Thefactorialfunctionoccurswidelyinfunctiontheory;especiallyinthedenomi - PDF document

Thefactorialfunctionoccurswidelyinfunctiontheory;especiallyinthedenominatorsofpowerseries expansionsofmanytranscendentalfunctions.Italsoplaysanimportantroleincombinatorics[Section2:14]. Becausetheytooariseinthecontextofcombinatorics,StirlingnumbersofthesecondkindarediscussedinSection 2:14[thoseofthefirstkindfindahomeinChapter18]. 2:6THEFACTORIALFUNCTION n ! 23 Setting m � n inequation2:5:2providesa duplicationformula forthefactorialfunction,enabling(2 n )!tobe expressedwiththehelpofaPochhammerpolynomial.Alternativeduplicationformulasareavailablefromequations 2:12:3and2:12:4,whichmayberewrittenas �� � � �� �� 1 1 2 2 /2 11 1 2 22 (1)/2 2!2,4,6, ! 2!1,3,5, n n n n nn n nn � �­ � �˜�˜�˜ �° � �® �� �˜�˜�˜ �° �¯ 2:5:4 Thereareanalogoustriplicationformulasthatcanbedevelopedfromtheequationsin2:12:5. Thefrequentoccurrenceoffactorialsascoefficientsofpowerseriespermitsthesummationofsuchseriesas 0 1111 exp(1)271828182845905 0!1!2!! j j �f � ����˜�˜�˜� � � �¦ . 2:5:5 and �� �� �� �� 0 2222 0 1111 I(2)227958530233607 0!1!2!! j j �f � ����˜�˜�˜� � � �¦ . 2:5:6 whereI 0 isthemodifiedBesselfunction[Chapter49].Thecorrespondingserieswithalternatingsignssumsimilarly toexp( � 1)andtotheparticularvalueJ 0 (2)ofthezero-orderBesselfunction[Chapter52].Thereiseventhe intriguingasymptoticresult[seeequation37:13:4] 0 0 exp() 0!1!2!()!d0596347362323194 1 j j t jt t �f �f � � ����˜�˜�˜� �� � �¦ �³ . ~ 2:5:7 Moreover,theseries � ( � 1) n /(2 n )!sumstocos(1)andthereareseveralanalogoussummations. 2:6EXPANSIONS Stirlingsformula [seealsoSection43:6] 23 11139 !2exp()1 1228851840 n nnnnn nnn �ª�º �S������˜�˜�˜�o�f �«�» �¬�¼ ~ 2:6:1 providesanexpansionforthefactorialfunction.Thoughtheyaretechnicallyasymptotic[Section0:6],this expansionandasimilaroneforthelogarithmofthefactorialfunction 1 35 B 1 ln(!)ln2ln()1 (1) ln(2)111 ln() 2123601260 j j j nnnn jjn n nnnn nnn � �ª�º �§�· �S��� �«�» �¨�¸ � �©�¹ �«�» �¬�¼ �S � �������˜�˜�˜�o�f �¦ ~ 2:6:2 areremarkablyaccurate,evenforsmall n .B j isthe j th Bernoullinumber[Chapter4]. 2:7PARTICULARVALUES 0!1!2!3!4!5!6!7!8!9!10!11!12! 112624120720 504040320362880362880039916800479001600 24 THEFACTORIALFUNCTION n !2:8 2:8NUMERICALVALUES Thedecimalintegerrepresenting n !hasexactlyInt( n /5)+Int( n /25)+Int( n /125)+ �#�#�# terminalzeros;forexample 31!endswithsevenzeros.Thisruleisusefulincalculatingexactnumericalvaluesoflargefactorials.Intisthe integer-valuefunctiondescribeinChapter8. Equator s factorialfunction routine(keyword ! )providesvaluesof n !.Forintegerinputintherange0 n �d 170,asimplealgorithmbasedonrecursion2:5:1,followedbyrounding,isusedtocompute n !.Exactoutputis reportedupto20! � 2 . 43290200817664E+18.For21 �d n �d 170, Equator providesafloatingpointapproximation of n !preciseto15digits. Separately,valuesofthenaturalanddecadiclogarithms,ln( n !)andlog 10 ( n !),areprovidedbythe logarithmic factorialfunction and logarithmtobase10ofthefactorialfunction routines( ln! and log10! ).Suchlogarithmic valuesareusefulwhen n islargebecause n !itselfisthenprohibitivelyhuge.Forinputupto n � 170,ln( n !)is computedbysimplytakingthelogarithmoftheoutputfromtheroutinedescribedabove.Forintegerinputinthe range171 �d n �d 1E305, Equator usesStirlingsformulainthetruncatedandconcatenatedform 222 ln(2)112 ln(!)1ln()11 212307 n nnn nnn �§�· �S�§�· �§�· � ����� �¨�¸ �¨�¸ �¨�¸ �©�¹ �©�¹ �©�¹ 2:8:1 Divisionby2 . 30258509299405generateslog 10 ( n !). 2:9LIMITSANDAPPROXIMATIONS Forlargeargument,thelimitingformula !2 n n nnn e �§�· �o�S�o�f �¨�¸ �©�¹ 2:9:1 applies, e beingthebaseofnaturallogarithms[Section1:7].Therelatedapproximation !Round(112) 72 n n nn ne �­�½ �S �°�° �§�· �|� �®�¾ �¨�¸ �©�¹ �°�° �¯�¿ 2:9:2 issurprisinglygood,andevenexact,forsmallpositiveintegers.Roundistheroundingfunction,describedin Section8:13. 2:10OPERATIONSOFTHECALCULUS Nooperationsofthecalculusarepossibleonafunctionsuchas n !thatisdefinedonlyfordiscretearguments. 2:11COMPLEXARGUMENT Inviewofrelation2:12:1,thegammafunctionformulasgiveninSection43:11maybeusedtoascribemeaning to( n + im )!. 2:12THEFACTORIALFUNCTION n ! 25 2:12GENERALIZATIONS Thefactorialfunctionisaspecialcaseofthegammafunction[Chapter43] !(1)0,1,2, nnn � �*�� �˜�˜�˜ 2:12:1 andofthePochhammerpolynomial[Chapter18] !(1)0,1,2, n nn � � �˜�˜�˜ 2:12:2 Thelatteridentitypermitsustowrite 1 2 (2)!4()(1) n nn n � 2:12:3 and 3 2 (21)!4(1)() n nn n �� 2:12:4 Similarly �� �� �� �� �� �� 333 5 12244 333333 (3)!3(1),(31)!3(1),(32)!3(1) nnn nnn nnnnnn nnn � �� �� 2:12:5 andsoon. 2:13COGNATEFUNCTIONS:multiplefactorials See6:3:4forthecloserelationshipbetweenthefactorialfunctionandbinomialcoefficients. The doublefactorial or semifactorialfunction isdefinedby 11,0 !!(2)(4)5311,3,5, (2)(4)6422,4,6, n nnnnn nnnn � � �­ �° � �u��u��u�˜�˜�˜�u�u�u� �˜�˜�˜ �® �° �u��u��u�˜�˜�˜�u�u�u� �˜�˜�˜ �¯ 2:13:1 Forevenargumentitreducesto �� 2 !!2/2!0,2,4, n nnn � � �˜�˜�˜ 2:13:2 whileforodd n itmaybeexpressedintermsoffactorials,orasagammafunction[Chapter43]orasaPochhammer polynomial[Chapter18] 2 (1)/2 (1)/2 (1)/2 !21 !!2121,3,5, 1 22 2! 2 n n n n nn nn n � � � �§�·�§�· � � �*�� � �˜�˜�˜ �¨�¸�¨�¸ � �§�·�S �©�¹�©�¹ �¨�¸ �©�¹ 2:13:3 Equivalenttothelastequationis (21)! (21)!! 2! n n n n � �� 2:13:4 Theseformulasareusedby Equator s doublefactorialfunction routine(keyword !! )tocomputevaluesof n !!for integersintherange � 1 �d n �d 300.Ofcourse !!(1)!!!0,1,2, nnnn �� � �˜�˜�˜ 2:13:5 Someearlymembersofthedouble-factorialfamilyarelistedbelow. ( � 1)!!0!!1!!2!!3!!4!!5!!6!!7!!8!!9!!10!!11!!12!!13!!14!! 1112381548105384945 38401039546080135135645120 26 THEFACTORIALFUNCTION n !2:14 Notethat,apartfrom0!! � 1,thedoublefactorial n !!sharestheparityof n .Alsonoticethat,toaccordwiththe n � � 1instanceofthegeneralrecursionformula (2)!!(2)!! nnn �� � 2:13:6 ( � 1)!!isassignedthevalueofunity.Withasimilarrationale,onesometimesencountersthevalues( � 3)!! � � 1, ( � 5)!! � ,etc. 1 3 Offrequentoccurrence[forexampleinSections6:4,32:5,61:6and62:12]istheratio( n � 1)!!/ n !!ofthedouble factorialsofconsecutiveintegers.Forodd n ,theratioisexpressiblebytheintegral 2 /2 1 0 (1)!!21 !sin()d1,3,5, !!!2 n n nn ttn nn �S � ��ª��º �§�· � � � �˜�˜�˜ �¨�¸ �«�» �©�¹ �¬�¼ �³ 2:13:7 whileforeven n itisgivenby Wallissformula (JohnWallis,Englishmathematicianandcryptographer,1616 � 1703) ���@ /2 2 0 (1)!!!2 sin()d0,2,4, !! 2(/2)! n n nn ttn n n �S � � � � �˜�˜�˜ �S �³ 2:13:8 Thisimportantratiohastheasymptoticexpansion 2 2 211 1even 432 (1)!! !! 11 1odd 2432 n nnn n n n nnn �­ �ª�º ����˜�˜�˜�o�f �° �«�» �S � �°�¬�¼ �® �S �ª�º �° ����˜�˜�˜�o�f �«�» �° �¬�¼ �¯ ~ 2:13:9 Finitesumsofsomesuchratiosobeythesimplerule 0 1315(21)!!(21)!!(21)!! 10,1,2, 2848(2)!!(2)!!(2)!! n j njn n njn � ��� �����˜�˜�˜�� � � �˜�˜�˜ �¦ 2:13:10 andthereistherelatedinfinitesummationduetoRoss: 1 1315105(21)!! ln(4) 2161441536(2)!! j j jj �f � � �����˜�˜�˜� � �¦ 2:13:11 The triplefactorial isdefinedanalogously 12,1,0 (3)(6)7411,4,7, !!! (3)(6)8522,5,8, (3)(6)9633,6,9, n nnnn n nnnn nnnn � �� �­ �° �u��u��u�˜�˜�˜�u�u�u� �˜�˜�˜ �° � �® �u��u��u�˜�˜�˜�u�u�u� �˜�˜�˜ �° �° �u��u��u�˜�˜�˜�u�u�u� �˜�˜�˜ �¯ 2:13:12 andfindsapplicationinconnectionswithAiryfunctions[Chapter56].Someearlyvaluesare: ( � 2)!!!( � 1)!!!0!!!1!!!2!!!3!!!4!!!5!!!6!!!7!!!8!!!9!!!10!!!11!!!12!!! 111123410182880162280880 1944 Theextensiontoa quadruplefactorialn !!!!isobvious;itisusefulinSections43:4and59:7. 2:14RELATEDTOPIC:combinatoricsandStirlingnumbersofthesecondkind Thefactorialfunctionappearsveryofteninapplicationsinvolving combinatorics .Forexample,thenumber 2:14THEFACTORIALFUNCTION n ! 27 of permutations (arrangements)of n objects,alldifferent,is n !.Ifnotallofthe n objectsaredifferent,thenumber ofpermutationsisreducedto 12 12 ! ()!()!()! J J n nnnn nnn � ���˜�˜�˜� �˜�˜�˜ 2:14:1 wherethereare n 1 samplesofobject1, n 2 samplesofobject2, �#�#�# , n J samplesofobject J ,sothat � n j � n . Iffromagroupof n objects,alldifferent,onewithdraws m objects,oneatatime,thenumberof variations (possiblewithdrawalsequences)is ! ()! n mn nm �d � 2:14:2 Ifoneignorestheorderofwithdrawal,2:14:2isreducedto ! ()!! n mn nmm �d � 2:14:3 whichthenrepresentsthenumberofwaysinwhich m objectscanbechosenfromamong n ,alldifferent,andis knownasthenumberof combinations .Expression2:14:3is,infact,thebinomialcoefficientaddressedinChapter6. Thenumberof partitions (differentwaysinwhich n distinctobjectsmaybeplacedin m identicalboxessothat eachboxcontainsatleastoneobject)isgivenbya Stirlingnumberofthesecondkind .Thereisnostandardized notationforsuchfunctions;this Atlas usesthesymbol.Clearly,nopartitioningispossibleif m � 0orif n m () m n �V andaccordinglythesecondStirlingnumberiszeroinsuchcircumstances.Otherwise,ageneralformulais () 0 () ()!! mjn m m n j j mjj � � � �V� � �¦ 2:14:4 Non-zerovaluesofStirlingnumbersof thesecondkindare,ofcourse,positive integersandsomeareshownin Figure2-2.Othersmaybecalculated viatherecursionformula ()()(1) 11 mmm nnn m � �� �V� �V��V 2:14:5 for m � 1,2,3, �#�#�# and n � 1,2,3, �#�#�# .This recursionformsthebasisof Equator s Stirlingnumberofthesecondkind routine(keyword sigmanum ).First, andareinitializedtozero (1) 0 m �o �V (0) 1 n �o �V whileissetequalto1.Then (0) 0 �V ,, �#�#�# ,arecalculated (1) 1 m �o �V (1) 2 m �o �V (1) m n �o �V viarecursion2:14:5.BecauseStirling numbersareintegers,roundingensures thatthe15digitsthat Equator generates areexact.