UNIV.BEOGRAD.PUBL.ELEKTROTEHN.FAK.Ser.Mat.Fiz.No577-No598(1977),86-88.595.ACOMBINATORIALIDENTITYOFL.N.DORDEVIC*L.Carlitz1.L.N.DORDEVIC[1]hasprovedthefollowingidentity:mmlmn-l(1.1)...mI-mz'...,mn-I-mn,mn)Jnl=om2=OLLmn=OL(-l)mj(m-mpm!(Pj+...+Pn)m+1'wherethePiarerealnumberssatisfyingPi>-1and(1.2)(m-mI'mI-mz'...,mn-I-mn,mn)=m!(m-mj)!(mj-m2)!...(mn-I-mn)!mn!.The,proofemploysmultipleintegrationoveracertaininfiniteregioninRn+I'Inthepresentnoteweproveaslightgeneralizationof(1.1)byamoreelementarymethod,namely(1.3)mIn!mn-lmn)ml~oL(m-ml'mI-mz\"'\"mn-I-mn,m2~OL'\"mn~OL(-l)mjx(pj)ml-m2'.'(Pn-l)mn_j-mn(Pn)mn-(A-Pj-'\"-Pn+l)m(A)mj+1(A)m+I'whereAisnotequaltozerooranegativeintegerbutisotherwisearbitrary.2.Putm=kO+mIandmI=kI+kz+'\"+kn-I+knmz=kz+...+kn-I+knkn-I+knm=nkn,Then(1.3)becomes(2.1)ko+kl+...+kn~mL(-lll+'\"Hn(ko,kI,...,kn-I'k).n(P1)kl(pzkz\".(PIl)kn(A)kl+...+kn+1(A-P1-\"'-PIl+l)m(A)m+I*PresentedJune1,1977byD.S.MITRINOVIC.86AcombinatorialidentityofL.N.Dordevic87Forbrevitydenotethelefthandsideof(2.1)byAn(m).Then{2.2)-m~OLxmko+kl+...+kn~mL(_I)kl+...+kn(P,)kl\"'(Pll)knkl!...kn!.(I-.+k,+...+kn+l)koko!=s~oL00(-1)'XS(1-x)-f.-s-Ikl+...+kn~sL(Pl)kl'..(Pnknkl!...kn!Bythemultinomial{2.2)becomestheorem,theinnersumisequalto(PI+...+Pn)s,sothats!~s~o(-1)'(P,+...+Pn)ss!=(1-X)-A-IxS(1-x)-A-S-I(1+~)I-x..-Pn+l)-Pl-\"'-Pn=(1-X)-(A-Pl-'=LThuswehaveprovedthatm~OCA-p1-\"'-Pm!n+1)mxm.andtherefore(2.3)ThisevidentlyAn(m)=proves(1.3).(A-PI-\"'-pn+l)m(A)m+l3.Multiplyingbothsidesof(2.1)by(A)m+Iwegettheidentity(3.1)L+kn~mko+kl+'\"(-l)k!+...+k,,'(ko'k1,...,kn)(Pl)kl...(Pn)kn88L.CarIitzwhichholdsforallA.Inparticular,forA=0,(3.1)becomes(3.2)-(-p'-\"'-Pn+I)mm!Itisnotdifficulttogiveadirectproofof(3.2)REFERENCE1.L.N.DORDEYIC:Proofofacombinatorialidentityusingmultipleintegration.SamePublications,pp.63-.DukeDepartmentUniversityofMathematicsDurham,N.C.27706,USA
