REVISTA INVESTIGACION OPERACIONAL Vol. 25, No. 1, k n ),


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1 REVISTA INVESTIGACION OPERACIONAL Vol 25, No, 24 RECURRENCE AND DIRECT FORMULAS FOR TE AL & LA NUMBERS Eduardo Pza Volo Cero de Ivesgacó e Maemáca Pura y Aplcada (CIMPA), Uversdad de Cosa Rca ABSTRACT I hs paper we sudy some properes cocerg he al ( ) ad Lah umbers ( L ), whch are hose ha arse he heory of combaorcs assocao wh he upward ad dowward facoral polyomals ad By usg geerag fuco echques ad smple bomal expasos we esablsh some ve resuls for he al umber ( ), le he recurrece formula, he geerag fuco ad he drec (closed) formula, amog ohers properes Fally we compare hese resuls wh he correspodg formulas for he Lah umbers ( L ) We esablsh he correspodg recurrece formulas, he geerag fucos ad he closed formulas for hs famly of umbers, amog ohers properes Key words: al umbers, Lah umbers, combaorcs, geerag fucos MSC: 5A TE AL NUMBERS RESUMEN E ese rabajo se esuda alguas propedades relavas a los úmeros de al ( ) y de Lah ( L ), que so aquellos úmeros que surge e combaora asocados co los polomos facorales haca arrba y haca abajo Se esablece las fórmulas por recurreca, las fucoes geeradoras y las fórmulas cerradas para esa famla de úmeros, ere oras propedades For ay real umber x ad ay aural umber we defe he upward facoral polyomal o be : x(x + ) (x + ) I assocao wh hese polyomals we have he al umbers ha sasfy he relao ( ), whose defo are hose umbers [x] : + + L, () ad we exed hs defo spulag ha whe >, or, or The ermology employed here ( al umbers) has bee chose o emphasze he aalogy bewee hs umercal famly wh he more famlar Lah umbers ( L ) for he dowward facoral polyomals ( hal s lah bacwards), as we see laer See also [2, 3] Noce ha by subsuo of x by x he relao () we oba he equvale relao : [x] + [ x] + L [ x] (2) As a cosequece from he Frs Iverso Formula (see appedx, or []) we medaely deduce ha m δ m,, s s m m Emal: 48
2 The, he square marx ( ), wh, {,,N} s verse of self, for all N IN* Furhermore, also as a cosequece of he same Frs Iverso Formula we deduce ha ay oe of he equaos a b, b a mples he oher Le s see how o ge a recurrece formula ad a drec (closed) formula for hese al umbers ( ) 2 RECURRENCE FOR TE AL NUMBERS Theorem + ( + ) (3) Proof: Cosder he followg polyomals of degree + : + A(x) {( + ) }, + + B(x) We wll prove ha hese polyomals are equal, obag he resul by comparg he correspodg coeffces I fac, we have: A(x) + {( + ) } Bu + (x + ), ad so we ca smplfy he las expresso: A(x) + (x + ) ( x + ) (x + )[x] [x] + B(x) g
3 The las recurrece maes easy o do fas calculaos of al umbers, whose frs erms are llusraed he Table I ca be observed ha eve colums have oly posve umbers, whle odd colums have oly egave umbers (below he prcpal dagoal) Besdes, all he rows have alerag umbers, begg by! These facs wll be easly proved wh he help of he drec formula whch we wll deduce laer 3 GENERATING FUNCTION FOR AL NUMBER Table Frs erms of he al umbers ( ) \ Followg a smlar approach o ha used by Rorda (958), we deduc he geerag fuco for he al umber Theorem 2 For all IN, he expoeal geerag fuco () of he al umbers ( ) IN s  ():!!  (4) Proof: From he dey [x] (x)(x + ) (x + ) () x x(x ) (x  + ) () ad usg he bomal expaso ( + u) β β u, vald for u <, we deduce ha: [x]! () x ( ) x + x x ( )!! The, ()!!! [ x]! !  The exchage bewee sum symbols s jusfed jus because all he sums over are fe, due o for > The resul s obaed by comparg he correspodg coeffces g 5
4 4 CLOSED FORMULA FOR TE AL NUMBERS Alhough he recurrece (3) for he al umbers s a lear ad frs degree formula boh ad, hs recurrece s o easy o resolve order o derve a drec formula for he al umbers ( ) I fac, we use a dffere approach o reach he closed formula Theorem 3 For : ()!! (5) Proof: Mag he bomal expaso of {/( )}, we oba: () ( ) ( )! ( )! ( ) + + ( ) ( )! ( )! ( )! ( )! (6)! Whe comparg he correspodg coeffces he formulas (4) ad (6) we oba he resul g From hs closed formula for al umbers we ca deduce, amog oher hgs, ha he umbers ( ) are always posve (for {,,}): he rows of he able of al umbers have alerag eres, begg by he egave facorals! Also oe ha he pary of al umber depeds oly o dex ad does o deped o dex 5 TE LA NUMBERS For ay real umber x ad for ay aural umber we defe he dowward facoral polyomal o be The Lah umbers are defed as he coeffces : x(x ) (x + ) ha sasfy he dey L 2 ] 2 [x] L [ x] + L [ x] + K + L [ x (7) We also exed he defo for oher dexes, spulag ha L whe >, or, or By smlar mehods used for he al umbers, we ca deduce ha he Lah umbers also form square marces whch are verse of self: m L L δ m,, s s m m 5
5 The complee aalogy bewee Lah ad al umbers wll be saed he followg resul Theorem 4 The Lah umbers L ) sasfy he followg recurrece ad drec formula: ( ( + )L L (8) L + L! ( ) (9)! Proof: Frs we fd he coeco bewee he polyomals [x] ad : [x] (x)(x ) (x +) () x(x + ) (x + ) (), ha s, () [x] Applyg hs formula () we oba ha also mples ha The dey (7) he mples ha () [x] () + +(), [x] () + + +() + L ( ) + () Therefore, he recurrece (8) ad he drec formula (9) are a cosequece of heorems () ad (3) ad he las dey () g APPENDIX: Frs Iverso Formula The followg s a well ow resul combaorcs Theorem 5 Le ϕ (x) ad ψ (x) be famles of polyomals of degree ad le α,β (wh ) be ay colleco of real umbers Suppose ha he followg relaos are sasfed : ϕ (x) α ψ (x), (,, ) ψ (x) β ϕ (x), (,, ) b If a, a,,a, b, b,,b are umbers ha sasfy he relaos a β a, for,, Furhermore, α b, for,,,, he 52
6 α m β δ m s s m m Prof: May be looed up he boos of Berge (97) or Pza (22) g REFERENCES BERGE, CLAUDE (97): Prcples of Combaorcs, Academc Press, New Yor PIZA, EDUARDO (22): Combaora Eumerava, Edoral de la Uversdad de Cosa Rca, Sa José (22): Sobre los úmeros de al y Lah Memoras del XIII Smposo de Méodos Maemácos Aplcados a las Cecas, Revsa de Maemáca: Teoría y Aplcacoes, 9() RIORDAN, JON (958): A Iroduco o Combaoral Aalyss, Joh Wley & Sos, Ic, New Yor 53
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