÷ƒ’À; è TeX output 2000.12.12:1655‹ ÿÿÿÿ y ý£ ‘> ïcolor push Blackïhtml:ïcolor push gray 0ï color popï html:Ž’Y ï color popŽŽ ? ‘> ýÚ ïhtml:ï html:Ÿ.¸M‘õÃïcolor push Blackï color popŽŽ‘õÃï4PSfile=logo29.eps llx=0 lly=0 urx=99 ury=16 rwi=2880ŽŽŽŸ(sç‘ç·óÂÖN G® cmbx12ºThe›záAkiy–ÿuÂama-T‘þaGaniga“w“a˜algorithm˜forŽŸ\'‘]êƒBernoulli‘zán–ÿuÂum“b‘ Š=ersŽŸ6Û‘{P×óX«Q cmr12»Masanobu‘ê¨Kanek¬roŽŸ ‘dz¡óKñ`y
cmr10²Graduate–UUSc¸ãhoGol“of“MathematicsŽ¤ ’ ƒ¼lKyush•¸ãu‘UUUniv“ersit“yŽ¡‘sj¡F‘ÿ*ªukuok‘ÿqÇa–UU812-8581,“JapanŽŸ ‘FÎ)Email‘UUaddress:‘qÇï2html:ïcolor push cmyk 0 1 0 0mk‘ÿqÇanek•¸ão@math.kyush“u-u.ac.jpï color popï html:ŽŸS!Ã’ –Ûóò"V
cmbx10¼AbstractŽ© óý ':
cmti10½A‘w±dir–ÿ}'e“ct›wìpr“o“of˜is˜given˜for˜A¾“kiyama˜and˜T‘ÿ;¼anigawa's˜algorithm˜for˜c“omputingŽ¡Bernoul‘ ‚Øli–Inumb›ÿ}'ers.‘ )»The“pr˜o˜of“uses“a“close˜d“formula“for“Bernoul‘ ‚Øli“numb˜ersŽ¡expr–ÿ}'esse“d–¹•in“terms“of“Stirling“numb›ÿ}'ers.‘
¡The“outc˜ome“of“the“same“algorithmŽ¡with–“çdier›ÿ}'ent“initial“values“is“also“brie
y“discusse˜d.ŽŸ‹ïhtml:ï html:Ÿ9óÂÖN ff cmbx12¿1Ž‘LËThe‘ffAlgorithmŽŸç²In–ûÖtheir“study“of“v‘ÿqÇalues“at“non-pGositivš¸ãe“in˜teger“argumen˜ts“of“m˜ultiple“zeta“func-Ž¡tions,‘8VS.–×¼Akiyš¸ãama“and“Y.“T‘ÿ*ªaniga˜w˜a“[ïhtml:ïcolor push cmyk 0 1 0 01ï color popï html:Ž‘ ]“found“as“a“spGecial“case“an“am˜usingŽ¡algorithm–`±for“computing“Bernoulli“n•¸ãum“bGers–`±in“a“manner“similar“to“\P¸ãascal'sŽ¡triangle"–UUfor“binomial“coGecien¸ãts.Ž¡‘ Their–¢…algorithm“reads“as“folloš¸ãws:‘&Start“with“the“0-th“ro˜w“1ó
b>
cmmi10µ;Ÿü‘ˆƒóÙ“ R cmr7±1Ž‘ˆƒŸ£&‰ fe üsŸ¿˜2ŽŽŽŽ– ¸)µ;Ÿü‘ˆƒ±1Ž‘ˆƒŸ£&‰ fe üsŸ¿˜3ŽŽŽŽ“µ;Ÿü‘ˆƒ±1Ž‘ˆƒŸ£&‰ fe üsŸ¿˜4ŽŽŽŽ“µ;Ÿü‘ˆƒ±1Ž‘ˆƒŸ£&‰ fe üsŸ¿˜5ŽŽŽŽ“µ;–ª¨:“:“:ŽŽ¡²and–9dene“the“rst“roš¸ãw“b˜y“1–°©ó
!",š
cmsy10¸“²(1“¸ Ÿü‘ãܱ1Ž‘ãÜŸ£&‰ fe üsŸ¿˜2ŽŽŽŽ›‚²)µ;‘UP²2“¸“²(Ÿü‘33±1Ž‘33Ÿ£&‰ fe üsŸ¿˜2ŽŽŽŽ˜¸ Ÿü‘ãܱ1Ž‘ãÜŸ£&‰ fe üsŸ¿˜3ŽŽŽŽ˜²)µ;‘UP²3“¸“²(Ÿü‘33±1Ž‘33Ÿ£&‰ fe üsŸ¿˜3ŽŽŽŽ˜¸ Ÿü‘ãܱ1Ž‘ãÜŸ£&‰ fe üsŸ¿˜4ŽŽŽŽ˜²)µ;–ª¨:“:“:Ž‘0²whic¸ãh‘9proGducesŽ¡the› sequenceŸü‘SA±1Ž‘SAŸ£&‰ fe üsŸ¿˜2ŽŽŽŽ‘ ‚çµ;Ÿü‘ÝÛ±1Ž‘ÝÛŸ£&‰ fe üsŸ¿˜3ŽŽŽŽ–
µ;Ÿü‘ÝÛ±1Ž‘ÝÛŸ£&‰ fe üsŸ¿˜4ŽŽŽŽ“µ;–ª¨:“:“:Ž‘ÿ÷:˜²Then˜dene˜the˜next˜ro•¸ãw˜b“y˜1–ÎR¸“²(Ÿü‘33±1Ž‘33Ÿ£&‰ fe üsŸ¿˜2ŽŽŽŽ–1+¸ Ÿü‘…±1Ž‘…Ÿ£&‰ fe üsŸ¿˜3ŽŽŽŽ“²)µ;›UP²2–ÎR¸“²(Ÿü‘33±1Ž‘33Ÿ£&‰ fe üsŸ¿˜3ŽŽŽŽ–1+¸ Ÿü‘…±1Ž‘…Ÿ£&‰ fe üsŸ¿˜4ŽŽŽŽ“²)µ;˜²3‘ÎR¸Ž¡²(Ÿü‘33±1Ž‘33Ÿ£&‰ fe üsŸ¿˜4ŽŽŽŽ–½U¸ Ÿü‘¯±1Ž‘¯Ÿ£&‰ fe üsŸ¿˜5ŽŽŽŽ“²)µ;–ª¨:“:“:Ž‘ÿ÷;–æ"²thš¸ãus“givingŸü‘U±1Ž‘UŸ£&‰ fe üsŸ¿˜6ŽŽŽŽ‘ Hûµ;Ÿü‘ÝÛ±1Ž‘ÝÛŸ£&‰ fe üsŸ¿˜6ŽŽŽŽ‘
µ;Ÿü‘ܱ3Ž‘ÝÛŸ£&‰ fe ø柿˜20ŽŽŽŽ‘ ôµ;–ª¨:“:“:Ž‘æ²as“the“second“ro˜w.‘L¶In“general,‘ü`denoting“theŽ¡µm²-th–Öç(“µm–Ÿ²=“0µ;–ª¨²1µ;“²2µ;“:“:“:Ž‘ÿ÷²)›Öçn•¸ãum“bGer˜in˜the˜µn²-th˜(µn–Ÿ²=“0µ;–ª¨²1µ;“²2µ;“:“:“:Ž‘ÿ÷²)˜ro•¸ãw˜b“y˜µaŸÿó 0e—r cmmi7´n;mŽ‘ê²,‘÷LtheŽ¡µm²-th›UUn•¸ãum“bGer˜in˜the˜(µn–8à²+“1)-st˜ro•¸ãw˜µaŸÿ´n±+1´;mŽ‘_q²is˜determined˜recursiv“ely˜b“yŽ¦‘]„”µaŸÿ´n±+1´;mŽ‘Ñ4²=‘Ç(µm–8à²+“1)“¸“²(µaŸÿ´n;mŽ‘"ù¸ “µaŸÿ´n;m±+1Ž‘
²)µ:ŽŽŸ ‘> ïcolor push Black’ ª ²1Ž’Y ï color popŽŽŒ‹ * y ý£ ‘> ïcolor push Blackïhtml:ïcolor push gray 0ï color popï html:Ž’Y ï color popŽŽ ? ýä ‘> ²Then–2§the“claim“is“that“the“0-th“compGonenš¸ãt“µaŸÿ´n;±0Ž‘ ˜²of“eac˜h“ro˜w“(the“\leading“diag-Ž¤ ‘> onal")–UUis“just“the“µn²-th“Bernoulli“n•¸ãum“bGers–UUµBŸÿ´nŽ‘q~²,“whereŽŸ‘Ÿóý’ ›¾ƒóO!â… cmsy7·1ŽŸ€’ ˜‚Ÿöüó ú±u
cmex10«XŽŽŸ†’ ˜25´n±=0ŽŽ’ ¨î^µBŸÿ´nŽŸù<$‘¤±µxŸü^ÿ´nŽŽ‘¤±Ÿw‰ fe (šŸ (Ö‘0rµn²!ŽŽŽŽ‘Ç–=Ÿù<$‘ »-µxeŸü^ÿ´xŽŽ‘úKŸw‰ fe éÄŸ (ÖµeŸýr´xŽ‘Aĸ ‘8à²1ŽŽŽŽ‘!ÁêŸñæ^«Ž‘)^²=Ÿù<$‘“ŸµxŽ‘úKŸw‰ fe éÄŸ (ÖeŸýr´xŽ‘Aĸ ‘8à²1ŽŽŽŽ‘"P"+‘8àµxŸñæ^«ŽŽ‘kóÔµ:ŽŸ R‘> ²Note–É that“wš¸ãe“are“using“the“denition“of“the“Bernoulli“n˜um˜bGers“in“whic˜h“µBŸÿ±1Ž‘C‹²=Ÿü‘úK±1Ž‘úKŸ£&‰ fe üsŸ¿˜2ŽŽŽŽ‘ )ñ².Ž¡‘> This–+-is“the“denition“used“bš¸ãy“Bernoulli“(and“indepGenden˜tly“Seki,‘3›published“oneŽ¡‘> yš¸ãear–jÊprior“to“Bernoulli).‘²%Inciden˜tally‘ÿ*ª,‘°&this“is“more“appropriate“for“the“EulerŽ¡‘> form¸ãula›{0µ‘ ¼ã²(1–R¸ “µkP—²)–/=“¸ µBŸÿ´kŽ‘ëµ=k‘ËDz(µk‘VƲ=“1µ;–ª¨²2µ;“²3µ;“:“:“:Ž‘ÿ÷²)˜for˜the˜v‘ÿqÇalues˜of˜the˜Riemann˜zetaŽ¡‘> function.Ž y`8‘> þ¯¯ïcolor push Black PìQŸè9‘ú˜g þË Þóü<˜
lcircle10®
‘à Ÿð Ž‘ð Ÿð ŽŽŽŽ þÇv²1Ž þ‘+á ±1Ž‘+á Ÿ£&‰ fe üsŸ¿˜2ŽŽŽŽŽ þ‘VŽÍ1Ž‘VŽÍŸ£&‰ fe üsŸ¿˜3ŽŽŽŽŽ þÂ’ <š1Ž’ <šŸ£&‰ fe üsŸ¿˜4ŽŽŽŽŽ þÂ’ «êg1Ž’ «êgŸ£&‰ fe üsŸ¿˜5ŽŽŽŽŽ þÂ’ Ö˜41Ž’ Ö˜4Ÿ£&‰ fe üsŸ¿˜6ŽŽŽŽŽ þÂ’F1Ž’FŸ£&‰ fe üsŸ¿˜7ŽŽŽŽŽ þÂ’+óÎ1Ž’+óΟ£&‰ fe üsŸ¿˜8ŽŽŽŽŽ þÂ’V¡›1Ž’V¡›Ÿ£&‰ fe üsŸ¿˜9ŽŽŽŽ’g{ê þÇv¸–ª¨“ŽŽ‘‰) þÙŽŸóäO£
line10¬JŽ‘
@ þà ŠJŽ‘
@ þà Š^ŽŽ‘36ö þÙŽŸJŽ‘7íØ þà ŠJŽ‘7íØ þà Š^ŽŽ‘]äà þÙŽŸJŽ‘b›¥ þà ŠJŽ‘b›¥ þà Š^ŽŽ’ ˆ’ þÙŽŸJŽ’ Ir þà ŠJŽ’ Ir þà Š^ŽŽ’ ³@] þÙŽŸJŽ’ ·÷? þà ŠJŽ’ ·÷? þà Š^ŽŽ’ Ýî* þÙŽŸJŽ’ ⥠þà ŠJŽ’ ⥠þà Š^ŽŽ’›÷ þÙŽŸJŽ’
RÙ þà ŠJŽ’
RÙ þà Š^ŽŽ’3IÄ þÙŽŸJŽ’8 ¦ þà ŠJŽ’8 ¦ þà Š^ŽŽ‘!*À þÙŽŸ
Ž‘sÞ þà Š
Ž‘sÞ þà ŠŽŽ‘KØ þÙŽŸ
Ž‘G!« þà Š
Ž‘G!« þà ŠŽŽ‘v†Z þÙŽŸ
Ž‘qÏx þà Š
Ž‘qÏx þà ŠŽŽ’ ¡4' þÙŽŸ
Ž’ œ}E þà Š
Ž’ œ}E þà ŠŽŽ’ Ëáô þÙŽŸ
Ž’ Ç+ þà Š
Ž’ Ç+ þà ŠŽŽ’ öÁ þÙŽŸ
Ž’ ñØß þà Š
Ž’ ñØß þà ŠŽŽ’!=Ž þÙŽŸ
Ž’†¬ þà Š
Ž’†¬ þà ŠŽŽ’Kë[ þÙŽŸ
Ž’G4y þà Š
Ž’G4y þà ŠŽŽ‘€ú þñ!?®
‘à Ÿð Ž‘ð Ÿð ŽŽŽŽ þ艑Š±1Ž‘ŠŸ£&‰ fe üsŸ¿˜2ŽŽŽŽŽ þ艑A7æ1Ž‘A7柣&‰ fe üsŸ¿˜3ŽŽŽŽŽ þ艑kå³1Ž‘k峟£&‰ fe üsŸ¿˜4ŽŽŽŽŽ þ艒 –“€1Ž’ –“€Ÿ£&‰ fe üsŸ¿˜5ŽŽŽŽŽ þ艒 ÁAM1Ž’ ÁAMŸ£&‰ fe üsŸ¿˜6ŽŽŽŽŽ þ艒 ëï1Ž’ ë&‰ fe üsŸ¿˜7ŽŽŽŽŽ þ艒œç1Ž’œçŸ£&‰ fe üsŸ¿˜8ŽŽŽŽŽ þ艒AJ´1Ž’AJ´Ÿ£&‰ fe üsŸ¿˜9ŽŽŽŽ’R% þì}¸–ª¨“ŽŽ‘à þþ‹¦¬JŽ‘"–ñ ÿ‘JŽ‘"–ñ ÿ‘^ŽŽ‘HÜ þþ‹¦JŽ‘MD¾ ÿ‘JŽ‘MD¾ ÿ‘^ŽŽ‘s;© þþ‹¦JŽ‘wò‹ ÿ‘JŽ‘wò‹ ÿ‘^ŽŽ’ év þþ‹¦JŽ’ ¢ X ÿ‘JŽ’ ¢ X ÿ‘^ŽŽ’ È—C þþ‹¦JŽ’ ÍN% ÿ‘JŽ’ ÍN% ÿ‘^ŽŽ’ óE þþ‹¦JŽ’ ÷ûò ÿ‘JŽ’ ÷ûò ÿ‘^ŽŽ’òÝ þþ‹¦JŽ’"©¿ ÿ‘JŽ’"©¿ ÿ‘^ŽŽ‘6¦ þþ‹¦
Ž‘1ÊÄ ÿ‘
Ž‘1ÊÄ ÿ‘ŽŽ‘a/s þþ‹¦
Ž‘\x‘ ÿ‘
Ž‘\x‘ ÿ‘ŽŽ’ ‹Ý@ þþ‹¦
Ž’ ‡&^ ÿ‘
Ž’ ‡&^ ÿ‘ŽŽ’ ¶‹
þþ‹¦
Ž’ ±Ô+ ÿ‘
Ž’ ±Ô+ ÿ‘ŽŽ’ á8Ú þþ‹¦
Ž’ Üø ÿ‘
Ž’ Üø ÿ‘ŽŽ’æ§ þþ‹¦
Ž’/Å ÿ‘
Ž’/Å ÿ‘ŽŽ’6”t þþ‹¦
Ž’1Ý’ ÿ‘
Ž’1Ý’ ÿ‘ŽŽ‘%×à ÿF®
‘à Ÿð Ž‘ð Ÿð ŽŽŽŽ ÿ
‘+á ±1Ž‘+á Ÿ£&‰ fe üsŸ¿˜6ŽŽŽŽŽ ÿ
‘VŽÍ1Ž‘VŽÍŸ£&‰ fe üsŸ¿˜6ŽŽŽŽŽ ÿ
’ ƒ:Ô3Ž’ <šŸ£&‰ fe ø柿˜20ŽŽŽŽŽ ÿ
’ è¡2Ž’ «êgŸ£&‰ fe ø柿˜15ŽŽŽŽŽ ÿ
’ Ø–n5Ž’ Ö˜4Ÿ£&‰ fe ø柿˜42ŽŽŽŽŽ ÿ
’D;3Ž’FŸ£&‰ fe ø柿˜28ŽŽŽŽŽ ÿ
’-ò7Ž’+óΟ£&‰ fe ø柿˜72ŽŽŽŽ’@Ê ÿÿ„¸–ª¨“ŽŽ‘36ö ÿ#ˆ¬JŽ‘7íØ ÿ*š˜JŽ‘7íØ ÿ*š˜^ŽŽ‘]äà ÿ#ˆJŽ‘b›¥ ÿ*š˜JŽ‘b›¥ ÿ*š˜^ŽŽ’ ˆ’ ÿ#ˆJŽ’ Ir ÿ*š˜JŽ’ Ir ÿ*š˜^ŽŽ’ ³@] ÿ#ˆJŽ’ ·÷? ÿ*š˜JŽ’ ·÷? ÿ*š˜^ŽŽ’ Ýî* ÿ#ˆJŽ’ ⥠ÿ*š˜JŽ’ ⥠ÿ*š˜^ŽŽ’›÷ ÿ#ˆJŽ’
RÙ ÿ*š˜JŽ’
RÙ ÿ*š˜^ŽŽ‘KØ ÿ#ˆ
Ž‘G!« ÿ*š˜
Ž‘G!« ÿ*š˜ŽŽ‘v†Z ÿ#ˆ
Ž‘qÏx ÿ*š˜
Ž‘qÏx ÿ*š˜ŽŽ’ ¡4' ÿ#ˆ
Ž’ œ}E ÿ*š˜
Ž’ œ}E ÿ*š˜ŽŽ’ Ëáô ÿ#ˆ
Ž’ Ç+ ÿ*š˜
Ž’ Ç+ ÿ*š˜ŽŽ’ öÁ ÿ#ˆ
Ž’ ñØß ÿ*š˜
Ž’ ñØß ÿ*š˜ŽŽ’!=Ž ÿ#ˆ
Ž’†¬ ÿ*š˜
Ž’†¬ ÿ*š˜ŽŽ‘: ÿ9÷ó®
‘à Ÿð Ž‘ð Ÿð ŽŽŽŽ‘@³ ÿ5ü‹²0Ž ÿ2—‘mãí±1Ž‘k峟£&‰ fe ø柿˜30ŽŽŽŽŽ ÿ2—’ ˜‘º1Ž’ –“€Ÿ£&‰ fe ø柿˜20ŽŽŽŽŽ ÿ2—’ Ã?‡2Ž’ ÁAMŸ£&‰ fe ø柿˜35ŽŽŽŽŽ ÿ2—’ ííT5Ž’ ë&‰ fe ø柿˜84ŽŽŽŽŽ ÿ2—’›!5Ž’œçŸ£&‰ fe ø柿˜84ŽŽŽŽ’+s© ÿ5ü‹¸–ª¨“ŽŽ‘HÜ ÿH…´¬JŽ‘MD¾ ÿO—ŸJŽ‘MD¾ ÿO—Ÿ^ŽŽ‘s;© ÿH…´JŽ‘wò‹ ÿO—ŸJŽ‘wò‹ ÿO—Ÿ^ŽŽ’ év ÿH…´JŽ’ ¢ X ÿO—ŸJŽ’ ¢ X ÿO—Ÿ^ŽŽ’ È—C ÿH…´JŽ’ ÍN% ÿO—ŸJŽ’ ÍN% ÿO—Ÿ^ŽŽ’ óE ÿH…´JŽ’ ÷ûò ÿO—ŸJŽ’ ÷ûò ÿO—Ÿ^ŽŽ‘a/s ÿH…´
Ž‘\x‘ ÿO—Ÿ
Ž‘\x‘ ÿO—ŸŽŽ’ ‹Ý@ ÿH…´
Ž’ ‡&^ ÿO—Ÿ
Ž’ ‡&^ ÿO—ŸŽŽ’ ¶‹
ÿH…´
Ž’ ±Ô+ ÿO—Ÿ
Ž’ ±Ô+ ÿO—ŸŽŽ’ á8Ú ÿH…´
Ž’ Üø ÿO—Ÿ
Ž’ Üø ÿO—ŸŽŽ’æ§ ÿH…´
Ž’/Å ÿO—Ÿ
Ž’/Å ÿO—ŸŽŽ‘Mô ÿaÏ~®‘Ø Ÿì Ž‘ì Ÿì ŽŽŽŽ‘Iú ÿZù’¸ Ÿü‘1m±1Ž‘33Ÿ£&‰ fe ø柿˜30ŽŽŽŽŽ‘t§Û ÿZù’¸ Ÿü‘1m±1Ž‘33Ÿ£&‰ fe ø柿˜30ŽŽŽŽŽ’ ŸU¨ ÿZù’¸ Ÿü‘/¦±3Ž‘33Ÿ£&‰ fe õYŸ¿˜140ŽŽŽŽŽ’ Ç+ ÿZù’¸ Ÿü‘/¦±1Ž‘33Ÿ£&‰ fe õYŸ¿˜105ŽŽŽŽŽ’ Î ÿZù’²0‘ª©¸–ª¨“ŽŽ‘]äà ÿm‚»¬JŽ‘b›¥ ÿt”¦JŽ‘b›¥ ÿt”¦^ŽŽ’ ˆ’ ÿm‚»JŽ’ Ir ÿt”¦JŽ’ Ir ÿt”¦^ŽŽ’ ³@] ÿm‚»JŽ’ ·÷? ÿt”¦JŽ’ ·÷? ÿt”¦^ŽŽ’ Ýî* ÿm‚»JŽ’ ⥠ÿt”¦JŽ’ ⥠ÿt”¦^ŽŽ‘v†Z ÿm‚»
Ž‘qÏx ÿt”¦
Ž‘qÏx ÿt”¦ŽŽ’ ¡4' ÿm‚»
Ž’ œ}E ÿt”¦
Ž’ œ}E ÿt”¦ŽŽ’ Ëáô ÿm‚»
Ž’ Ç+ ÿt”¦
Ž’ Ç+ ÿt”¦ŽŽ’ öÁ ÿm‚»
Ž’ ñØß ÿt”¦
Ž’ ñØß ÿt”¦ŽŽ‘eÜ”Ÿƒ`S®
‘à Ÿð Ž‘ð Ÿð ŽŽŽŽ‘j²€ ÿö™²0Ž’ ‰þÁ ÿö™¸ Ÿü‘1m±1Ž‘33Ÿ£&‰ fe ø柿˜42ŽŽŽŽŽ’ ´¬Ž ÿö™¸ Ÿü‘1m±1Ž‘33Ÿ£&‰ fe ø柿˜28ŽŽŽŽŽ’ Üø ÿö™¸ Ÿü‘/¦±4Ž‘33Ÿ£&‰ fe õYŸ¿˜105ŽŽŽŽ‘h¸–ª¨“ŽŽ‘s;©Ÿ’¬JŽ‘wò‹Ÿ™‘JŽ‘wò‹Ÿ™‘^ŽŽ’ évŸ’ÂJŽ’ ¢ XŸ™‘JŽ’ ¢ XŸ™‘^ŽŽ’ È—CŸ’ÂJŽ’ ÍN%Ÿ™‘JŽ’ ÍN%Ÿ™‘^ŽŽ’ ‹Ý@Ÿ’Â
Ž’ ‡&^Ÿ™‘
Ž’ ‡&^Ÿ™‘ŽŽ’ ¶‹
Ÿ’Â
Ž’ ±Ô+Ÿ™‘
Ž’ ±Ô+Ÿ™‘ŽŽ’ á8ÚŸ’Â
Ž’ ÜøŸ™‘
Ž’ ÜøŸ™‘ŽŽ‘|Ÿ¬Ÿ©ÉŒ®
‘à Ÿð Ž‘ð Ÿð ŽŽŽŽŸ¡¬’ ƒ:Ô±1Ž’ <šŸ£&‰ fe ø柿˜42ŽŽŽŽŽŸ¡¬’ è¡1Ž’ «êgŸ£&‰ fe ø柿˜42ŽŽŽŽŽŸ¡¬’ Ú”§1Ž’ Ö˜4Ÿ£&‰ fe õYŸ¿˜140ŽŽŽŽ’ ïkiŸ¤ó ¸–ª¨“ŽŽ’ ˆ’Ÿ·|ɬJŽ’ IrŸ¾Ž´JŽ’ IrŸ¾Ž´^ŽŽ’ ³@]Ÿ·|ÉJŽ’ ·÷?Ÿ¾Ž´JŽ’ ·÷?Ÿ¾Ž´^ŽŽ’ ¡4'Ÿ·|É
Ž’ œ}EŸ¾Ž´
Ž’ œ}EŸ¾Ž´ŽŽ’ ËáôŸ·|É
Ž’ Ç+Ÿ¾Ž´
Ž’ Ç+Ÿ¾Ž´ŽŽ’ ø´ŸÍì®
‘à Ÿð Ž‘ð Ÿð ŽŽŽŽ’ •`MŸÉð§²0ŽŸÆ ³’ ÁÓV±1Ž’ ¿ÕŸ£&‰ fe ø柿˜30ŽŽŽŽ’ Ô«ÞŸÉ𧸖ª¨“ŽŽ’ évŸÜyЬJŽ’ ¢ XŸã‹»JŽ’ ¢ XŸã‹»^ŽŽ’ ¶‹
ŸÜyÐ
Ž’ ±Ô+Ÿã‹»
Ž’ ±Ô+Ÿã‹»ŽŽ’ ¡Q¼ŸõÚ®‘Ø Ÿì Ž‘ì Ÿì ŽŽŽŽ’ ŸU¨Ÿîí®¸ Ÿü‘1m±1Ž‘33Ÿ£&‰ fe ø柿˜30ŽŽŽŽ‘ õ¸–ª¨“ŽŽŽŽŽŽŸ ‘ZuJïcolor push Black²Figure›UU1:‘qÇïhtml:ïcolor push gray 0ï color popï html:Akiy•¸ãama-T‘ÿ*ªaniga“w“a˜triangleï color popŽŽŽï color popŽ‘> ŸÄïhtml:ï html:ŽŸ ‘> ïcolor push Black’ ª 2Ž’Y ï color popŽŽŒ‹ f y ý£ ‘> ïcolor push Blackïhtml:ïcolor push gray 0ï color popï html:Ž’Y ï color popŽŽ ? ýä ‘> ¿2Ž‘VLËPros3ofŽŸç‘> ²The–proGof“is“based“on“the“folloš¸ãwing“iden˜tit˜y“for“Bernoulli“n˜um˜bGers,‘Nîa“v‘ÿqÇarian˜tŽ¤ ‘> of–ìúwhicš¸ãh“goGes“as“far“bac˜k“as“Kronec˜k˜er“(see“[ïhtml:ïcolor push cmyk 0 1 0 04ï color popï html:Ž‘ ]).‘8µHere“w˜e“denote“b˜y“Ÿ÷æb«ŽŸû
‘
Õà´nŽŸâ¾‘ ÂQmŽŽ‘ÚìŸ÷æb« ŽŽŽ‘=²theŽ¡‘> Stirling›UUn•¸ãum“bGer˜of˜the˜second˜kind:ŽŸV’ Àà#µxŸûÞÿ´nŽ‘8–²=Ÿóý‘ꨴnŽŸ€‘*ƒŸöü«XŽŽŸ†‘Ç´m±=0ŽŽ‘ª^Ÿñæ^«ŽŸù<$‘îµnŽŸ
Ÿå‘*_mŽŽ‘%òŸñæ^«ŽŽŽ‘-rµxŸûÞÿ´mŽŸ\$‰ W ˜›Ž‘˜›µ;ŽŸ²‘> ²where›³˜µxŸü^ÿ´mŽŸ\$‰ W ˜›Ž‘
üͲ=‘d2µx²(µx–w·¸ “²1)–ª¨¸““Ž‘ÿ÷²(µx“¸ “µm“²+“1)˜for˜µm–d2¸“²1˜and˜µxŸü^ÿ±0ŽŸ\$‰ W |sŽ‘ॲ=“1.‘Œ‘(W‘ÿ*ªe˜use˜Kn¸ãuth'sŽ¡‘> notation–y[ïhtml:ïcolor push cmyk 0 1 0 07ï color popï html:Ž‘ ].‘Y)F‘ÿ*ªor“the“Stirling“n•¸ãum“bGer›yiden“tities˜that˜w“e˜shall˜need,‘?the˜reader˜isŽ¡‘> referred–UUfor“example“to“[ïhtml:ïcolor push cmyk 0 1 0 05ï color popï html:Ž‘ ].)Ž‘> Ÿ€ ïhtml:ï html:Ÿ`½‘û ïcolor push Black‘ ¼Theorem‘ÕT1ï color popŽŽŽŸ!N‘U™µBŸÿ´nŽ‘8–²=Ÿóý‘ꨴnŽŸ€‘*ƒŸöü«XŽŽŸ†‘Ç´m±=0ŽŽŸ÷Ñ}‘Ý‘²(¸ ²1)Ÿü^ÿ´mŽ‘˜›µm²!Ÿ÷æb«ŽŸû
‘èæ´n±+1ŽŸâ¾‘ÕW´m±+1ŽŽ‘
õŸ÷æb« ŽŽŽŽ‘Ý‘Ÿáµ‰ fe D˜øŸ (Ö‘L1µm–8à²+“1ŽŽŽŽ‘\©¼µ;‘h¸8µn–Ǹ“²0µ:Ž©€
²W‘ÿ*ªe–º6shall“givš¸ãe“later“a“proGof“of“this“iden˜tit˜y“for“the“sak˜e“of“completeness.‘ kOnceŽ¡w•¸ãe›UUha“v“e˜this,˜the˜next˜propGosition˜ensures˜the˜v‘ÿqÇalidit“y˜of˜our˜algorithm.ŽŸñÇïhtml:ï html:Ÿîö‘û ïcolor push Black‘ ¼PropQÇosition‘ÕT2ï color popŽŽ‘Hìµ½Given–ýSan“initial“se–ÿ}'quenc“e›ýSµaŸÿ±0´;mŽ‘òa²(µm–UÕ²=“0µ;–ª¨²1µ;“²2µ;“:“:“:Ž‘ÿ÷²)½,‘W®dene˜theŽ¡se–ÿ}'quenc“es›“çµaŸÿ´n;mŽ‘~ ²(µn–Ǹ“²1)˜½r–ÿ}'e“cursively˜byŽ¤à½‘3eȵaŸÿ´n;mŽ‘±1²=›Ç(µm–8à²+“1)“¸“²(µaŸÿ´n· ±1´;mŽ‘_m¸ “µaŸÿ´n· ±1´;m±+1Ž‘#F²)‘
8à(µn˜¸˜²1µ;‘ª¨m˜¸˜²0)µ:ŽŽŽ’L8áïhtml:ï html:²(1)ŽŽŽŽŽ¡½ThenŽŸdN‘aOªµaŸÿ´n;±0Ž‘• ²=Ÿóý‘ꨴnŽŸ€‘*ƒŸöü«XŽŽŸ†‘Ç´m±=0ŽŽ‘ÿ¶²(¸ ²1)ŸûÞÿ´mŽ‘˜›µm²!Ÿñæ^«ŽŸù<$‘ãµn–8à²+“1ŽŸ
Ÿå‘€µm–8à²+“1ŽŽ‘!€—Ÿñæ^«ŽŽŽ‘) ˜µaŸÿ±0´;mŽ‘
õµ:ŽŽŽ’L8áïhtml:ï html:²(2)ŽŽŽŽŽ¦½Pr–ÿ}'o“of.‘
ŽP²PutŽŸuX’ €~ñµgŸÿ´nŽ‘q~²(µt²)–Ç=Ÿóý‘fõ·1ŽŸ€‘*ƒŸöü«XŽŽŸ†“´m±=0ŽŽ‘ª^µaŸÿ´n;mŽ‘êµtŸûÞÿ´mŽ‘˜›µ:ŽŸÄ²By–UUthe“recursion“(ïhtml:ïcolor push cmyk 0 1 0 01ï color popï html:)“wš¸ãe“ha˜v˜e“for“µn–Ǹ“²1ŽŸäN‘.üzµgŸÿ´nŽ‘q~²(µt²)ŽŽ‘N–’=ŽŽŸóý‘dý·1ŽŸ€‘aÁŸöü«XŽŽŸ†‘`]°´m±=0ŽŽ‘q–N²(µm–8à²+“1)(µaŸÿ´n· ±1´;mŽ‘_m¸ “µaŸÿ´n· ±1´;m±+1Ž‘#F²)µtŸûÞÿ´mŽŽŽŽŸ!1‘N–’²=ŽŽŸù<$‘c_µdŽ‘aãŸw‰ fe ÐäŸ (ÖdtŽŽŽŽ‘k”ú²(Ÿóý‘ŸÝ·1ŽŸ€‘ckŸöü«XŽŽŸ†´m±=0ŽŽ–ãFµaŸÿ´n· ±1´;mŽ‘&µtŸûÞÿ´m±+1Ž›¸ž²)‘8ภŸù<$‘:LµdŽ‘lŸw‰ fe ÐäŸ (ÖdtŽŽŽŽ‘
p*²(Ÿóý‘ŸÝ·1ŽŸ€‘ckŸöü«XŽŽŸ†´m±=0ŽŽ“µaŸÿ´n· ±1´;m±+1Ž‘#FµtŸûÞÿ´m±+1Ž˜²)ŽŽŽŸc)‘N–’=ŽŽŸù<$‘c_µdŽ‘aãŸw‰ fe ÐäŸ (ÖdtŽŽŽŽ‘k”ú²(µtgŸÿ´n· ±1Ž–ò²(µt²))›8ภŸù<$‘:LµdŽ‘lŸw‰ fe ÐäŸ (ÖdtŽŽŽŽ‘
p*²(µgŸÿ´n· ±1Ž“²(µt²)˜¸ ˜µaŸÿ´n· ±1´;±0Ž‘
e²)ŽŽŽ¤‘¬‘N–’=ŽŽ‘`]°µgŸÿ´n· ±1Ž›ò²(µt²)–8à+“(µt“¸ “²1)Ÿù<$‘lµdŽ‘33Ÿw‰ fe ÐäŸ (ÖdtŽŽŽŽ‘7J²(µgŸÿ´n· ±1Ž˜²(µt²))ŽŽŽ¡‘N–’=ŽŽŸù<$‘c_µdŽ‘aãŸw‰ fe ÐäŸ (ÖdtŽŽŽŽ‘k”ú²((µt–8ภ“²1)µgŸÿ´n· ±1Ž‘ò²(µt²))µ:ŽŽŽŽŸ ‘> ïcolor push Black’ ª ²3Ž’Y ï color popŽŽŒ‹ )È y ý£ ‘> ïcolor push Blackïhtml:ïcolor push gray 0ï color popï html:Ž’Y ï color popŽŽ ? ýä ‘> ²Hence,–UUb¸ãy“putting“(µt–8ภ“²1)µgŸÿ´nŽ›q~²(µt²)–Ç=“µhŸÿ´nŽ˜²(µt²)–UUw¸ãe“obtainŽŸ5£’ ™ßœµhŸÿ´nŽ‘q~²(µt²)–Ç=“(µt–8ภ“²1)Ÿù<$‘lµdŽ‘33Ÿw‰ fe ÐäŸ (ÖdtŽŽŽŽ‘7J²(µhŸÿ´n· ±1Ž‘ò²(µt²))‘
(µn–Ǹ“²1)µ;ŽŸÍБ> ²and‘UUth¸ãusŽŸ
³’ ©œYµhŸÿ´nŽ‘q~²(µt²)–Ç=“Ÿñæ^«Ž‘
#Œ²(µt–8ภ“²1)Ÿù<$‘lµdŽ‘33Ÿw‰ fe ÐäŸ (ÖdtŽŽŽŽ‘7JŸñæ^«ŽŽ‘9S¹ŸóøÛ´nŽ‘@oß²(µhŸÿ±0Ž‘|s²(µt²))µ:ŽŸ ‘> ²Applying–UUthe“form¸ãula“(½cf.›qDz[ïhtml:ïcolor push cmyk 0 1 0 05ï color popï html:Ž‘ ,“Ch.˜6.7“Exer.˜13])ŽŸ‘’ ¡—4Ÿñæ^«Ž’ ¨ó¨µxŸù<$‘ÁdŽ‘33Ÿw‰ fe
뎟 (ÖdxŽŽŽŽ‘
QôŸñæ^«ŽŽ’ ÃY,ŸóøÛ´nŽ’ ˑ²=Ÿóý‘ꨴnŽŸ€‘*ƒŸöü«XŽŽŸ†‘Ç´m±=0ŽŽ‘ª^Ÿñæ^«ŽŸù<$‘îµnŽŸ
Ÿå‘*_mŽŽ‘%òŸñæ^«ŽŽŽ‘-rµxŸûÞÿ´mŽ‘ CCŸñæ^«ŽŸù<$‘®xµdŽ‘ÒêŸw‰ fe
뎟 (ÖdxŽŽŽŽ‘ñ«Ÿñæ^«ŽŽ‘%NŸóøÛ´mŽŽ¤ŸM‘> ²for›UUµx–Dz=“µt–8ภ“²1,˜w•¸ãe˜ha“v“eŽ©uX’ ’u8µhŸÿ´nŽ‘q~²(µt²)–Ç=Ÿóý‘ꨴnŽŸ€‘*ƒŸöü«XŽŽŸ†“´m±=0ŽŽ‘ª^Ÿñæ^«ŽŸù<$‘îµnŽŸ
Ÿå‘*_mŽŽ‘%òŸñæ^«ŽŽŽ‘-r²(µt–8ภ“²1)ŸûÞÿ´mŽ‘ CCŸñæ^«ŽŸù<$‘¡#µdŽ‘ÒêŸw‰ fe ÐäŸ (ÖdtŽŽŽŽ‘ןñæ^«ŽŽ‘#3uŸóøÛ´mŽ‘,v¸µhŸÿ±0Ž‘|s²(µt²)µ:Ž¡‘> ²Putting›UUµt–Dz=“0˜w¸ãe˜obtainŽ¦‘JEŠ¸ µaŸÿ´n;±0ŽŽŽ‘m#ɲ=ŽŽŸóý’ …w´nŽŸ€’ €NRŸöü«XŽŽŸ†‘~êç´m±=0ŽŽ’ ‘Î-Ÿñæ^«ŽŸù<$’ š±½µnŽŸ
Ÿå’ ™N.mŽŽ’ ¢åŸñæ^«ŽŽŽ’ ©•æ²(¸ ²1)ŸûÞÿ´mŽ‘˜›µm²!(µaŸÿ±0´;m· ±1Ž‘jb¸ ‘8àµaŸÿ±0´;mŽ‘
õ²)ŽŽŽŸ"°a‘m#É=ŽŽŸóý‘ð=´n· ±1ŽŸ€’ €NRŸöü«XŽŽŸ†‘~êç´m±=0ŽŽ’ ‘Î-Ÿñæ^«ŽŸù<$’ £N,µnŽŸ
Ÿå’ ™N.m–8à²+“1ŽŽ’ ³NÄŸñæ^«ŽŽŽ’ ºÎŲ(¸ ²1)ŸûÞÿ´m±+1Ž‘¸ž²(µm–8à²+“1)!µaŸÿ±0´;mŽ‘-î¸ Ÿóý‘\p´nŽŸ€‘œKŸöü«XŽŽŸ†“´m±=0ŽŽ‘&Ÿñæ^«ŽŸù<$‘ÿ¶µnŽŸ
Ÿå‘œ'mŽŽ‘%cÞŸñæ^«ŽŽŽ‘,ãß²(¸ ²1)ŸûÞÿ´mŽ‘˜›µm²!µaŸÿ±0´;mŽŽŽŽ¤!1‘m#ɲ=ŽŽ‘~êç¸ Ÿóý‘Î8´nŽŸ€‘Ÿöü«XŽŽŸ†‘ª¨´m±=0ŽŽ‘ãF²(¸ ²1)ŸûÞÿ´mŽ‘˜›µm²!µaŸÿ±0´;mŽ‘Ÿ¶Ÿñæ^«Ž‘ü*²(µm–8à²+“1)Ÿñæ^«ŽŸù<$‘ÿµnŽŸ
Ÿå‘€m“²+“1ŽŽ‘!€—Ÿñæ^«ŽŽŽ‘+9x²+“Ÿñæ^«ŽŸù<$‘pµnŽŸ
Ÿå‘ ¸ámŽŽ‘€˜Ÿñæ^«ŽŽŽ‘ ™Ÿñæ^ŽŽŽŽŽ¡‘m#ɲ=ŽŽ‘~êç¸ Ÿóý‘Î8´nŽŸ€‘Ÿöü«XŽŽŸ†‘ª¨´m±=0ŽŽ‘ãF²(¸ ²1)ŸûÞÿ´mŽ‘˜›µm²!Ÿñæ^«ŽŸù<$‘ãµn–8à²+“1ŽŸ
Ÿå‘€µm–8à²+“1ŽŽ‘!€—Ÿñæ^«ŽŽŽ‘) ˜µaŸÿ±0´;mŽ‘
õµ:ŽŽŽŸ Q‘> ²(W‘ÿ*ªe›‰‘ha•¸ãv“e˜used˜the˜recursion˜Ÿ÷æb«ŽŸû
‘
rw´n±+1ŽŸâ¾‘ ^è´m±+1ŽŽ‘—†Ÿ÷æb« ŽŽŽ‘#‹²=‘&(µm–[²²+“1)Ÿ÷æb«ŽŸû
‘øç´nŽŸâ¾‘ÕWm±+1ŽŽ‘
õŸ÷æb« ŽŽŽ‘>þ²+“Ÿ÷æb«ŽŸû
‘ D˜´nŽŸâ¾‘1 mŽŽ‘I¤Ÿ÷æb« ŽŽŽ‘û².)‘{This˜pro•¸ãv“es˜theŽ¤ ‘> propGosition.Ž¡¡‘> ½Pr–ÿ}'o“of–ÿ]of“The–ÿ}'or“em‘ÿ]1.‘ û²W‘ÿ*ªe–àfshoš¸ãw“the“generating“series“of“the“righ˜t“hand“sideŽ¡‘> coincide–UUwith“that“of“µBŸÿ´nŽ‘q~².‘qÇT‘ÿ*ªo“do“this,“wš¸ãe“use“the“iden˜tit˜yŽ¦Ÿù<$’ ¤¤3µeŸü^ÿ´xŽ‘ä²(µeŸü^ÿ´xŽ‘Aĸ ‘8à²1)Ÿü^ÿ´mŽŽ’ ¤¤3Ÿw‰ fe 3úbŸ (Ö‘5ǵm²!ŽŽŽŽ’ ܘà=Ÿóý‘áz·1ŽŸ€‘¥Ÿöü«XŽŽŸ€‘Ç´n±=´mŽŽ‘ŸiŸñæ^«ŽŸù<$‘‚ùµn–8à²+“1ŽŸ
Ÿå‘jµm–8à²+“1ŽŽ‘8 Ÿñæ^«ŽŽŽŸù<$‘@Ó4µxŸü^ÿ´nŽŽ‘@Ó4Ÿw‰ fe (šŸ (Ö‘0rµn²!ŽŽŽŽŽŽŽ’Š8áïhtml:ï html:(3)ŽŽŽŽŽŸ ‘> whicš¸ãh–±‚results“from“the“w˜ell-kno˜wn“generating“series“for“the“Stirling“n˜um˜bGers“(½cf.Ž¡‘> ²[ïhtml:ïcolor push cmyk 0 1 0 05ï color popï html:Ž‘ ,‘UU(7.49)])ŽŸ‘Ÿù<$’ ²(µeŸü^ÿ´xŽ‘Aĸ ‘8à²1)Ÿü^ÿ´mŽŽ’ ²Ÿw‰ fe *I}Ÿ (Ö‘]Uµm²!ŽŽŽŽ’ à\Þ=Ÿóý‘áz·1ŽŸ€‘¥Ÿöü«XŽŽŸ€‘Ç´n±=´mŽŽ‘ŸiŸñæ^«ŽŸù<$‘‚ùµnŽŸ
Ÿå‘jmŽŽ‘&ç!Ÿñæ^«ŽŽŽŸù<$‘/šUµxŸü^ÿ´nŽŽ‘/šUŸw‰ fe (šŸ (Ö‘0rµn²!ŽŽŽŽŽŽŸ ‘> ïcolor push Black’ ª 4Ž’Y ï color popŽŽŒ‹ 8‚ y ý£ ‘> ïcolor push Blackïhtml:ïcolor push gray 0ï color popï html:Ž’Y ï color popŽŽ ? ýä ‘> ²bš¸ãy–=replacing“µm“²with“µm–s²+“1–=and“dieren˜tiating“with“respGect“to“µx².‘iµWith“this,‘Aöw˜eŽ© ‘> ha•¸ãv“eŽŸíöŸóý‘w2·1ŽŸ€‘sÏÀŸöü«XŽŽŸ†‘sä´n±=0ŽŽ’ „<
Ÿîæ\« ŽŸóý’ ’JI´nŽŸ€’ Š$Ÿöü«XŽŽŸ†’ Œ&¹´m±=0ŽŽŸ÷Ñ}’ =2²(¸ ²1)Ÿü^ÿ´mŽ‘˜›µm²!Ÿ÷æb«ŽŸû
‘èæ´n±+1ŽŸâ¾‘ÕW´m±+1ŽŽ‘
õŸ÷æb« ŽŽŽŽ’ =2Ÿáµ‰ fe D˜øŸ (Ö‘L1µm–8à²+“1ŽŽŽŽ’ æ ]Ÿîæ\«!ŽŽŸù<$’ ðÑäµxŸü^ÿ´nŽŽ’ ðÑäŸw‰ fe (šŸ (Ö‘0rµn²!ŽŽŽŽŽŽŽ¤!1‘a¸Æ=ŽŽŸóý‘xÁ·1ŽŸ€‘tãOŸöü«XŽŽŸ†‘sä´m±=0ŽŽŸù<$’ ‡–]²(¸ ²1)Ÿü^ÿ´mŽ‘˜›µm²!Ž’ ‡–]Ÿw‰ fe 'µ¬Ÿ (Ö‘Ú‹µm–8à²+“1ŽŽŽŽŸóý’ ·DF·1ŽŸ€’ ´ÔŸöü«XŽŽŸ€’ ²)ä´n±=´mŽŽ’ Æ5Ÿñæ^«ŽŸù<$’ Îåŵn–8à²+“1ŽŸ
Ÿå’ Í‚6µm–8à²+“1ŽŽ’ ç‚ÌŸñæ^«ŽŽŽŸù<$’ ð6 µxŸü^ÿ´nŽŽ’ ð6 Ÿw‰ fe (šŸ (Ö‘0rµn²!ŽŽŽŽ’ ÿXå=Ÿóý‘fõ·1ŽŸ€‘*ƒŸöü«XŽŽŸ†‘Ç´m±=0ŽŽŸù<$‘Ý‘²(¸ ²1)Ÿü^ÿ´mŽ‘˜›µm²!Ž‘Ý‘Ÿw‰ fe 'µ¬Ÿ (Ö‘Ú‹µm–8à²+“1ŽŽŽŽŸù<$‘@ù£µeŸü^ÿ´xŽ‘ä²(µeŸü^ÿ´xŽ‘Aĸ ‘8à²1)Ÿü^ÿ´mŽŽ‘@ù£Ÿw‰ fe 3úbŸ (Ö‘5ǵm²!ŽŽŽŽŽŽŽ¡‘a¸Æ=ŽŽ‘säµeŸûÞÿ´xŽŸóý‘Si·1ŽŸ€‘÷Ÿöü«XŽŽŸ†‘³Œ´m±=0ŽŽŸù<$‘ʲ(1–8ภ“µeŸü^ÿ´xŽ‘ä²)Ÿü^ÿ´mŽŽ‘ÊŸw‰ fe *I}Ÿ (Ö‘$tµm–8à²+“1ŽŽŽŽ‘I
Í=Ÿù<$‘–»µeŸü^ÿ´xŽŽ‘úKŸw‰ fe éÄŸ (Ö²1–8ภ“µeŸýr´xŽŽŽŽŽŸóý‘&aÇ·1ŽŸ€‘#%UŸöü«XŽŽŸ†‘!Áê´m±=1ŽŽŸù<$‘5Øc²(1–8ภ“µeŸü^ÿ´xŽ‘ä²)Ÿü^ÿ´mŽŽ‘5ØcŸw‰ fe *I}Ÿ (Ö‘ÀãµmŽŽŽŽŽŽŽŸñ‘a¸Æ²=ŽŽŸù<$‘}O‡µeŸü^ÿ´xŽŽ‘t³Ÿw‰ fe éÄŸ (Ö²1–8ภ“µeŸýr´xŽŽŽŽŽ’ ’z¶²(Ž’ –^E¸ ‘ª¨²logŽ‘?ü(Ž‘#‹1–8ภ“²(1“¸ “µeŸûÞÿ´xŽ‘ä²))ŽŽ‘KðÛ)ŽŽ’ ðÀå=Ÿù<$‘ »-µxeŸü^ÿ´xŽŽ‘úKŸw‰ fe éÄŸ (ÖµeŸýr´xŽ‘Aĸ ‘8à²1ŽŽŽŽ‘ Bµ:ŽŽŽŸ‘> ²This›UUpro•¸ãv“es˜Theorem˜1.ŽŸ`Ë‘> ¼Remark.‘
²A‘•|referee–•Ïsuggested“the“folloš¸ãwing“in˜terpretation“of“the“algorithmŽ¦‘> using–UUgenerating“function:Ž¦‘M SuppGose–UUthe“rst“ro¸ãw“is“µaŸÿ±0Ž–|sµ;›ª¨aŸÿ±1Ž“µ;˜aŸÿ±2Ž“µ;˜:˜:˜:Ž‘ÿ÷;–UU²with“ordinary“generating“functionŽŸãE’ Ä¢õµA²(µx²)–Ç=Ÿóý‘Sf·1ŽŸ€‘ôŸöü«XŽŽŸ†“´n±=0ŽŽ‘ƒAµaŸÿ´nŽ–q~µxŸûÞÿ´nŽ“µ:ŽŸ
:‘> ²Let–ãCthe“leading“diagonal“bGe“µbŸÿ±0Ž‘ ÚÀ²=‘^MµaŸÿ±0Ž–|sµ;›ª¨bŸÿ±1Ž“µ;˜bŸÿ±2Ž“µ;˜:˜:˜:Ž‘ÿ÷;–ãC²with“expGonen¸ãtial“generatingŽ¦‘> functionŽŸƒ‘’ ÄY¯óˆ¶È
msbm10ÅB²(µx²)–Ç=Ÿóý‘Sf·1ŽŸ€‘ôŸöü«XŽŽŸ†“´n±=0ŽŽ‘ƒAµbŸÿ´nŽŸù<$‘¤±µxŸü^ÿ´nŽŽ‘¤±Ÿw‰ fe (šŸ (Ö‘0rµn²!ŽŽŽŽ‘ ~µ:ŽŸzÛ‘M ²Then–UUwš¸ãe“ha˜v˜eŽ¦’ ¾ltÅB²(µx²)–Ç=“µeŸûÞÿ´xŽ›äµA²(1–8ภ“µeŸûÞÿ´xŽ˜²)µ:ŽŸÛŽ‘> ²This–kfollo¸ãws“from“(ïhtml:ïcolor push cmyk 0 1 0 02ï color popï html:)Ž‘Ïõand“(ïhtml:ïcolor push cmyk 0 1 0 03ï color popï html:)Ž‘ËŠ,‘šthe“calculation“bšGeing“parallel“to“that“of“the“pro˜ofŽ¦‘> of–ý•Theorem“1.‘T‡T‘ÿ*ªo“get“the“Bernoulli“n•¸ãum“bGers›ý•w“e˜tak“e˜µaŸÿ±0Ž–C‹²=‘Ç1µ;›ª¨aŸÿ±1Ž“²=Ÿü‘úK±1Ž‘úKŸ£&‰ fe üsŸ¿˜2ŽŽŽŽ‘ )ñµ;˜aŸÿ±2Ž“²=Ÿü‘úK±1Ž‘úKŸ£&‰ fe üsŸ¿˜3ŽŽŽŽ‘ )ñµ;˜:˜:˜:ŽŽ¦‘> ²with›UUµA²(µx²)–Ç=“¸ ‘ª¨²logŽ‘•T(1–8ภ“µx²)µ=x²,˜and˜nd˜ÅB²(µx²)–Ç=“µxeŸü^ÿ´xŽ‘äµ=²(µeŸü^ÿ´xŽ‘Aĸ ‘8à²1).Ž‘> ŸôYïhtml:ï html:Ÿ€ ¿3Ž‘LËP•ŒÌoly-Bernoulli‘ffn“um“bs3ersŽŸç²If–Ä/wš¸ãe“replace“the“initial“sequence“1µ;Ÿü‘ÝÛ±1Ž‘ÝÛŸ£&‰ fe üsŸ¿˜2ŽŽŽŽ–
µ;Ÿü‘ÝÛ±1Ž‘ÝÛŸ£&‰ fe üsŸ¿˜3ŽŽŽŽ“µ;Ÿü‘ÝÛ±1Ž‘ÝÛŸ£&‰ fe üsŸ¿˜4ŽŽŽŽ“µ;–ª¨:“:“:Ž‘Ä&²b˜y›Ä/1µ;Ÿü‘
k±1Ž‘ÝÛŸ£&‰ fe [’ŸßG2Ÿþ óO
Ú\ cmmi5³kŽŽŽŽŽ–l µ;Ÿü‘
k±1Ž‘ÝÛŸ£&‰ fe [’ŸßG3Ÿþ ³kŽŽŽŽŽ“µ;Ÿü‘
k±1Ž‘ÝÛŸ£&‰ fe [’ŸßG4Ÿþ ³kŽŽŽŽŽ“µ;–ª¨:“:“:Ž‘Ä&²and˜apply˜theŽŸ@same–[Ïalgorithm,›mthe“resulting“sequence“is“(¸ ²1)Ÿü^ÿ´nŽ‘q~µDŸúÎ:G±(´k+B±)ŽŸ]"´nŽŽ‘Î~²(µn–|Œ²=“0µ;–ª¨²1µ;“²2µ;“:“:“:Ž‘ÿ÷²),˜whereŽ¤
ñƵDŸúÎ:G±(´k+B±)ŽŸ]"´nŽŽ‘&3²is–³„a“v‘ÿqÇarianš¸ãt“of“\pGoly-Bernoulli“n˜um˜bGers"“([ïhtml:ïcolor push cmyk 0 1 0 06ï color popï html:Ž– ],›Ë[ïhtml:ïcolor push cmyk 0 1 0 02ï color popï html:Ž“],˜[ïhtml:ïcolor push cmyk 0 1 0 03ï color popï html:Ž“]):‘.&F‘ÿ*ªor–³„anš¸ãy“in˜teger“µkP—²,Ž¡wš¸ãe–UUdene“a“sequence“of“n˜um˜bšGers“µDŸúÎ:˜±(´k+B±)ŽŸ]"´nŽŽ‘Ȳb¸ãyŽŸãEŸù<$‘o`ÓµLiŸÿ´kŽ‘ë²(1–8ภ“µeŸü^ÿ· ´xŽ‘Hå²)Ž‘o`ÓŸw‰ fe 8¡Ÿ (Ö‘™oµeŸýr´xŽ‘Aĸ ‘8à²1ŽŽŽŽ’ «w¿=Ÿóý‘Sf·1ŽŸ€‘ôŸöü«XŽŽŸ†‘Ç´n±=0ŽŽ‘ƒAµDŸûÞÿG±(´k+B±)ŽŸ™á´nŽŽŸù<$‘¥âµxŸü^ÿ´nŽŽ‘¥âŸw‰ fe (šŸ (Ö‘0rµn²!ŽŽŽŽ‘¯µ;ŽŸ7²where›W‡µLiŸÿ´kŽ‘ë²(µt²)–uj=“Ÿøü«PŽŸúøÞ‘¥·1ŽŸ%‘¥´m±=1ŽŽŸü‘$`´tŸüûr³mŽŽ‘#šŸ£&‰ fe wºŸßG´mŸþ ³kŽŽŽŽŽ‘4œ’²(µkP—²-th˜pGolylogarithm˜when˜µk‘Ƹ“²1).‘x^The˜abGo•¸ãv“e˜as-Ž¦sertion–Õis“then“a“consequence“of“the“follo¸ãwing“(or,‘õis“just“a“spGecial“case“of“theŽ¦preceding‘UUremark)ŽŸ€ ïhtml:ï html:ŽŸ ‘> ïcolor push Black’ ª 5Ž’Y ï color popŽŽŒ‹ G> y ý£ ‘> ïcolor push Blackïhtml:ïcolor push gray 0ï color popï html:Ž’Y ï color popŽŽ ? ýä ‘9 ïcolor push Black‘ ¼PropQÇosition‘ÕT3ï color popŽŽ’ †ìµ½F‘ÿ;¼or–“çany“µk‘¯¸2›Ç¼Z“½and“µn˜¸˜²0½,“we“haveŽŸƒ’ Î4µDŸûÞÿG±(´k+B±)ŽŸ™á´nŽŽ‘9Dz=‘Ç(¸ ²1)ŸûÞÿ´nŽŸóý‘
?¶nŽŸ€‘‘Ÿöü«XŽŽŸ†‘&´m±=0ŽŽŸ÷Ñ}‘2Ÿ²(¸ ²1)Ÿü^ÿ´mŽ‘˜›µm²!Ÿ÷æb«ŽŸû
‘èæ´n±+1ŽŸâ¾‘ÕW´m±+1ŽŽ‘
õŸ÷æb« ŽŽŽŽ‘2ŸŸáµ‰ fe D˜øŸ (Ö‘òÚ²(µm–8à²+“1)Ÿýr´kŽŽŽŽŽ‘`þʵ:ŽŸP‘> ½Pr–ÿ}'o“of.‘o²The–©ÅprošGof“can“b˜e“givš¸ãen“completely“in“the“same“w˜a˜y“as“the“proGof“ofŽ© ‘> Theorem–UU1“using“generating“series,“and“hence“will“bGe“omitted.Ž¤ øø‘> ¿Ac•ŒÌkno“wledgemen“tsŽŸç‘> ²I–UUshould“likš¸ãe“to“thank“the“referee“for“sev˜eral“commen˜ts“and“suggestions.Ž¡‘> ¿ReferencesŽ‘> Ÿçïhtml:ï html:Ÿã4ïcolor push Black‘ ²[1]ï color popŽŽ‘Žï html:Ÿ
c4ïcolor push Black‘ [2]ï color popŽŽ‘Žï html:Ÿ
c4ïcolor push Black‘ [3]ï color popŽŽ‘Žï html:Ÿ
c4ïcolor push Black‘ [4]ï color popŽŽ‘Žï html:Ÿ
c4ïcolor push Black‘ [5]ï color popŽŽ‘Žï html:Ÿ
c4ïcolor push Black‘ [6]ï color popŽŽ‘Žï html:Ÿ
c4ïcolor push Black‘ [7]ï color popŽŽ‘Ž