; TeX output 2006.07.21:0651 KE&:9color push Blackhtml:color push gray 0 color pop html:G color pop3ڍ:9|&html: html:.MTvkcolor push Black color popZVg5PSfile=logo129.eps llx=0 lly=0 urx=99 ury=16 rwi=2880Bs獍color push Black color pop
N q cmbx12ConZverting
BetweenGeneralizedBell,Lah,W[Stirling,
andT=anhNumZb ers.s獒 ?XQ ff cmr12Giacomo/DellaRiccia LDepartment/ofMathematicsandComputerScience &University/ofUdine [Via/delleScienze206 ıf33100/Udine cuItaly %html:color push cmyk 0 1 0 0߆T ff cmtt12dlrca@uniud.it html: color pop)
č vcolor push Black color pop r"V
3
cmbx10AbstractMƍlcolor push Black color pop-̻K`y
3
cmr10WeeR^deneaRiordantriangleforgeneralizedBelln!umbMersR^andw!eestablishgeneral
_iden!tities<.connectingLah,QmStirling,Teanh<.andthegeneralizedBellnumbMers.uSeveralfa-_miliarԄin!verserelationsconvertingbMetweenspMecialsequencesareshowntobMeparticular_casesfofthegeneraliden!tities.6ghtml: html:N G cmbx121(Inutro =ductionb#XQ cmr12WVeconsider,intheconrtextofthe(expSonential)Riordangroup(!",
cmsy10R;g cmmi12?)&G,inverserelations html: html:8Uc(n;m)UR=u
cmex10X
㇍
2 cmmi8ia(n;i)b(i;m),b(n;m)=X
㇍
iA(n;i)c(i;m);
n;m0;(1.1)$lconrvertingWbSetweenarraysofnumbSers.+TheconceptofRiordangroupwasintroSducedby Shapiro#etal.[!html:color push cmyk 0 1 0 015 html: color pop]asageneralizationofastudyofRogers[ html:color push cmyk 0 1 0 012 html: color pop]on+@ cmti12rffenewalgarrays!hand#usedin_sevreralapplications,including[#html:color push cmyk 0 1 0 016 html: color pop,#html:color push cmyk 0 1 0 017 html: color pop,!html:color push cmyk 0 1 0 014 html: color pop,"html:color push cmyk 0 1 0 019 html: color pop, html:color push cmyk 0 1 0 01 html: color pop?,#html:color push cmyk 0 1 0 010 html: color pop]. Thelastreference,pSoinrtedouttousbrythereferee,|containsanapplicationoftheconceptofaRiordanarraytotheAkiyrama-TVanigawa DtransformationanalogoustotheformalismdevrelopSedinthepresentpapSer.
T6AnoelemenrtRT\2;(R;m?)isdenotedbyRT\=;(qn9(u);mRJ(u))whereq(u)isanex-pSonenrtialRgeneratingfunctionsuchthatqn9(0) =1RandRJ(u)isanexpSonentialgeneratingfunction|qsucrhthatRJ(0)Mx=0|qandR2K cmsy80烹(0)Mx=1.