshavebeenstudiedbymanymathematiciansforalongtime(fordetails,see[1-30]).
Eulerianpolynomialsequence{An(x)}n≥0isgivenbythefollowingsummation:∞∑l=0lnxl=An(x)(1x)n+1,|x|0in(15),becomesAn(a,b)=1a1n1∑k=0nkAk(a,b)(1a)nk(lnb)nk.
(16)Wewanttonotethattakinga=xandb=ein(16)reducestoAn(x)=1x1n1∑k=0nkAk(x)(1x)nk(17)(see[5]and[25]).
Weseethat(17)isproportionalwithBernsteinpolynomialswhichwestateinthefollowingtheorem:c2014NSPNaturalSciencesPublishingCor.
J.
Ana.
Num.
Theor.
2,No.
2,59-63(2014)/www.
naturalspublishing.
com/Journals.
asp61Theorem2.
ThefollowingidentityAn(x)=n1∑k=0Ak(x)Bk,n(x)xk+1xkistrue.
Letusnowconsiderlimt→0dkdtkin(14),thenwereadilyarriveatthefollowingtheorem.
Theorem3.
Letb∈R+anda∈C,thenwehaveAk(a,b)=limt→0dkdtk1abt(1a)a.
(18)By(18),weeasilyconcludethefollowingcorollary.
Corollary1.
ThefollowingCauchy-typeintegralholdstrue:11aAk(a,b)=k!
2πiCtk1bt(1a)adtwhereCisaloopwhichstartsat∞,encirclestheoriginonceinthepositivedirection,andthereturns∞.
By(14),wediscoverthefollowing:∞∑n=0Ana2,b2tnn!
=1abt(1+a)(1a)a1+abt(1a)(1+a)a=∞∑n=0(1+a)nAn(a,b)tnn!
∞∑n=0(1a)nAn(a,b)tnn!
.
ByusingCauchyproductontheaboveequality,thenwegetthefollowingtheorem.
Theorem4.
ThefollowingequalityAna2,b2=∑nk=0nk(1+a)kAk(a,b)Ank(a,b)(1a)nk(19)istrue.
Afterthebasicoperationsin(19),wediscoverthefollowingcorollary.
Corollary2.
Thefollowingpropertyholds:Ana2,b2=n∑k=01+1akBk,n(a)Ak(a,b)Ank(a,b).
Nowalso,weconsidergeometricseriesin(14),thenwecomputeasfollows:∞∑n=0An(a,b)tnn!
=1aet(1a)lnba=1a11a1et(1a)lnb=11a∞∑j=0ajejt(1a)lnb=11a∞∑j=0aj∞∑n=0jn(1a)n(lnb)ntnn!
=∞∑n=011a∞∑j=0ajjn(1a)n(lnb)ntnn!
.
Bycomparingthecoefcientsoftnn!
ontheaboveequation,thenwereadilyderivethefollowingtheorem.
Theorem5.
Thefollowing1a1nAn(a,b)=(lnb)na(lnb)n∞∑j=1ajjnistrue.
TheabovetheoremisrelatedtoPoly-logarithmfunction,asfollows:1a1nAn(a,b)=(lnb)na(lnb)nLina1.
(20)In[27],itiswell-knownthatLin(x)=xddxnx1x=∑nk=0k!
S(n+1,k+1)x1xk+1(21)whereS(n,k)aretheStirlingnumbersofthesecondkind.
By(20)and(21),wehavethefollowinginterestingtheorem.
Theorem6.
Thefollowingholdstrue:aAn(a,b)=(lnb)nn∑k=0k!
S(n+1,k+1)1a1kn.
3FurtherRemarksNow,weconsider(14)forevaluatingata=1,asfollows:∞∑n=0An(1,b)tnn!
=2b2t+1(22)whereAn(1,b)arecalledEulerianpolynomialswithparameterb.
By(22),wederivethefollowingequalityincomplexplane:∞∑n=0inAn(1,b)tnn!
=2b2it+1=2e2itlnb+1.
Fromthis,wediscoverthefollowing:∞∑n=0inAn(1,b)tnn!
=∞∑n=0En2nin(lnb)tnn!
(23)whereEnaren-thEulernumberswhicharedenedbythefollowingexponentialgeneratingfunction:∞∑n=0Entnn!
=2et+1,|t|0,thenwehaveAn(1,b)=2n+1(lnb)n∞∑j=1(1)jjn.
(26)Asiswellknown,Euler-zetafunctionisdenedbyζE(s)=2∞∑j=1(1)jjs,s∈C(see[3]).
(27)From(26)and(27),weobtaintheinterpolationfunctionofnewgeneralizationofEulerianpolynomialsata=1,asfollow:An(1,b)=2n(lnb)nζE(n).
(28)Equation(28)seemstobeinterpolationfunctionatnegativeintegersforEulerianpolynomialswithparameterb.
LetusnowconsiderWitt'sformulaforourpolynomialsata=1,soweneedthefollowingnotations:Imaginethatpbeaxedoddprimenumber.
Throughoutthispaper,weusethefollowingnotations.
ByZp,wedenotetheringofp-adicrationalintegers,Qdenotestheeldofrationalnumbers,Qpdenotestheeldofp-adicrationalnumbers,andCpdenotesthecompletionofalgebraicclosureofQp.
LetNbethesetofnaturalnumbersandN=N∪{0}.
Thenormalizedp-adicabsolutevalueisdenedby|p|p=1p.
Letqbeanindeterminatewith|q1|pb2t+1=∞∑n=0An(1,b)tnn!
.
(31)By(31)andusingTaylorexpansionofe2tυlnb,weobtainWitt'sformulaforourpolynomialsata=1,asfollows:Theorem11.
Thefollowingholdstrue:An(1,b)=(lnb)n2nXυnd1(υ).
(32)Equation(32)seemstobeinterestingforourfurtherworksintheconceptofp-adicintegrals.
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c2014NSPNaturalSciencesPublishingCor.