On Stirling Numbers for Complex Argument and

0 downloads 0 Views 699KB Size Report
ݳÞ te ß Dibex d”be‚Bi§ àkáY â“àiàƒãka â“àiàkâg ämÞ twÃ¥ em Þ thæ«ç scè ugéhê etç tCë©ìÞ ss æfiÀçÞ thæ«î ïgð} ñiò Eâña ó e“ôj õ¼î ö ÷ s Þïf÷ pӌföÞ sesqð ...
On Stirling Numbers for Complex Argumentand Hankel Contours Philippe Flajolet, Helmut Prodinger

To cite this version: Philippe Flajolet, Helmut Prodinger. On Stirling Numbers for Complex Argumentand Hankel Contours. [Research Report] RR-3373, 1998.

HAL Id: inria-00073316 https://hal.inria.fr/inria-00073316 Submitted on 24 May 2006

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destin´ee au d´epˆot et `a la diffusion de documents scientifiques de niveau recherche, publi´es ou non, ´emanant des ´etablissements d’enseignement et de recherche fran¸cais ou ´etrangers, des laboratoires publics ou priv´es.

        !"# $ %&   "# $  

'

(

)+*-,./,(10

( 254 3 EK(HL

6879.: ;=. ?@`tdgi W ;* W*fFaešgWMj—`–jcei W v `PjFiŸabšBW+jF`P} X=d v W*`PjFi m v jcPW £ i#WY`Pn§aedg`–d— žgiŸa v c ae`tf `DmtXqPW v n(dFMh dgX=v |DceW v £¾j v fBmtv X=WY`Si nz@d v  U U alh U X=dSnpih#cljBn n§a8hYjcv a8}DWY`Dv ipaeipaeWn=j v Wnpiparcrc n j—ipa8nJP· W}¦¸7¹ W Wnpi alh idBm npW cešgWni#d_i WX=dSnŸi1h dgX=X=dg` | dg|PW ipaeWnceWjšgae`tf2aei i d‰iv U W=aeX–UƒjFf—’ ab`Pj—ipaedgU ` d— i U W v Wjg}DW v i d_v fgd @m v i U W v ¦«º v ae™gWae`»v €M‚9ƒ ¤dgm v jF|t| v dSjBh U npi j i¥n * aei j—mPh v o•U h dDW ”•U h#aeWY`DU iab`Di WYf v jc8nY¦˜zd^* WYšgW gae`ƒh dg`Si jgnpi*i d€‹‚Cƒ ¼v * WmPnpWv j‰˜jF`t™gW ch dg`Di dgm i jFi jgn(i WX=W aeidF§¦}ž;'š q¡W ae`tf }DWY|¡WY`P}DWY`SiJdg`›|Pj ipa8h mDclj

ae`P}Dv W £ šFjcemtWnY¦ T*U a8n(ae`Si v ab`Pnxalh‰h U j v jBh i W v dF*i U W‰h dg`Si#dgm v | v Wh#a8npW ceo WY`PnŸm v WnJi U W |¡W { X–jF`tWY`Ph WJv dF¨a8v }DWY`DipaeipaeWnYv ¦ U W ceWYšFjF`Si W ¢W WY`Ph Wn‹¢d i WMh#c8jgn#n§a8hYjcIi U WYd v oj v W§€l„ D[ g f½;ƒ Fi U W clj—i i W v |PjF|¡W v q¡W ab`tf‰jF`‰W £Ph W crcbWY`Di U a8npi d v alhYjkc¼jghYh dBmt`Si'd—žgipa v crae`tf‰`SmtXqPW v nY¦ [t¦¾¼t?~*uÀ¿^~Ár|ÂÃr?z?Ä%Å+Ƽu+Ç vyÈÉ{ÀvyÄʼ¿pÆ;Ë»~Ár?x?Æ;Ë Ì o‰}DSW ·P`DabiŸabdB` ;i U WZžgipa v crab`tf_nŸmtqPnpWYiM`DmtXqPW v “n ¸]Í d—¨i U WnŸWh dg`P}‰™Bae`P}À„«º …fÎÏ;ˆ j v W ¢d gv Ð0ÑiÒÓ Ô fFaešgWY`2qSo Õ Ð Ð0Ø Ï ÒyÖ× ÒÀØ €‹Ù Î ƒp¸ÛÚkܐ f„ º Ñ Ý³Þ ±e¬ ß ,i>AN @^>AR*E§àká;â]àiàSãk= â]àiàkâgämÞ ±MåF¬ ¯ Þ ±hæ«çp­?èg®Céhêp¬Jç ±+ë©ì Þ ­§­Qæ íÀç Þ ±hæ«î ïCð}ñiò -â ñ = ó ¬]ôjõ¼î ö÷ ­ Þ ïf÷‹³^åfö Þ ­x¬p­Gð kObLMRbT`LMNCP%N @Cd-efHGR*VGá ÀFSdj[ TMHGl§>AR*PS@CdjHGN O*VGá Y >ANCZSHJTwBGFSN O*Fi@9R*VG= ø

      

d v SqDo ’ jFmPh U oÀ„ n'h dDW ”•Õh#aeWY`Siz¢d v XmDc8j Ð Ð0Ø „ !" Ï$# Ù ¸¥‰„ º Ò³Ö × ÒÀØ [  ¸QÚ Ü  „fº Ù Î% ø Ñ  U W v W i U W‰ae`Si#WYf v jFipaedg` h dg`Di dgm v'& a8n‰j npXjcrc(h dg`Di dgm v WY`Ph#a v h#crae`tf i U W d v aefFae`;¦ yn Ð a8n`tdg`t`tWYfSj—ipaešgWU ae` ¸ „fº v i U W=v h dg`Di dgm v(& hYjF`›q¡U W–}DW @d v X=W}2ae`Di d_j*v v)m¡;¦,+^œCª ¨Sž;¦+š*žpŸ5¤(- ¸xnŸWYWW €M–9ƒ*º i j—i npi¥j i¥n¤ dgX´/. q¡W ced^ i W`tWYfSjFiŸabšBW+j£Da8n tnpm dBmt`P}tn i U Wd v aefFae` h dgmtv`Di W v h#ced h#™©v *a8npW_jF`P} v vWYi m v `Pn(i d’/. ae` i U W U jkc |Dclj—`tU W0nÙ132t¦ WYi¥jka c8n*d—4- j v WdF h U dBm v npWaeX=X–jFi#W a8jc¨jF`P}*WM`tWYW} dg`Dceo•jgn#npmtX=Wi jFi>aei*a8n'jFi }Ba8npT*i¥j—U `Ph 6W 5 „ dgX U i W Wjc j£Da8nY¦ a8n'nŸmtfgfgWnpi¥n i W@d—c cefd *ae`tf‰}DSW ·P`DabiŸabdB` ! 798: Œ :@BA4C D(EFHG v dgX W £t|PjF`P}Bae`tfi U

Ú 

“

ø



× Ï  

¸QÚ Ü  „fº Ï ÒÀØ

      

ˆ :‚8 Š : Š ;    eŽ @ *T U 8a nh#c8jgn n§a8hYjc ¢d v XmDc8j € 9[ ƒ jc8npdfgWY`tW v jkc aeœYWnY¦ ¶ n *W1W £  |PjF`P} !

;

Ú 

*WfgWYi

“

ø

ÏÜ Ú  ø × Ú Ï   ÀÒ Ø

ŽØ „ ׆ Ú Ï   ÀÒ Ø

“

„

ÏÜ # Ù Ú [a ن%ø ” T*U W>ae`Di WYf v jcPhYjF``tdf qPW*WYškjkcbmPj—i W}qDo1mPnpW*dF;˜MjF`t™BW cÁ„ n¤@d ¢mt`Ph ipaedg`¸pnpmtqPnpipaei mti W Ò Ù × º PjF`P} „ !w ÚÏÜ #Ù ø Ò † [  Ù † % × Ž Ø 

! w

v XmDc8j @d v i U W*fSj—X=X–j

U WY`Ph W ¡i U WfgWY`tW v jkc aeœYW} dgqDae`Pnp™Ba;¢d v XmDc8j ¸Q‚wº

҆ “ ø  Ú † × Ï   ÒÀØ ”



  8 ;  : 8  ƒˆ  Fv Š @(]'aešgWY`›i U jFÌ ii v U W a v W £ |¡dg`tWY`SiŸv aljkc fBWY`tW vv jFipae`tf‰¢mt`Ph ipaedgv ` a8n Ù p¸QÚ Ü 2„fº F aei¼aln¼`PjFi m jkcPi d1W £t|PWh i W `tdgmDcrc a‹`SmtXqPW n i dq¡W W c8jFi#W}i d=žgiŸa c ae`tf v ’ v v U v Ð T*U 

`DmtXqPW n*d— ipoD|PWÕ … “ † ø ˆt¦ gd `Pn§a8}DW · npi'i W1hYjgnŸWdF¼j—`ab`Di WYfgW ae`P}DW £ ¦ WY` Ð Ð0Ø „ ! w ‡f“ ø # Ù Á ¸ k Ú g Ü “  f „ º ‘“„ Ö × ¸Q‘ „fº Ø [a Ù Î% ø ” ygn ‘ $2 ©*W U jšgWÕ qSo ’ jFPm h U o|„ n¼¢d v XmDc8j=jF`P} i U W jBh i i U jFim¸Q‘ „fº Ø

‡ø Ð

T*U a8n

‘ „ Ö v W clj—ipaedg`_fFaešgWn Õ

Fj `P}ZX=d v W'fgWY`tW v jcrceo ti U Wg¦ f¡¦n€‹–fƒ*º

‘ Ð0Ø €‹Ù Î% ø ƒ Ú Ü Ù 

„

‘ Ð Î~ % ø„ × 

”

ø  Ð Ñ ‘ ‘ Ö  ‡  “ ø × Ð Î ~“% „ × ¸S º j—`t™ n¼i d˜jF`t™gW c7„ n v WY| v WnŸWY`Si¥j—ipaedg`–d—Ëi U Õ # Ž  ¸igŽÀº ” # ‘ ‘ Ö  ‡  “ ø × #

#

Ð



W  ¢mt`Ph ipaedg`¸pnpWYW

¶ `1dgi U W v * d v }tn Ì W v `tdgmDcrc at`DmtXq¡W v n¤dF h dgX=|DceW £ae`P}DW £i U jFi;j v W `PjFi m v jcrcbo1}DWS·P`tW} qDo  † % ø — × ¸QŽ‹v ~„fº  ¸igŽÀv ºj v W_Õ jc8npd dgv qti¥jae`tW} qDoÕ j›n§aeX=|DceW2c aeXaeipae`tf | v d h Wn#n jF|t|DcraeW}2i dfgWY`tW jcraeœYW} žgipa crae`tf‰`SmtXqPW n'dF¤ab`P}DW £.„;— # Ž  ø c aeX „ Ž  † % Ž ~ „ ×     „q~ Ö × # ‘ ‘ Ö  ‡  “ ø ” 

o| Õ v dFšga8}DWjcbdBfSj v aei U Xa8h+@d v X d— žgipa v crae`tf_“ npmt“qPnpWYiz`SmtXqPW v n † ø  ‡f“ ø #  Ñ Ž ¸QŽ „‰º Ø „ ! w  cbdBf „ ¸Q–wº ‘ Ö × ¸Q‘ „fº Ø [  „‰~    U W v W- aln*j˜ v jae`P} v dg|h dg`Si#dgm v µi U jFi‹a8ni U W*aeX–jFfgWdF-/qSoÙ  × Ú Ü _„B¦ ¸ T U mPn =- npi¥j v i njFi „ae` i U W=cedf* W vU jcrM|Dc8jF`tW npm v#v dgmt`P}tnƒ22jF`Dipa8h#cbdth#™5*a8npW jF`P} • v WYi U m v `Pn*U i d v; U „ae`_i ’ U WmtU|t|¡W vU jcr |Dc8jF`tWg¦«º v `v i W>dgi W j—`P} j—mPv h o|„ nIh dDW ”•h#aeWY`Di‹@d v XmDc8j'j—|t|Dc aeW}Ji d*i U W>W £ |¡dg`tWY`SiŸaljkc fgWY`tW jFipae`tf¢mt`Ph ipaedg`_dF*žgipa crae`tf_h o h#ceWJ`SmtXqPW n'fFaešgWn Л Ð0Ø ! w  Ï

¸ €pº  „‰º Î “ Ï ÒÀØ [„  e c g d ¼¥¸ „‰~“º   #  ø f ™ Ò × ¸i Î % ” v U U v y v }Ba Wv h ih dg`PnpWsBmtWYv `Ph W'dF?¸Q–©;º jF`P}' ¸ €;¡º a8n‹i jFi‹i Wv žgipa crae`tfh voth#cbWz`SmtXqPW v n‹dFPae`Di vW  f jcIj fgmtXZWY`Si¥n j a8npW+jBn‹craeXaeipae`tf–hYjgnpWn¼dFÕ ;fgWY`tW jcraeœYW}2žgipa crae`tf nŸmtqPnpWYi*`SmtXqPW n jgnz}DWSP· `tW}ae`x¸ [wº }x¸ ©º Л  Ò ~    c e a X ™ Ò ×    Ð ~  Ö ” • `tWWY`Ph dgmt`Di W v nJdg`Ph WX=d v W–j—`ƒae`Pnpi¥jF`Ph W‰dF'i U W‰}DmPjcraeipo v W c8jFipaedg` ŠM†‡iŒ … ““w†‡ ˆ × i U j—i gi dBfgWYi U W v  abi U ¸Q–wº Sh dg`©· v X–n>i U jFi'žgiŸa v c ae`tf=`DmtXqPW v n>WYšgWY`Di mPjcrceo v W}DmPh Wi d j=nxab`tf—cbW( j—Xarcbo p¸ npWYW€  ©½;ƒ* º ¦



» S· FÒ

!#"$%&#')(+* zTWVYv-,bdl/.&,C^uv-vmk [W[ b lme \ nVZp[ ^ l e01emTW\ &bdpWq%z\fl j \32 v \ ^ l mT9Slmbh`c\ ce5476983:=@?BA DC  ]djB7DkLMZf=U0N ObLMRbT`LMNCP'N @Cd-efHGR*V>ANCX ¼@kTMHQRV*@CdjVG=QP î CöJï Þ ì+îIRë¼î Þ hæ«î ï Þ ì Þ ïf÷ S ;ìMæ 7÷ ämÞ Må Þ hæ«ç áAD^>Adá ³= À=^]0N @9O*D;áw>ANCX% =5)¼>AO7>]V*D9N LMZf= ë¼î ïfç]ö ämÞ Må Þ hæ«ç è 7ç7î ïf÷7_À÷ æa` hæ«î ï =hY ] X9N X @9LMO*V*D;FS=Nc gb Àf HQF%VbTMNCHGESFSá¼O*HQâ]VjàiàdFSC[ N= NCFSO7>]ObLMFSN;=S K [\L y  =  =e0 ]öJæoç Þ ï ä§Þ Må Þ hæ«ç Þ ì ä î ï ‹å9ìMô á5àSà ß [ ñ VWUh[ X)X á â]àià X = K ò Lj3 i =d} LMB7D9djFiN9X>]N9X y  =% " HGRbT`LMN Lh= k ObLMRbT`LMNCP%N @Cd-efHGR*VÀchFSR|BGFSdj[ TMHGl§>AR*PS@CdjHGN O*VG= èBk SyälP î CöJï Þ ì Ý æ Gç]ö ä§Þ ‹å Þ hæ«ç áfâ ñkß TV3CU ã X á©âUàSà T = î ï KYm L   =W³ =^ b D LMO*O7>AZiHGRg>ANCXW/ = ³  =$W b >AO*V*FiNp=nS ä S Þ = 3 >]d eCRbLMXCPSH ?NkLYo \SHGR*VbLMOEm)¼R*HGV*VGá9c FS@CR*O*D%HGX LMObLMFSN;á}âUà X kT =WÀ HG[CRbLMë¼N O*î HGCXö â]jà)TîIR kV = îA÷ ]ö7ï ï ìMô Qæ

³Y³ r¬ ± ´

± FY¬ ²¯ ± º¬p­ Y¯

­ 8¬#±e¬ ± F¬ ¯ ± p­

± p­

#­ ­



)©=opTo>3p*F]TMHGOGáW3TMPiFSRbLMO*DCdjV-) R*F\p*HGBQOGáo   á oDTSã â òiñ ÀF B3q @9HQN9BGFi@9R*OGá 9R7>]N9BGH _r`@¯ Þ æMì Þ ÷ ÷ 8ö ¬p­§­i!ß st!u)vFu3w