SPECIAL FUNCTIONS EarlD.Rainville,Ph.D. PROFESSOROFMATHEMATICS INTHEUNIVERSITYOFMICHIGAN THEMACMillANCOMPANY New York SPECIALFUNCTIONS . .s0 TilECmlPANY :--;1-.\\'YORK.CHICAGO DALl.ASATl.A::-.iTASASFRA,SnSCO INCANADA LTD. GALT, EarlD.Rainvill{'1960 Allrightsreservr,d-nopart ofthis bookmayhe illally formwithoutinwritingfromtheexceptbya reviewerwhowishestoquotebriefpassagesinCOllnectionwitha reviewwrittenforinelusioninmagazineornewspaper. FirstPrinting Libraryof Conaresscataloacarr!number:60-5115 The.'\facmillanCompany,:\ewYork Brett-.'\lacmillanLtd.,Galt,Ontario Printed intheUnitedStates ofAmerica Preface Ihaveattempted to 'writethisbookinsuchawaythat it canbe read not only by professionalmathematicians,physicists, engineers, andchemists,butalsoby\yell-trainedgraduatestudentsinthose and closely allied fields.Even the research worker in special functions may notice, however, some results or techniques with which he isnot already familiar. Many ofthe standardconceptsandmethodswhichare usefulin the detailedstudy ofspecial functionsare included.The readerwill alsofindhereothertools,suchastheShefferclassificationofpoly-nomialsetsandSisterCeline'stechniqueforobtainingrecurrence relations,whichdeservetobecomemorewidelyused. Those who know me will not be surprised to findacertain empha-sisongeneratingfunctionsandtheirw,efulness.That functionsof hypergeometriccharacterpervadethebulkofthebookisbutn reflectionoftheir frequentoccurrenceinthesubject itself. Morethanfiftyspecialfunctionsappearinthiswork,someof them treated extensively, others barely mentioned. There are dozens oftopics,numerousmethods,andhundredsofspecinlfunrtiolls \vhichcouldwellhavebeenincludedbut whichhavebeenomitted. Thetemptationtoapproachthesubjectontheencyclopediclevel intended by the late Harry Bateman was great. To meit seems that suchanapproach\vouldhaveresultedinless,ratherthanmore, usefulness;theworkwouldneverhavereachedthestageofpubli-cation. Theshortbibliographyattheendofthebookshouldgivethe reader ample material \'lith which to start onamore thorough study ofthe field. Thisbookisbaseduponthe lecturesonSpecialFunctionswhich Ihavebeengivingat TheUniversityofMichigansince1946.The enthusiasticreceptionaccordedthecourseherehasencouragedme topresentthematerialinaformwhichmay facilitatethe teaching ofsimilarcourseselsewhere. v VIPREFACE Iwishto acknowledgethe assistance givenme inthe way ofboth corrections and comments OIlthe manuscript byProfessor Phillip E. BedientofFranklinandMarshallCollege,ProfessorJackR. BrittonofTheUniversityofColorado,andProfessorRalphL. Shivelyof'WesternReserveUniversity.Iv,asalsoaidedand encouragedbythelateFredBrafmanwhowasAssociatcProfessor ofIVlathematicsat ThcLnivcrsityofOklahomaatthetimeofhis death.Professor Drafmanread the firstten chapters criti('allyand discussed some ofthe latcr material withme.Scvcral studentsnow taking the coursehavebeenhelpfulincatching crrorsandpointing out roughspots inthepresentation. ProfessorBedienthasalsoaidedmebyanindepcndentreading oftheproofsheets. EAHLD.RAlr\VILLE AnnArbor,Michigan Contents Chapter1:INFINITE PRODUCTS 1.Introduction 2.Definitionofaninfiniteproduct 3.AnecessaryconditionforC'ol1\'crgcnce 4.The associated"criesoflogarithms 5.AbsolutecOI1\'ergence 6.Uniformconvergence Chapter2:THEGAMMAANDBETAFUNCTIONS 7.The Eulerorl'.Iascheroniconstan tI 8.The Gamma function 9,Aseriesforr'(z)/r(z) 10,Evaluationofr(1)andr'(l) 11.The Eulerproduct forr(z) 12.The differenceequationr(z + 1)=zr(z) 13.The order symbols 0and0 14.Evaluation ofcertaininfiniteproducts 15.Euler'sintegralforr(z) 16.The Beta function 17.The\'alueofr(z)r(l- z) 18.The factorialfunction 19.Legendre's duplicationformula 20.Gauss'multiplicationtheorem 21.AsummationformuladuetoEuler 22.Thebehavior oflogr(z)forlarge! z I Chapter3:ASYMPTOTICSERIES 23.Definitionofanasymptoticexpansion 24.Asymptoticexpansions about infinity 25.Algebraicproperties 26.Term-by-termintegration 27.Uniqueness 28.'Watson'slemma Chapter 4:THE HYPERGEOMETRICFUNCTION 29.The functionF(a,b;c;z) 30.Asimpleintegralform Page 1 1 2 2 3 5 8 9 10 10 11 12 12 13 Li 1." Hl 2:2 28 24 2G 29 33 36 3" 39 40 41 4.') 47 \"11 VlllCONTENTS 31.F(a,b;C;1)as afunctionof the parameters 32.Evaluation ofF(a,b;c;1) 33.The contiguousfunctionrelations 34.The hypergeometric differentialequation 35.Logarithmic solutions ofthehypergeometricequation 36.F(a,b;C;z)as afunctionofitsparameters 37.Elementary seriesmanipUlations 38.Simple transformations 39.Relation between functionsofz and1- 2 40.Aquadratictransformation 41.Other quadratictransformations 42.Atheoremdueto Kummer 43.Additionalproperties Chapter5:GENERALIZEDHYPERGEOMETRICFUNCTIONS 44.The functionpF q 45.The exponentialandbinomialfunctions 46.Adifferentialequation 47.Other solutions ofthe differentialequation 48.The contiguousfunctionrelations 49.Asimple integral 50.ThepF qwith unit argument 51.Saalschiitz'theorem 52.Whipple's theorem 53.Dixon's theorem 54.Contour integrals ofBarnes'type 55.The Barnes integrals andthefunctionpF q 56.Ausefulintegral Chapter6:BESSEL FUNCTIONS 57.Remarks 58.DefinitionofJ n(z) 59.Bessel's differential equation 60.Differential recurrencerelations 61.Apure recurrencerelation 62.Agenerating function 63.Bessel's integral 64.Index half an oddinteger 65.ModifiedBesselfunctions 66.Neumannpolynomials 67.Neumannseries Chapter7:THECONFLUENTHYPERGEOMETRICFUNCTION 68.Basicproperties ofthe)F) 69.Kummer'sfirstformula 70.Kummer's secondformula Page 48 48 50 53 54 55 56 58 fi1 63 65 68 68 73 74 74 76 110 85 85 86 88 92 94 98 102 108 108 109 110 111 112 114 114 116 116 119 123 124 125 CONTENTS Chapter8:GENERATINGFUNCTIONS 71.The generating functionconcept 72.Generating functionsoftheformG(2xt- t2) 73.Setsgeneratedby etif;(xt) 74.ThegeneratingfunctionsA(t)exp[ -xtl(l- t) 1 75.Another classofgenerating functions 76.Boas andBuck generating functions 77.Anextension Chapter 9:ORTHOGONAL POLYNOMIALS 78.Simple setsofpolynomials 79.Orthogonality 80.Anequivalent conditionforortllOgonality 81.Zerosoforthogonalpolynorniab 82.Expansionofpolynomials 83.The three-termrecurrencerelation 84.The Christoffel-Darbouxformula 85.Normalization;Bessel'sinequality Chapter10:LEGENDREPOLYNOMIALS 86.Agenerating function 87.Differential recurrencerelations 88.The purerecurrencerelation 89.Legendre's differentialequation 90.TheRodriguesformula 91.Bateman's generating function 92.Additionalgenerating functions 93.Hypergeometric formsofP n(X) 94.Brafman's generating functions 95.Specialproperties ofPn(x) 96.More generating functions 97.Laplace'sfirstintegralform 98.SomeboundsonPn(x) 99.Orthogonality 100.An expansiontheorem 101.Expansionofxn 102.Expansionofanalyticfunctions Chapter11:HERMITEPOLYNOMIALS 103.DefinitionofIln(x) 104.Recurrencerelations 105.The Rodriguesformula 106.Other generating functions 107.Integrals 108.TheHermitepolynomial asa2Fo IX Page 129 131 132 13.5 13i 1-10 143 14i 14i 14S 149 15t) 1.51 153 155 15i 15S 15U lGO 161 162 163 16;') 16i 16S 169 IiI 172 1i3 1i6 1i9 lSI lSi ISS lSU 190 190 191 x 109.Orthogonality 110.Expansionofpolynomials 111.More generating functions Chapter12:LAGUERREPOLYNOMIALS 112.ThepolynomialLn(a)(x) 113.Generatingfunction" 114.Recurrencerelations 115.The Rodriguesformula lUi.The differentialequation 117.Orthogonality 118.Expansionofpolynomials 119.Specialproperties 120.Other generating functions 121.The simpleLaguerrepolynomials Chapter13:THE SHEFFERCLASSIFICATION ANDRELATEDTOPICS 122.Differential operators andpolynomial sets 123.Sheffer'sA-type classification 124.Polynomials of ShefferA-typezero 125.Anextension ofSheffer'sclassification 126.Polynomials ofo--typezero CHAPTER14:PURE RECURRENCERELATIONS 127.Sister Celine'stechnique 128.A mildextension Chapter15:SYMBOLICRELATIONS 129.Notation 130.Symbolic relations among classicalpolynomials 131.Polynomials ofsymbolicformLn{ y(x) Chapter16:JACOBIPOLYNOMIALS 132.The Jacobipolynomials 133.Bateman's generating function 134.TheRodriguesformula 135.Orthogonality 136.Differentialrecurrencerelations 137.The purerecurrencerelation 138.Mixedrelations 139.Appell'sfunctionsoftwovariables 140.Anelementarygenerating function 141.Brafman's generating functions 142.Expansioninseriesofpolynomials CONTENTS Page 191 193 196 200 201 202 203 204 204 206 209 211 213 218 221 222 226 228 233 240 246 247 249 254 256 257 258 261 263 263 265 269 271 272 CONTENTS Chapter17:ULTRASPHERICAL AND GEGENBAUER POLYNOMIALS 143.Definitions 144.The Gegenbauerpolynomials 145.Theultrasphericalpolynomials Chapter18:OTHER POLYNOMIALSETS 146.Bateman's Zn(X) 147.Rice's Hn(t,p,v) 148.Bateman'sFn(z) 149.SisterCeline'spolynomials 150.Besselpolynomials 151.Bedient's polynomials 152.Shively'spseudo-Laguerre andotherpolynomials 153.Bernoullipolynomials 154.Eulerpolynomials 155.Tchebicheffpolynomials Chapter19:ELLIPTICFUNCTIONS 156.Doublyperiodicfunctions 157.Ellipticfunctions 158.Elementaryproperties 159.Orderofan ellipticfunction 160.TheWeierstrassfunctionP(z) 161.Other ellipticfunctions 162.AdifferentialequationforP(z) 163.Connectionwith ellipticintegrals Chapter20:THETA FUNCTIONS 164.Definitions 165.Elementaryproperties 166.The basicpropertytable 167.Location ofzeros 168.Relations among squaresoftheta functions 169.Pseudoadditiontheorems 170.Relationtotheheat equation 171.The relation81'=8283114 172.Infiniteproducts 173.The valueof G Chapter21:JACOBIANELLIPTICFUNCTIONS 174.Adifferentialequationinvolvingtheta functions 175.The functionsn(u) 176.The functionsen(u)and dn(u) 177.Relations involving squares XI Page 276 277 283 285 287 289 290 293 297 298 299 300 301 305 306 306 308 309 311 311 313 314 315 316 319 322 325 328 329 332 334 339 342 343 344 xii 178.Relations involving derivatives 179.Addition theorems Bibliography Index CONTENTS Pa!!:e 3,15 3.fi 3,19 359 SPECIALFUNCTIONS CHAPTER1 Infinite Products 1.Introduction.Twotopics,infiniteproductsandasymptotic series,whichare seldom included instandard coursesaretreatedto someextent inshortpreliminarychapters. Thevariablesandparametersencounteredaretobeconsidered complex except\vhereit isspecificallystipulated thatthey arereal. Exercisesareincludednotonlytopresentthereaderwithan opportunitytoincreasehisskillbutalsotomakea\'ailableafew resultsforwhichthereseemedto beinsufficientspaceinthetext. Ashortbibliographyisincludedattheendofthehook.All references aregiveninaformsuch asFasenmyer[2J,meaning item number two under thelisting ofreferencestotheworkof~ i s t e r 1\1. CelineFasenmyer,orBrafman[1;944J,meaningpage944ofitem number one under the listing ofreferencesto the workofFred Braf-man.Ingeneral,specificreferencetomaterialacenturyormore oldisomitted.Theworkofthegiantsinthefield,Euler,Gauss, Legendre, etc.,iseasilylocated either in standard treatises or inthe collected\yorksofthepertinent mathematician. 2.Definitionofaninfiniteproduct.Theelementarytheoryof infiniteproductscloselyparallelsthatofinfiniteseries.Gi\'ena sequenceakdefinedforallpositiveintegralk,considerthefinite product n (1) 2INFINITEPRODUCTS[Ch.l If LimP nexistsandisequaltoP~ 0,wesaythattheinfinite product (2) converges to the value P.If at least one of the factors of the product (2)iszero,ifonly afinitenumber ofthefactorsof(2)arezero,and iftheinfiniteproduct,,-iththezerofactorsdeletedconyergestoa valueP~ 0,wesay thattheinfiniteproduet convergestozero. If the infiniteproductisnotconYergent,it issaid to be divergent. If thatdivergenceisduenottothefailureofLimP ntoexistbut n+oo to the fact that the limit iszero,the product issaid to divergetozero. Wemakenoattempttotreatproduetswithaninfinityofzero factors. The peeuliar rolewhichzeroplaysinmultiplicationisthereason forthe slight differeneebetweenthedefinitionofconyergenceofan infiniteproduetandtheanalogousdefinitionofconyergeneeofan infinite series. 3.Anecessaryconditionforconvergence.Thegeneralterm ofaconvergentinfiniteseriesmustapproachzeroastheindexof summationapproaehesinfinity.Asimilarresultwillnowbeob-tainedforinfiniteproducts. m THEOREM1.If II (1+ an)converges, n=l Proof:If theproductCOll\-ergestoP~ 0, P 1=P LimII (1+ ak) _ ~ " , _ k ~ ~ ____ n-l Lim II (1+ ak) n+ock ~ 1Lim(1+ a,,). n+oo ReneeLim an=0,asdesired.If theproductconyergestozero, removethezerofactorsand repeattheargument. 4.Theassociatedseriesoflogarithms.Anyproductwithout zerofactorshasassociatedwithittheseriesofprincipal\"aluesof thelogarithms ofthe separatefactorsinthefollowingsense. 5]ABSOLUTECONVERGENCE 3 =co THEOREM2.If noan=- 1,II (1+ an)andLLog(1+ an) n=ln=l convergeordivergetogether. Proof:Letthepartialproductandpartialsumbeindicatedas follows: n Sn=LLog(!+ ak). k=! Then * exp Sn=P n.'Ve know from the theory of complex variables thatLim exp Sn=exp Lim Sn.ThereforePn approachesalimit n+ron+oo ifandonlyifSnapproachesalimit,andP n cannotapproachzero because the exponentialfunctioncannot take onthevaluezero. co 5.Absoluteconvergence.Assumethat the productII (1+ an) n-l hashaditszerofactors,ifany,deleted.Wedefineabsolutecon-vergence of the product by utilizing the associated series of logarithms. co TheproductII (1+ an),withzerofactorsdeleted,issaid tobe n=l00 absolutelyconvergentifandonlyiftheseriesL Log(1+ an)IS absolutelyconvergent.n-l = THEOREM3.TheproductII (1+ an),withzero factorsdeleted,is 11==1 'n absolutelyconvergentif andonly 1jLOnisabsolutelyconvergent. Proof:Firstthrowout anyan'swhicharezero;they contribute only unit factorsinthe product andzeroterms in the sum and thus havenobearingonconyergence. Weknowthat ifeithertheseriesortheproduct inthetheorem conyerges,Lim an=O.Let us then consider n large enough, n> no, n+oo so thatIan I < !foralln> no.'Vemay nowwrite (1) Log(1+ an)_f(-I)kank --0-;-- - k=Ok+---Y-' fromwhich it followsthat I Log (I+ On)- 11sf~ 0, notingthat inthisinstancethemannerofapproachisimmaterial. EXAMPLE(b):Forrealx,leos xl~ 1,fromwhiehit iseasyto concludethat cos x- 4x=O(x),asx-->00,xreal. EXAMPLE(c):InChapter 3weshallshowthat if Sn(X)=L k!xk, k=O li"'e-t dtI - - ~ - Sn(X) .aI-xl ~ (n+ I)!Ixjn+1,forReex)~ O. Fromtheprecedinginequalitywemayeoncludethat,forfixedn, i"'e-t dt ---- - Sn(X)=o(xn), oI-xl as x-->0inRe(x)~ O. 14.Evaluationofcertaininfiniteproducts.TheWeierstrass infiniteproductforr(z)yieldsasimpleeyaluationofallinfinite productswhosefactorsarerationalfunctionsoftheindexn.The mostgeneralsuchproductmusttaketheform (1) becauseconvergencerequiresthat thenth factorapproachunity as n~ 00,whiehinturn forcesthenumerator anddenominatorpoly-14THEGAMMAANDBETAFUNCTIONS[Ch.2 nomialstobeofthesamedegreeandtohaveequalleadingcoeffi-cients.Nowthe nth factorinthe right member of(1)may be put inthe form so that wemust alsoinsist,toobtainconyergence,that 3 (2) L: ak=L: bk k=! If (2)isnotsatisfied,theproductin(1)diverges;wegetabsolute convergenceornoconvergence. We nowhaveanabsolutelyconvergentproduct(1)inwhichthe a'sand b'ssatisfythecondition(2). Since wemay,withoutchangingtheyalueoftheproduct(1),insertthe appropriateexponentialfactorstowrite (3) The Weierstrassproduct,page9,forl/r(z)yields g[( 1 + exp( -;i) J=z exp(=-r(z + IJ exp(Thus weobtainfrom(3)theresult P_IT1'(1+ bk )exp(- k=lr(1+ (h) expl THEOREM5. integer, 88 IfL: a k=L: b k,and1fnoa korb kisanegative k=lk=l 15]EULER'SINTEGRALFORr(z) IT(n+aI)(n+a2)' .. (n+a.)=r(1 +bI) r(1 +b2) .. r(1 +b.) ,,=1(n+bI)(n+b2) (n+b.)r(1+aI) r(I +a2)". r(I+a8) 15 If oneormoreoftheakisanegativeinteger,theproduct onthe left iszero,whichagreeswiththeexistenceofoneormorepolesof the denominator factorsontheright. EXAMPLE:Evaluate co(c- a+ n- 1)(c- b+ n- 1) IT (c+ n- 1) (c- a- b+ ~ n - 1)' Since (c- a- 1)+(c- b- 1)=(c- 1)+(c- a- b- 1), wemayemployTheorem5ifnooneofthequantitiesc,c- a, c- b,c- a- biseitherzerooranegativeinteger.Withthose restrictionsweobtain (4) IT (c- a+n- 1)(c- b+n- 1)=r(c)r(c- a- b). n=I(c+n- 1)(c- a- b+n- 1)r(c- a) r(c- b) 15.Euler'sintegralforr(z).Elementarytreatmentsofthe Gammafunctionareusuallybasedonanintegraldefinition. Theorem6connectsthefunctionr(z)definedbytheWeierstrass product withthat definedby Euler's integral. THEOREM6.If Re(z)> 0, (1)r(z)=iCOe-ttz-ldt. Weshallestablishfourlemmasintendedtobreaktheproofof Theorem6into simpleparts. Lemma 1.If 0~ a0 and R---->(X)and use 0< Re(z) 0, (3)logr(z)=(z- Log z - z+Log(27f)_:x, in whichP(x)=x- [xJ- asin Section21. Let us nextconsider the integralontherightin(3).Since f P(x)dx= P2(X)+ c, wemayusec=- -h and integratebyparts tofindthat (roP(x)dx=1 [P2(X)-+ 1 ('"'[P(x)- -l:ddx Jo z+x2z+x02Jo (Z+X)2 =__1_+ 1 (CO[P(x)- -l2"Jdx. 12z2Jo (z+ X)2 Nowthemaximum\'alueof[P2(X)- 112Jisiand,intheregion Iarg z I7f- 0,0> 0, Iz+ Xl2 X2+IZ!2,forRe(z) 0, 22]THEBEHAVIOROFLOGr(z)FORLARGEIzl31 Iz+ Xl2 [x+ Re(z)J2+ Izl2 sin2 0, It followsthat forRe(z)< o. ("'lp2(x)-dx_0(_1) Jo (z+ X)2- Izl' asIz IcoinIarg z I7r- 0,0> O. Wehave shownthat asIzl coinlarg zl 7r- 0,0> 0, (4)log r(z)=(z- !) Log z - z + !Log(27r)+ 0(1). Indeedweshowedalittlemorethanthat,but(4)isitselfmore precisethan isneededlater inthisbook. From(4)weobtainatoncetheactualresulttobeemployedin Chapter 5. THEOREM13.AsIz Iro1'ntheregionwhereIarg z I7r- andIarg(z+ a) I7r- 0,0> 0, (5)log r(z+ a)=(z+ a- !) Log z- z + 0(1). EXERCISES .r'(z)1 ro(11) 1.Start WIth- =-'Y- - - L-- - - , r(z)z Z+ nn provethat 2r'(2z)r'(Z)f'(z+ 1) r(2z)- fez)- 1'(z+1) =2Log2, andthus deriveLegendre's duplicationformula,page24. 2.Showthat f'C!)=- ('Y+ 2Log2h/;':. 3.Use Euler's integral formfez)=dtto showthat f(z+ 1)=zf(z). 4.Showthat fez)=Lim n'B(z,n). n+ro 5.DerivethefollowingpropertiesoftheBeta function: (a)pB(p, q + 1)=qB(p+ 1,q); (b)B(p, q)=B(p + 1,q)+ B(p, q + 1); (e)(p+ q)B(p, q + 1)=qB(p, q); (d)B(p, q)B(p+ q,r)=B(q,r)B(q+ r, pl. 6.Showthat forpositiveintegral n,B(p, n+ 1)=n!/(P)n+1' 7.Evaluate1:(1+ x)p-I(1- X)q-Idx. Ans. 2p+q-1B(p,q). 32THEGAMMAANDBETAFUNCTIONS[Ch.:8 8.Show that for0 k n ()_(-I)k(a)n an-k- (l- a- nh Noteparticularlythe special case a=l. 9.Showthat ifaisnot aninteger, r(l-a-n)(_l)n -r(I=-a) - =Taj-;:-' In Exs.10-14,thefunctionP(x)isthat of Section21. 10.Evaluate('P(y)dy. Jo Ans.- i. 11.Useintegrationbyparts andthercsult ofEx.10toshowthat < _. n1 + x=8(1+ n) 12.Withthe aidofEx.11provetheconvergenceof:x . 13.Showthat (ro P(x) dx =f=fn+lp(X)dx=(I(y - dy. Jo1+ x n1+ xn_OJo1+ n+ y Thenprovethat Limn2(\y - dJL= n+roJo1+ n+ y12 (ro P(x)dx. andthusconcludethat Jol-+x ISconvergent. 14.ApplyTheorem11,page27,tothefunctionf(x)=(1+ X)-I;letn-->00 andthusconcludethat 'Y= - iroy-2P(Y)dy. 15.Usetherelationf(z)f(l- z)=1r/sin 1rZandtheelementaryresult sin xsin y= [cos(x- y)- cos(x+ y)] toprovethat 1_f(c)f(1- c)f(c- a- b)f(a + b +1 - c) f(c- a)f(a + 1- c)f(c- b)l'(b+ 1 - c) 1'(2- c)f(c- l)f(c- a- b)f(a + b + 1- c) f(a)f(1- a)f(b)f(l- b) CHAPTER3 Asymptotic Series 23.Definitionofanasymptoticexpansion.Letusfirstrecall thesensein,vhichaconvergentpowerseriesexpansionrepresents thefunctionbeingexpanded.WhenafunctionF(z),analyticat Z=0,isexpanded inapower seriesaboutz=0,wewrite (1) Izl< r. Defineapartial sum ofthe seriesby Sn(Z)=LCkZk k ~ OThen the seriesontheright in(1)representsF(z)inthe sensethat (2) Lim[F(z)- Sn(Z) ]= n+co foreachz intheregionIz I < r.That is,foreachfixedztheseries in(1)canbemadetoapproximateF(z)ascloselyasdesiredby taking asufficientlylargenumberoftermsoftheseries. We nowdefinean asymptoticpowerseriesrepresentationofafunc-tionf(z)asz~ insomeregionR.We,,,rite co (3)fez)~ Lanzn, z ~ inR, Tl_O if andonly if 33 34 (4)Lim .+0inR ASYMPTOTICSERIES[Ch.3 Ifez)- Snez)I Izin=0, foreachfixedn,with (5)Sn(Z)=Lakzk

Byemployingtheordersymboldefinedin 13,wemay writethecondition(4)intheform (6)fez)- Sn(Z)=o(zn),asZ---->0inR. Hereweseethattheseriesin(3)representsthefunction fez)in thesensethatforeachfixedn,thesumofthetermsouttothe termanzn canbemadetoapproximatefCz)morecloselythan I z I napproximates zero, in the sense ofC4), by choosing Zsufficiently closetozerointheregionR. It isparticularlynoteworthythat inthedefinitionofanasymp-toticexpansion,thereisnorequirementthattheseriesconverge. Indeedsomeauthorsincludetheadditionalrestrictionthatthe seriesin(3)diverge.Mostasymptoticexpansionsdodi verge,but it seemsartificialto insistuponthat behavior. Asymptotic series are of great value inmany computations.They play an important rolein the solution of linear differential equations about irregular singular points. Such series were used by astronomers morethanacenturyago,longbeforethepertinentmathematical theorywasdeveloped. EXAMPLE:Showthat (7) i""e-t dt"" 1_t"-'Ln!xn, oxn=O x---->0inRe(x) O. Let usput Sn(X)=Lk!xk

IntheregionRe(x) 0,theintegralontheleftin(7)isabso-lutelyanduniformlycOll\'ergent.Toseethis,notethatt 0so that Re(l- xt) 1.Hence11- xt I1,andwehave I J:""e-t dt=1. For kanon-negativeinteger, 23]DEFINITION OFAN ASYMPTOTIC EXPANSION35 (8)J:"'e-ttk dt=r(k + 1)=kL Hence f"" e-t dtf"" e-t dtnf"" -::--- - Sn(X)=--- - Le-ttkxk dt o1- xt01- xtk_00 =J""e-t[-_l- - t(xt)k]dt. o1- xtFrom elementary algebrawehaye r=;6.1. Therefore f""e-tdtf""e-t(xt) n+ldt -- - S(x)=----o1- xtn01- xt' fromwhich,since11- xt I1,weobtain I- Sn(X) I Ix I n+li""e-'tn+1 dt,inRe(x) o. Wemayconcludethat (9)1.[""- Sn(X) I (n+ I)! Ix I n+ 1,inRe(x) o. From(9)itfollowsatoncethatthecondition(4),page34,is satisfied,whichconcludestheproof.Actually(9)moreinfor-mationthanthat.LetEn(x)betheerrormadeincomputing thesumfunctionbydiscardingalltermsafterthetermn!xn ThenIEn(x) I istheleftmemberof(9),andtheinequality(9) showsthatIEn(x) I issmallerthanthemagnitudeofthefirstterm omitted.Thisproperty,althoughnotpossessedby allasymptotic series,isoneoffrequentoccurrence. Theprecedingexamplegi\"eslittleindicationofmethodsfor obtaining asymptotic expansions.Later weshall exhibittwocom-monmethods,successi\"eintegrationbypartsandterm-by-term integration ofpower series. Extensionoftheconceptofanasymptoticexpansiontoonein whichthevariableapproachesany specificpointinthefiniteplane is direct.For finiteZowesaythat "" fCz)'" Lan(z- zo)n,asz-->ZoinR, n_O 36ASYMPTOTICSERIES[Ch.3 ifand only if,foreachfixedn, j(z)- Sn(Z)=o([z- Zo]n),as Z---+Zoin R, inwhich 24.Asymptoticexpansionsaboutinfinity.Asymptoticsenes areoftenusedforlarge1z I.Wesaythat ro (1)J(z),-....,L anz-n,asz---+00inR, n=O ifand only if,foreachfixedn, (2)J(Z)- Sn(Z)=o(z-n),asZ---+00inR, inwhich n (3)Sn(Z)=Lakz-k

Attimes,asinthesubsequentexample,wewishtoworkonly alongtheaxisofreals.'Vethenuse(1),(2),and(3)forareal variablex,withtheregionRreplacedby adirectionalongthereal aXiS. One last extensionofthe term asymptoticexpansionfollows.It maybethat J(z)itselfhasnoasymptoticexpansioninthesenseof the foregoingdefinitions.Wedo,however,write ro (4)J(z)'"" h(z)+ g(z)Lanz-n,asZ---+00inR, n=O ifand only if (5)asz ---+00inR, andsimilarlyforasymptoticexpansionsaboutapointinthefinite plane. Obtain,forreal x,as x---+00,an asymptotic expansion oftheerrorfunction (6) 2LX erf(x)=---=exp( _t2)dt. V7r0 From the factthatrCD=vi:;',it followsat oncethat 24]ASYMPTOTICEXPANSIONSABOUTINFINITY37 Limerf(x)=1. z+ro Let us write 2rro2fro erf(x)=---=Inexp(-[2)dt- ---=cxp(-t!)dt V7r 0V7rx 2fro =1- -:=exp( - t2)dt. V7rz N owconsiderthefunction B(x)=fooexp( _t2)dt andintegrateby parts toget Iterationoftheintegrationbypartssoonyields B(x)= [1113135(-1)n135 .. (2n-1)] exp( _X2)2x - 2Zx3 + 23x5 - 24X7+ ... +2n+lxZn+l (-1)n+l13 5 .. (2n+ l)jro +t-2n-2exp( - t2)dt 2n+1z., or (7) + (_l)n+l(!)f"'t-Zn-Zexp( - tZ)dt. 2n+lz Let Then,from(7), exp(x2)B(x)- Sn(X)=(_l)n+la)n+l exp(xZ) j"'t-Zn-2 exp( _t2)dt. x 38ASYMPTOTICSERIES[Ch.3 Thevariableofintegrationisneverlessthanx.Wereplacethe factort-2n-2 intheintegrandby tx-2n-3 andthusobtain r exp(x2)B(x)- Sn(X) I 0:>. n=O Surely weare interestedhereinlargex,sothat an integralwhichit is natural to consider isfCCj(x)dx.But f"'ao dx andf"'a1x-1 dx do vvv not exist.Thereforewerestrictourselvestotheconsiderationof anexpanSIOn co (2)g(x)"-'Lanx-n, n=2 40ASYMPTOTICSERIES[Ch.3 andseekj""g(x)dx.Ofcourseg(x)=f(x)- ao- alx-1 u Let Snex)=L akx-k k-2 Then g(x)- Snex)=o(x-n),X---'>co, and ~ f""l g(x)- Sn(X)I dx " < f'"'1 o(x-n) I dx y =o(y-n+l). But Hence (3) thedesiredresult. 27.Uniqueness.Sincee-X =O(Xk),asx---'>co,foranyrealk, wholeclassesoffunctionshavethesameasymptoticexpansion. Surely if ro f(x)ro../LAnx-n, n_O thenalso "" f(x)+ ce-X ro../LAnx-n, 71=0 andnumerous similar examplcsarc casilyconcocted. Ontheotherhandagi\"cnfunctioncannothavemorethanone asymptotic expansion as z--?Zo,finitcor infinite.Let us use Z--?co inaregionRasarepresentative example. THEOREM15.If co (1)fez)ro../LAnz-n,Z---'>coinR, n=O 28]WATSON'SLEMMA 41 and ro (2)fez)"-'LBnz-n, z - coin R, n_O thenAn=Bn. Proof:From(1)and(2)wehave fez)- LAkz-k=o(z-"), k-O fez)- LBkZ-k=oCz-"),

fromwhichit followsthat L(Ak- Bk)Z-k=o(z-,,) ,

or its equivalent n L(Ak- Bk)zn-k=0(1),z _00inR,

foreachn.ThereforeA k=B kforeachk.Theexpansion(1) associatedwithz - 00inaparticularregionRisunique.The function fez)may,ofcourse,ha\>eadifferentasymptoticexpansion asz- 00insomeregionother thanR. 28.Watson'slemma.Thefollowingusefulresultdueto Watson[1 ;236Jgivesconditionsunderwhichtheterm-by-term Laplacetransformofaseriesyieldsanasymptoticrepresentation forthetransformofthesum oftheseries.For detailsonLaplace transformsseeChurchill[1]. Sincerelativelycomplicatedexponentsappearinthefollowing fewpages,weshallsimplifytheprintingbytheintroductionofa notationsimilartothecommonone,expu=euThesymbol isdefinedby expxCm)=xm. Watson'sLemma.Let F(t)satisfythefollowingconditions: (1)FCt)=fan expt('/1,- 1), initl:;;:;a+ 0,witha,0,r> 0; n=!r (2)ThereexistpositiveconstantsKandb suchthat IF(t) \< Kebt,fort a. 42ASYMPTOTICSERIES[Ch.3 Then (3) as's'-4cointheregion'args I 3h- forarbitrarilysmall positive Notethat(1)impliesthat F(t)iseitheranalyticat t=orhas at mostacertaintypeofbranchpointthere. Proof:It isnotdifficulttoshow(Exs.1and2attheendofthis chapter)thatundertheconditionsofWatson'slemma,thereexist positiveconstants c and(3suchthat forallt 0,whethert3aor t> a, (4)IF(t)- - 1)/-1, Re(s)> 0. In orderto deriye(3),weneedtoshowthat foreachfixedn If(S)- t akr( 1 slniT=0(1), asIs I -4coinIarg s!3 -> 0. Now f(s)- i: =fcoe-'t[F(t)- i: ak- l)Jdt. k=lrJok=lr Hence,withtheaidof(4), Islnlrlf(S)- t(n+ 1)(n+l) < clslnlrr--r:- [Re(s)- (3r--;-, if Re(s)>(3.In theregionIargs I 3 -> 0,Re(s)>{3 28]WATSON'SLEMMA43 assoonaswechooseISI >(J(sinLl)-l.Therefore,asIs I ---+(X)ill theregionIarg s I!11"- Ll, Isln'rlf(S)- 1; =0(1), asdesired. EXAMPLE:Obtain an asymptoticexpansionof (COe-XI dt f(x)=Jo I+ t2' Ix I ---+coinIarg x I! 11"- Ll,Ll> O. Notethat theresultwillbevalidinparticular forrealx->co. We shallapply Watson'slemmawithF(t)=1/(1+ f2).Then coco F(t)=L(-1)nt2n=L(_1)n+ll2n-2, It I < 1, n=On=l sothat wemay write n=l inIt I inwhichaZn =0,aZn_l=(-l)n+l,andwehavechosenr=1, a=!,0=iinthenotationofWatson's lemma. For t?;!, et >1and1/(1+ t2)O. EXERCISES 1.With the assumptions ofWatson'slemma,page 41,show,withthe aidofthe convergence of the series in(1),that for0 t a,there exists apositive constant Clsuchthat 44ASYMPTOTICSERIESlCh.3 [F(t)- i:.ak- 1) I 0, and comparetermsofthe series (3) withcorresponding termsofthe series (4) knownto beconvergent.SinceIz I =1 and LimI n+co(C)nn. .I(a)n(b)n(n- l)!nc (n- 1)!n1HI = (n- 1) !na .(n- 1) !nb --cc3-n-.n!nc-a-b =Irta).rtb).ric) I Inc-Lb-ol=0, becauseRe(c- a- b- 0)=20- 0> 0,theseriesin(1)isab-solutelyconvergentonIzi=1 whenRe(c- a - b)> o. AmildvariationofthenotationF(a,b;c;z)isoftenused;itis (5) F[a,b; c, whichissometimes moreconvenient forprinting and whichhas the advantage of exhibiting the numerator parameters a and b above the denominatorparameterc,thusmakingiteasytorememberthe respectiverolesofa,b,andc.\Vhenwecometothegeneralized hypergeometricfunctions,weshallfrequentlyuseanotationlike that in(5). The seriesontheright in(1)or in (6) F[a,b; C", 30]ASIMPLEINTEGRALFORM47 iscalledthehypergeometricseries.Thespecialcasea=c,b=1 co yieldsthe elementary geometricseriesLzn;hencethetermhyper-n=O geometric.The functionin(6)or in(1)iscorrespondingly calledthe hypergeometricfunction.AlthoughEulerobtainedmanyproperties ofthefunctionF(a,b;c;z),weowemuchofourknowledgeofthe subject tothemore systematicand detailedstudymadebyGauss. 30.Asimpleintegralform.If nisanon-llegativeinteger, (b)nreb + n) r(c) (C)n=r(c+ n) reb) r(c) reb) r(c- b) r(b+ n) r(c- b) r(c + n) If Re(c)> Re(b)> 0,weknowfromTheorem7,page19,andthe integral definitionoftheBeta function,that reb+ n) r(c- b)=(ItHn-I(1_t)c-b-ldt. r(c + n)Jo Therefore,forIz I 0,the series (1)F(a,b;c;1) isabsolutelyconvergent. Let0 be any positive number.We shallshowthat intheregion Re(c- a- b);;:;20> 0,theseries(1)forF(a,b;c;1)isuni-formlyconvergent.Tofixtheideas,it maybedesirabletothink ofRe(c- a- b);;:;20> asaregioninthec-plane,\vithaand b chosen first.It isnot necessary to look onthe region in that way. The seriesofpositiveconstants (2) isconvergent because0> 0.We showthat fornsufficientlylarge andforalla,b,cintheregionRe(c- a- b);;:;20> 0,withc neitherzeronor anegativeinteger, (3) Now(seepage 46) LimI I ,,+00(C)nn. =Ir(c)ILim1 __11= rea) reb)n+oo nc-a-b-o , sinceRe(c- a- b- 0);;:;20- 0=0> 0.Hence(3)istrue fornsufficientlylarge,andtheWeierstrassM-testcanbeapplied tothe seriesin equation(1). THEOREM17.Ifcisneitherzeronoranegative7ntegerand Re(c- a- b)> 0,F(a,b;c;1)isananalyticfunctionof a,b,c. 32.EvaluationofF(a,b;c;1).IfRe(c- a- b)> 0, Theorem17permits ustoextend the integralformforF(a,b;c;z), page47,tothepointz=1inthefollowingmanner.Since Re(c- a- b)> 0,wemaywrite F(a,b;c;1)=i: (c)nn. IfwealsostipulatethatRe(c)>Rc(b)> 0,itfollowsbythe techniqueofSection30that 32]EVALUATIONOFF(a,b;c; 1)49 F(a,b;c;1)=r(c)f- t)c-b-ldt reb) r(c- b) n!0 =r(c)i1tH(1_t)C-H(1_t)-adt. reb) r(c- b)0 Therefore,ifRe(c- a- b)> 0,ifRe(c)> Re(b)> 0,andsince c isneitherzeronoranegativeinteger, F(abc1)=--. __-- -f.l/b-l (1- t)- c-a-b-ldt ,"I'(b)r(c- b)" r(c)reb) r(c- a- b) =r(b)f(c- b) r(c)r(c- a- b) =r(c-"=--a)r(c=--b)' Wenowresortto Theorem17andanalyticcontinuationtocon-cludethat theforegoingevaluationofF(a,b;c;1)isvalidwithout the conditionRe(c)> Re(b)> 0. THEOREM18.If Re(c- a- b)> 0andIf cisneitherzeronor a negativeinteger, r(c)r(c- a- b) F(a,b;c;1)=r(c--=:-a) r(c_b)' ThevalueofF(a,b;c;1)willplaya\'italroleinmanyofthe resultstobeobtained inthisandlaterchapters.Theorem18can beprovedwithouttheaidoftheintegralinTheorem16.For such aproof see-WhittakerandWatson[1;281-282]. EXAMPLE:ShowthatifRe(b)> 0andifnisanon-negative integer, [-!n,-!n + L F b+ t; By Theorem18weget [-1.n-1.n+1.. 2,22, F b +!; 1]=2n(b)n. (2b)n reb+ !)r(b + n) --'----,-reb+ !n) reb++ !) (b)n reb) reb+ D reb+ !n) reb++ 50THEHYPERGEOMETRICFUNCTION[Ch.4 Legendre'sduplicationformulafortheGammafunction,page24, yields reb) reb+ ~ ) =21-2b\/;r(2b), reb+ ~ n ) reb+ ~ n +) =21-2b-ny'; r(2b+ n). Therefore F[-!n,-!n + ~ ;b+!; as desired. 33.Thecontiguousfunctionrelations.Gaussdefinedascon-tiguousto F(a,b;c;z)each ofthe sixfunctionsobtained by increas-ing or decreasing one ofthe parameters by unity.For simplicity in printing,weusethe notations (1) (2) (3) F=F(a,b;c;z), F(a+)=F(a+ 1,b;c;z), F(a-)=F(a- 1,b;c;z), togetherwithsimilarnotationsF(b+),F(b-),F(c+),F(c-)for the other four of the six functions contiguous to F. Gaussproved,andweshallfollowhistechnique,that betweenF and any two of its contiguous functions,there exists a linear relation with coefficients at most linear in z.The proof isone ofremarkable directness;weprovethattherelationsexistbyobtainingthem. Thereare,ofcourse,fifteen(sixthingstakentwo at atime)such relations. Put sothat (4) and F(a+)=i:t ~ ) n On. , , ~ O (a)n Sincea(a+ l)n=(a+ n)(a)n,wemaywritetheSIXfunctions contiguoustoFintheform 33]THECONTIGUOUSFUNCTIONRELA TIONS51 (5) ena-I F(a-)=L----- On, a-I + n enb- 1 F(b-)=L---- On, b- 1+ n coc F(c+)=L--+ On, Cn F(c-)= ---- On. II-,(jC- 1 We also employ the differential operator0=operntor hastheparticularlypleasantpropertythatOZ"=nz",thusmaking ithandytouseonpower series. Since co (6)(0+ a)F=L(a+ n)On, 11cn OF(ak-) =L _O:i;- 12. -IO:k+ 11- 1n!' or Now,with the notationof(7), Cn +l -+-- =CnTn,k, O:kn p=q+1. whereTn,khas,forp~ q,itsnumeratoroflowerdegreethanitsde-nominator.Thus where ~ (3jWj ,k Tnk=~ - -,}=1(3j+n in which the Wj,k is as defined in(4). \Venowhave q OF(ak-) =(ak- l)x LWj ,kF(3j+);p~ q;k=1,2, . "p. j=1 But,from(12), OF(ak- )=(0:k- l)[F- F(o:k- )]. The elimination ofof(a k-) fromthe preceding two formulasyields theprelations: q (20)F=F(ak-) + xLWj,kF(3]+);p~ q; k=1,2". "p. Whenp=q+l, the fractionTn,khas its numeratoranddenomina-50]THEpFqWITHUNITARGUMENT85 tor ofequaldegree.Thenwewrite pq II (a,+ n)- II ({3.+ n) Tn.k=1+ 8=1.( k)s=1_____ q II ({3,+ n) })=1 and concludeinthesamemanner asbeforethat q eF(ak-) =(ak- l)xF+ (ak- l)x LlVi,kF({3j+). )1 Therefore,forp=q+l,(20)istobereplacedby q (21)(1- x)F=F(ak-) +xLlVi,kF({3j+);Ie=1,2"", p. )-1 Wehaveshownthat for inwhichnotwo{3'sareequalandno(3isanonpositiveinteger,a canonical set of(2p+q)contiguous functionrelations isas described below. If p Re(aJ)> 0,'f noone of bJ,b2 "', bq is zerooranegativeinteger,and ifIz I co, logr(am+ 8)=(pcos 0+ ipsin 0+ am- p+ 1:0)- pCos0 - i psin0+ 0(1), or logream+ 8)=[pcos0 + Re(am)- !JLogp- pcose - pOsin0 + PI + 0(1). 100GENERALIZED HYPERGEOMETRIC FUNCTIONS[Ch.5 AgainweusePItodenoteallpureimaginaryterms.Thereal term- OIm(am )isbounded by the constanthlm(am)and isthere-foreincluded in 0(1).Ofcourse logr(bj+s) =[pcos 0+ Re(bj)-p- pCos 0- posin o+PI +0(1), logr(l+s)=[pcosO+I-!]Logp-p cosO-pOsinO+PI+O(I), and loge -z)" =pcos0Logjzj- psin0arg( -z)+PI. It isanelementarymatter(confliderj sin'If.'!I -1)toshowalsothat log-;-1I'-- =-'lfPjflinol+PI+O(I). fllll'lfii For theif;(s)of(3)it followsthat (6)log if;(s)=[A -B-1 +(p cos 0- !)(p-q-l)]Log p - p( cos0 +0 sin 0) (p - q - 1) +pcos0 Log Iz I - psin 0 arg( - z) -'lfP Isin 01+PI +0(1), inwhich m=l BecauseofTheorem34,page97,weareinterestedonlyin p=q+1 and inp=q.Firstletp=q+1.Then(6)yields logif;(s)=(A-B-l)Logp+pcosOLoglzl-psinOarg(-z) -'lfplsin01+PI+0(1), fromwhich (7)if;(s)= O[pA-B-l exp{ p cos 0 Log Iz 1- p sin 0 arg( -z) -'lfP Isin 011], Since-!'Ifcos0 0.If wechooseIzl 0,Re(f3)> 0,and1fkisapositive integer,theninsidetheregionof convergenceof theresultantseries (5) etk a+f3a+f3+1a+f3+h-l b...b--- ~ - ~ ...-- -- ---' I,,q,k'k"/;, Thereadercaneasilywritethecorrespondingtheoremfork=0, s> 0. EXAMPLE:In equation(4)ofTheorem 37 choosep=0(no a's), q=1(oneb),b1 =1,a=1,/3=1,k=1,s=1,e=- t.The resultis ][ 1,1; - tx(t-x)dx=B(I,l)t 2F:, 1,! , ~ ;=toFI[-; 3 ~ ,56]AUSEFULINTEGRAL105 The reader may already know,orwilldiscoverinthe nextchapter, thatJo(z),theBesselfunctionofthefirstkindandofindexzero, isgivenby Jo(z)=oFl( - ; 1;- !Z2). The oFlinthe second formofthe right member of(6)iselementary. Indeed =2sinTherefore(6)mayberewrittenintheform (7)llJoC yxV-- x)dx=2 sintt. EXERCISES 1.Showthat OFI[-;XJoFI[-; a;b; ][la+ Ib!a+ Ib- J.. 22' 222, X=2F3 a,b,a+ b- 1; YoumayusetheresultinEx.6,page69. 2.Showthat i[1 t4'", o x!(t- x)-![1- X2(t- x)2]-idx="FI1. , 3.With theaidofTheorem8,page21,showthat andthat r(1+ r(1+ a) cos!7rar(1- a) r(1- !a) Thus put Dixon'stheorem,Theorem33,page92,intheform [a,b,c;] 3F2I I+ a- b,1 + a- c; cossin 7r(b-r(l- a)r(b- !a)r(l+ a- c)f(l+ - b - c) =-s[; 7r(b=- a)-- .r(I- - c)r(l + a- b- c) 106GENERALIZED HYPERGEOMETRIC FUNCTIONS[Ch.5 4.Usethe result inEx.3to showthat if nisanon-negative integer, [-2nCi1- '"- 2n-"fJ, Ji'2 1- Ci- 2n,(3; 5.Withthe aidofthe formulain Ex.4 proveRamanujan'stheorem: ][a;][a,(3- a; XIFI-x=2F. {:J;{:J,!(3,;(3+ !; 6.Let'Yn=sF;( -n,I- a- n,1- b- n;a,b;1).UsetheresultinEx.3 to showthat 'Y2n+1=0and (-I)n(2n) !(a+ b- 1)3n 'Y2n=-,---------. n.(a)n(b)n(a+ b- Ihn 7.With the aidofthe result inEx.6 showthat oF2(-; a,b;t)oF2(-; a,b;-t) [ !ea + b- 1), i(a + b),!ea + b + 1); -=J's a,b,!a, !a + !, !b, !b + !, !(a + b- 1),Ha+ b); 8.Provethat n(_l)n-k('Y- b-c)n_kC'Y-b)k(-y-chx,,-k[-k, b,c;x] t;ok!(n- k)!('YhSF2l--y+b-k,I--y+c-k; =('Y-b)n('Y-c),,(l-x)n sF2[-!n,-tn+t, l--y-n;-4X] n!('Y)n+b(l_X)2' 1-'1'-n,I-'Y+c-n; andnotethe specialcase-y=b + c,Whipple'stheorem,page 88. Exs.9-11below usethe notation of the Laplace transform as inEx.16,page 71. 9.Showthat 10.Showthat 11.Showthat 56]AUSEFULINTEGRAL107 12.Showthat 13.In Ex.19,page71,wefoundthat thecompleteellipticintegralofthefirst kindisgivenby Showthat CHAPTER6 Bessel Functions 57.Remarks.Nootherspecialfunctionshavereceivedsuch detailed treatment inreadily availabletreatises*ashavethe Bessel functions.Consequently weherepresentonly abriefintroduction tothesubject,includingthoseresultswhichwillbeusedinlater chapters ofthis book.f For adiscussion of orthogonality properties and ofzerosofBessel functions,seeChurchill[2JandWatson[1].Anextremely simple resultonzerosofBesselfunctionsappearsinEx.13attheendof thischapter. 58.DefinitionofIn(z).WealreadyknowthattheoFoisan exponential and that thelFoisabinomial.It isnatural to examine nextthemostgeneralof l,theonlyotherpF q withlessthantwo parameters.Thefunction"\veshallstudyisnotpreciselytheoFl but onethat hasan extra factorasin(1)below. We defineJ n(Z),fornnot anegativeinteger, (1) (z/2)n(Z2) In(z)=r(I+ n)of!-; 1+ n;-4 . If nisanegativeinteger,weput *Themoststrik;'lgexampleis\'latson's('xhaustive(804pages)work;Watson[1]. AnexpositionsufficientlythoroughformostreaderswillbefoundinChapter17of Whittaker andWatson[1]. 108 59]BESSEL'SDIFFERENTIALEQUATION 109 (2)In(z)=(-l)nJ_n(Z). Equations(1)and(2)togetherdefineJ n(Z)forallfinitezandn. The functionJ n(Z)iscalled"the Besselfunctionofthe firstkind of indexn." Since of1( - ;1 + n;-z4:)=fJ:-: (1+ n) kl.. ! ,>;r (1+ n) ( - 1) kZ2 k t=oITr(l--+-n--t=-k)22k' therelation(1)isequivalentto m(_1)kz2k+n (3)In(z)=622k1n1Jr(1+ n+ k) Note alsothe immediateresult (4) 59.Bessel'sdifferentialequation.Weknowadifferential equationsatisfiedbyanyoFJbyspecializingtheresultinSection 46.The equation (1)[0(0+ b- 1)- yJu=0; d 0- y-- dy' hasu=oFJ( -; b;y)asonesolution.Equation(1)canalsobe written d2udu y- + b -- - u=0. dyZdy (2) Wenowput b=1+ n,y=-zz/4 in(2)to obtain (3)zu"+ (2n+ l)u'+ zu=0, inwhichprimesdenotedifferentiationswithrespecttoz.One solution of(3)isu=oFJ( -; 1+ n;- z2/4).We seek an equation satisfied by w=znu.Hence in(3)wenow put u=z-nw and arrive at the differentialequation (4)z2w"+ zw'+ (Z2- n2)w=0, ofwhichone solutionisw=znOF1(-; 1+ n;-z2/4). Equation(4)isBessel'sdifferentialequation.IfnISnotan integer,two linearlyindependent solutionsof(4)are 110 (5) and (6) BESSELFUNCTIONS[Ch.6 WI=J n(Z) Wz=J _,,(z) andthey arevalidforallfinitez. If niszeroorapositiveinteger,(5)isstillavalidsolutionof Bessel'sequation,butthen(6)isnotlinearlyindependentof(5). For integraln,asecondsolutiontoaccompanyJ"(z)islogarithmic incharacterandcanbeobtainedbystandardprocedures. *The result is n-I(-l)k+I(n_I)!zzk-n (7)Wa=Yn(Z)=J n(Z)log Z + k22k,1-n/,;1(1- n)k 1m(-I)k+l(Hk+ Hk+n)z2k+n + "26, -- n) ,--forn>1.In(7)thecommonconventionIIo=hasbeenused. Forn=0,thefirstseriesontherightin(7)istobeomitted; forn=1,thefirstseriesontherightin(7)istobereplacedby the singleterm(-Z-I). 60.Differential recurrencerelations.Intheequation co(-I)kz2k+n J n(Z)=L--;;k+-;;- .,------- -_. kdJ2"". r(1+ n+ J ) (1) \yemay multiply both members by zn and then differentiate through-out \vithrespecttoztoobtain den(- 1) kz2k+2n-1 (2)dz[znJ,,(z)]=.6r(n+ Ie) inwhichwehavecanceledthefactor(2k+ 2n)innumeratorand denominator,usingr(I+ n+ k)=(n+ k)r(n + k).Equation (2)canberewrittenas d00(-1) kzZk+n-1 dz[znJ n(Z)]=zn(; 22k+n-1k!r(I+ n- 1+k)' andwethusseethattherightmemberisznJ n_I(Z).Weconclude that (3) 'InRainville[1],pertinenttechniquesareexplainedonpages285-291(forn=0) andonpages299-303(forn> 0). 61]PURERECURRENCERELATION111 Equation(3)may alsobeput intheform (4) whichiscalledadifferentialrecurrencerelation,differenti.albecause ofthedifferentiation recurrencehemuseofthepresenec ofdifferentindicesnand(n- 1). Next wereturnto equation(1),insert thefad.orz-n on each side, and againdifferentiate eachmemberwithrrsppcLtoztoobtain (5) dCD(-I)kzU-l dz[z-nJ n(Z)]=t; 22kI,,-=i(l..= 1)kY Ashiftofindexfromkto(I..+1)yields d00(_1 ) k1lZ2 HI dz[z-"J,,(z)]=6r(1+ n+- i fromwhichweobtain (6) Equation(6)canbeexpressedequallywellas (7) Inequations(7)and (4)zJ n'(z)=zJ n_l(Z)- nJ,,(z) wehaye two differential recurrence relations.From them it follows alsothat (8) 61.A purerecurrencerelation.EliminationofJ ,,'(z)fromthe relations(4)and(7)oftheprecedingsectiongivesusatoncethe purerecurrencerelation (1) It isalsoinstructive to obtain(1)fromthe solecontiguous function relationpossessedby theoFlfunction. Consider the set ofBessel functions J n(Z)fornon-negative integral index.If wewrite(1)intheform 2(n- 1) (2)J n(Z)=J n-1(Z)- J n_2(Z), Z 112BESSELFUNCTIONS[Ch.6 weobtaineachJ nofthesetintermsofthetwoprecedingit;J fromJo andJ1,J3 fromJ1 andJ2,etc.Inthis\vaywecan,for integral n,write (3) The coefficients An(z) and BnCz)are then polynomials in liz.These are simple special cases ofLommel's polynomials Rm.,(z)which may beencountered*byapplyingthesameprocessto(2),withnre-placedby(II+ m)toarriveat theresult (4)J,+m(Z)=Rm.,(z)J,(z)- Rm-I.,+I(Z)J._1(z). TheLommelpolynomialisa2F3(secWatson[1:207]): (5) - ~ ] .Z2 62.Ageneratingfunction.Ourapproach(Section58)tothe functionJ nCZ)wasfromthehypergeometricstandpoint,whichis naturalherebecausethishookislargelyconcernedwithfunctions ofhypergeometriccharacter.SomeauthorsapproachJ n(Z)by firstdefiningit,forintegralnonly,by meansofageneratingfunc-tiontrelationwhichweshallnowobtain. Lemma 12.Forn~ 1, [n/2][(n-1)/2] (1)LA (k,rn)=LA(k,n)+LA(n- k,n). k ~ O k ~ O k ~ OProof:First notethat forintegraln~ 1, (2)n=1+ [!nJ+ [!en- l)J, inwhich[]istheusualgreatestintegersymbol.Equation(2)is easilyverifiedseparatelyfornevenandfornodd. Next note that n[n/2]1+ [n /2] +[(n-l) /2] (3)LrA(k, n)=LA(k, n)+LA(k, n). k=Ok ~ O k=J.+[n/2] Inthelastsummationin(3)replacekby(n- k);thatis,kby 1+ [nI2]+ [en- 1)/2J- k.Then *SeeWatson[1 :294]. tSeeChapter 8forsomedetailonthe generatingfunctionconcept. 62]AGENERATINGFUNCTION 113 n[n/2] LA(k, n)=LA(k, n)+LA(n - k,n), fromwhichLemma12followsby reversingtheorder ofthe second summationontheright. THEOREM39.FortT"=0and forall finitez, (4)- DJ="tooJ,,(z)tn Proof:Let uscollectpowersofz inthesUlllmation co':n LJ ,,(z)t"=LJ,,(z)tn+ LJ ,,(z)tn Tt=-UJ'ft.,...-0011=0 0000 =LJ _n_I(Z)t-n-1 + LJ n(Z)tn. 7l=On=O We defined J m(Z)fornegative integral minSection58.Using that definitionweget rororo LJ n(Z)tn =L(-l)n+IJ n+I(Z)t-n-1 + LJ n(z)ln n=-ron=On=O ro(_ 1) n+k+ll-n-Izn+Zk+1ro(_1) ktnzn+Zk =+ nto2"+2kk!(n+ k)! ro[n/Z](_l),,-k+lt-n+Zk-lzn+1ro[n/Z](-l)ktn-Zkzn =?; {;o2n+lk!(n+1- k)! 2nk!(n- k)! ro[(n-ll/2](_1)n- ktk-(n-klznro[n/2](_1)kt n- k- kzn - LL. - + 1+ LL. -' - n-lk_Qk!(n- k)!2nn-I k!(n- k)!2" We nowuseLemma12to concludethat roron(_ 1) kln-k-kzn J,,(z)tn =1+(; k!(n_k)!2" it'"cxp[iz(t- l)J. n.22l SeealsoEx.23at theendofthischapter. 114BESSELFUNCTIONS[Ch.6 63.Bessel'sintegral.Theorem39oftheprecedingsection may be interpreted asgivingtheLaurent expansion,validnear the essentialsingularityt=0,forthefunctionexp - t-1)J.The Laurent seriescoefficientis known.Indeed, (1) 1 f(Ot)[1] In(z)=27riu-n-1 exp2z(u- u-1)du, inwhichthecontour(0+)isasimpleclosedpathencirclingthe originu=inthepositivedirection. In(1)let uschoosethepartieularpath u=eiO =cos0+ 'Isin0, o runningfrom(- 7r)to7r.Thenu-I =cos0- isin0,and(1) yields J() 1f"[...J d nZ=2;_:xp - mO+ tZsm 00 =217ri:cos(no- Zsin 0)do- f; - Zsin0)do. Inthelasttwointegralstheformerhasanevenfunctionof0as integrand,thelatter anoddfunctionof0 asintegrand.Hence ]i" J n(Z)=- cos (nO- Z sin 0)do, 7r0 whichisBessel's integralforJ n(Z). THEOREM40.Forintegraln, 1 i"(2)J n(Z)=- cos (no- Z sin 0)do. 7r0 Bessel's integral representationofJ n(z)canbe*extended to non-integraln.The result,calledSchlajl'i'sintegral,IS (3)J n(Z)=1.("cos(no-z sin 0)do- n7r("'exp( -nO-z sinh 0)do, 7rJ 07rJ0 validforRe(z)> O.Equation(3)willnotbeusedinourwork. 64.Index half an odd integer.Let usput into hypergeometric formtheelementary expansion *SeeWhittaker andWatson[1:362]. 641INDEXHALFANODDINTEGER ro(_1)kz2k+l (1)sm z=.6(2k+ 1)( Since(2k+ I)!=(2)2k)equation(1)yields or (2) Now Jl(z)=(z/2) I oFl( _. ;t._lZ2) ) 2,4 and=!y;.Hence (3) Ji(z)=(;zY sin z. or In much the samemannerthe elementary expansion = cos z=L-----(2n)!' (4)cos z=F(- . l - lZ2) oI,2,4 leadsus totherelation (5) ( ?)! J -1(Z)=;zcosz. In Section61wederivedthepurerecurrencerelation (6)J n(Z)=2(n-1)z-IJ n_I(Z)- J n_2(Z). In(6)replacenby(n+ !) toobtain (7)In+!(z)=(2n- l)z-lJn_l(z)-Let nbeapositiveintegeranditerate(7)to seethat (8) inwhichPIand P2 arepolynomials intheir arguments. From equations(3),(5),and(8)it followsthat forintegral n In+l(z)=A(z)cosZ+ B(z)sinz 115 116BESSELFUNCTIONS[Ch.6 inwhichA(z)and B(z)arepolynomialsinz-l. Besselfunctionsofindexhalfanoddintegerareoftencalled sphericalBesselfunctions.They,aswellasmostotherI3essel functions,are encountered in various physical problems.Spherical Bessel functions led to the definition and study of I3essel polynomials whichwediscussto some extent later inthis book. 65.ModifiedBesselfunctions.Manyphysicalproblemslead to the study ofBessel functionsofpure imaginary argument.This in turnleadstothe definitionofsuchfunctionsas ..(z/2) "(Z2) (1)I,,(z)=r"J,,(zz)=1'(1+ 71)uFI-; 1+ 71;4' 71notanegativeinteger.ThefunctionI n(Z)iscalledamodified Besselfunctionofthefirstkindofindexn.AstudyofJ,,(z)for complexzincludescorrespondingpropertiesofI ,,(z)bysimple changesofvariables.The functionI nisrelatedto J ninmuchthe samewaythat thehyperbolicfunctionsarerelatedtothetrigono-metricfunctions.SomeelementarypropertiesofI n(Z)willbe foundinthe exercisesbelow. 66.Neumannpolynomials.FromTheorem39,page113,we obtain,forw;;e0and forallfinitez, co (1)- w-I)]=LJ n(Z)Wn, n=-ro whichcan equally well(Ex.2,page120)bewritten co (2)exp[!z(w- w-I)]=Jo(z)+ LIn(z)[Wn + (-l)nw-n], n=l becauseJ _n(Z)=(-1) nJ n(Z).In(2)putw=t+ Vr + 1and notethat(-w-1)=t- Vt2+ 1.Theresultis co (3)ezt =Jo(z)+ LIn(z)[(t+ Vr + 1)n+ (t- Vt2+ I)n]. 7t=1 Letusdefinefn(t)by (4)fn(t)=(t+ Vt2-.f.l)n+ (t- Vt2+I)n,n O. Then fn(t)isapolynomial int and(3)nowappears as

(5)ed =Uo(t)Jo(z)+ Lfn(t)Jn(z). T/=1 66]NEUMANNPOLYNOMIALS 117 TheLaplacetransform(Churchill[1])ofapolynomialint isa polynomialinS-1.Let20n(s)betheLaplacetransformofour f,,(t): (6) Thenfrom(5),sinceLI eztl=(s- Z)-I,weobtain (7) The polynomialsO,,(s)arecalledNeumannpolynomials. Letusassumethattheseriesin(7)issufficientlywellbehaved (provedbelow)thatthemanipulationstobeperformedarelegit-imate.Differentiationof(7)yields m (8)-(s - Z)-2=Oo'(s)Jo(z)+ 2L.,On'(s)Jn(z) 7/.=1 and = (9)(s- Z)-2=Oo(s)Jo'(z)+ 2L.,On(S)J n'(z). 71 0 -1 Now2J n'(z)=J n_I(Z)- J n+I(Z)andJo'(z)=-J1(z),sothat(9) maybewrittenas co'" (s- Z)-2=-Oo(s)J1(z)+ L.,On(S)Jn_1(z)- L., On(S)Jn+1(z) n=l co(0 =-Oo(s)J1(z)+ L., On+l(S)J n(Z)- L.,O,,_I(S)J ,,(z), n=On=2 or OJ (10)(s- Z)-2=01(S)JO(z)+ L.,[O,,+I(S)- O,,-I(S)JJ n(Z). 71=1 From(8)and(10)it follmvsthat OJ [Oo'(s)+ 01(S)]JO(z)+ L.,[20n'(s)+ O,,+I(S)- O,,_I(S)JJn(z)=O. Sinceforeachnthefunctionz-nJ ,,(z)isnonzeroatz=0,it followsthat an expansionoftheform ISumque. 118BESSELFUNCTIONS[Ch.6 HenceOI(S)=-Oo'(s)and (11)On+I(S)=On_I(S)- 20/(s),n;;;:;1. WeknowthatOo(s)=g-Iandnowthat01(S)=g-2.The Neumann polynomials maynowbedescribedasfollows: (12) (13) Oo(S)=S-l,01(8)=g-2, On(8)=On_2(S)- 20:_1(8),n 2. The Ones)are uniquely determined bythe description(12)and (13). THEOREM 41.TheNeumann polynomials defined by(12)and (13) abovearegivenbyCMs)=S-Iand (14) n[n/2J(n- 1- k)!(2js)n+I-2k Ones)=4(;k! n 1. Proof:From(14),OI(S)=t(2/S)2=g-2.Also,forn;;;:;2, and [n-IJ n- 12(n- 2k)(n- 2- k)!( _2/s2) (2/s)n-I-Zk -.. . _-4k-Ok! n- 1[n/2](n- 2k)(n- 2- k)!(2/s)n+l-2k =--- L:,. 8k_Ok. Therefore,fortheOneS)of(14), On_2(8)-_2[n/2J[en- 2)k+ (n- l)(n- 2k)J(n- 2- k)!(2/8)n+l-2k - 4L:/'-' c. =On(8), as desired. By Theorem 41the dominating term in is 2n-In!8-n-1The dominating term inIn(z)is(!z)n/n!.Therefore,asn->00, 2n-In!zn On(S)J n(Z)= 2nn!(1+ En) 67]NEUMANNSERIES119 in which En-> O.ForIZ I r,chooselsi R where R> T.Thus fornsufficientlylarge, IOn(S)J n(Z)I 0 onthat interval,and if 147 148ORTHOGONALPOLYNOMIALS[Ch.9 fbW(X)tpn(X)cpm(X)dx=0, a mr!'n,(1) wesay that the polynomialstpn(X)are orthogonalwith respect to the weightfunctionw(x)ovcrtheintcrvala< x< b.Becausewe have taken w(x)> andCPn(X)real,it followsthat fbwex)cpn2(X)dxr!'0. a Withdueattentiontoconver!!;ence,eitherorbothendpointsof theintervaloforthogonalitymaybetakentobeinfinite.The conceptoforthogonalityusedhcrehasbeencxtendedinmany directions,butthesimpleversionabo\"cisallweuse.Alar!!;e number ofthc sets ofpolynomials encountcrcdlater inthcbookarc orthogonal sets.The limits ofintegration in(1)are important but theforminwhichtheintervaloforthogonalityisstated(opcnor closed)isnotvital. 80.Anequivalentconditionfororthogonality.Thefollowing theorem isofuseinour study ofpolynomialscts. THEOREM54.If theCPn(X)formasimplesetofrealpolynomials and w(x)> ona< x< b,anecessaryandsufficientconditionthat theset tpn(X)beorthogonal with respect tow(x) over theinterval a O.Therefore sFo( -k, !+n-k; - i 1)(,,=0k.(n- 2k). The finalresultis (6) Pn(X)=[I:J 2Fo(-k,!+ n- k;-; k!(n- 2k)! Next let us expand the Hermite polynomial in aseries of Legendre polynomials.By Theorem 65,page181, (7) Now (2x)n n! i: H n(X )t"_co[n/2](-1) k(2x) n-2kt" n=On!-?; t:okl(n- 2k)! -(-1)k(2n + 1)Pn(x)tn+2k+28. 'k'(lt) s..2n+8 Againwecollectpowersoft: 196HERMITEPOLYNOMIALS[Ch.11 i: n. =ft (-l)k-'(2n + 1)Pn (x)tn+2k 8!(k- 8)!0)n+. =:t t(-1)8k!. (-1)k(2n +l)Pn (x)tn+2k 81(1.;- 8)10+ n).k10)n =['f - k;+ n- 2ft';+ l)P n_2k(X)tn. "=0 k1CVn-2k Therefore, (8)Hn(x)= IFJ(-k;! + n- 2k; 1)(-1)kn!(2n- 41.;+ l)Pn_2k(x). k!q)n-2k The expansionoffunctionsother thanpolynomialsinto seriesof Hermitepolynomialsisomittedhere.Theoremsexistsimilarto theonesrelativetoexpansionsinseriesofLegendrepolynomials. 111.Moregenerating functions.WewishtoobtainforH n(X) aproperty similar tothat forP n(X)expressedinequation(7),page 169.Considerthe series _ Hn(x)(t+ v)n -L- r n. =exp[2x(t+ v)- (t+ V)2] =exp(2xt- t2)exp[2(x- t)v- V2] (2t =expx- L- I.. r k=OL. By equating coefficientsofvk /k!,weobtain 111]MOREGENERATINGFUNCTIONS197 Hn+k(x)tn ....---,- =exp(2xt- t2)H k(X- t). 11. (1) Asafirstexample inthe useofequation(1)let us derivewhat is sometimescalledabilineargeneratingfunction.Considerthe serles 00Hn(x)Hn(y)tn 00[n/2)(-1)k(2x)n-ZkHn(y)tn L---,--- =LL---.j - - n.Ii. (n- '21,). by(1).Since =i: k'!n! =i: exp( 4xyt- 4XZ[2)H - 2x[) ( -1) k[Zk k=Ok!' k(-1)s(2k)!(2v)2k-28 H2k(v)=L ---:r(2-k- 2-)-'-, s.s. and(2k)!=it followsthat 00(-1)k22s(J)(9y_4,,-t)Zkl2k+28 =exp(4xyt- 4x2t2)L'1k+,.-':-- k, 8 0s!( D k 00DO(!+ k)s22st2s(_ 1)ktZk(2y- 4xt)2k =exp( 4xyt- 4x2t2)LL2 hO 8!k! DO(-1) ktZk(2y- 4xl)2k =exp(4xyt- 4x2t2).6k!(l_4tz)!+k [-4t2(y- 2Xt)2] =(1- 4tz)-!exp(4xyt- 4xz[2)exp--1=4t-z - 198HERMITEPOLYNOMIALS[eh.11 Theexponentialfactorsmaybecombinedandthepreceding identity written as (2) fHn(x)Hn(y)tn =(1_4t2)-i exp[Y2_ (y- 2xt)2], n=On!1- 4t2 agenerating relationknownforabout acentury. We canapply equation(1)to any knowngenerating;relation and sometimesobtainanewresult.Onpage1DOweobtainedthe relation (3) [1C'}c+;(1- 2xt)-c 2Fo_; To(3)weapply(1)inthefollowingmanner.Considertheseries Becauseof(3)it nowfollmvsthat '" exp(2xt- t2)[1+ 2yt(x- t)J-c arelation obtained by Brafman[2Jwithcontour integration asthe maintool. 111]MOREGENERATINGFUNCTIONS EXERCISES 1.Usethe factthat exp(2xt- t2)=exp(2xt- x2t2)exp[t2(x2 + 1)] to obtainthe expansion [n12J'II(1)n-2k(2+ 1) k II(x)=L1l:.: __ =2". __xx. nk-Ok!(n- 2k)! 2.Usethe expansionofx"inaofHermitepolynomialsto showthat i: exp( -x")xnlf"n(X)dx=2-2kn!{t Note inpartieular the specialcasek=O. 199 3.Usethe integral evaluation in equation(4),page1D2to obtainthe result Looexp( -X2)II2k(X)H2..+1(x)dx=+ I- 2k). 4.Byevaluatingtheintegralontheright,usingequation(2),page187,and term-by-term integration, showthat (A) 2Lm Pn(x)=-- exp(-t2)tnll"(xt)dt, n!00 whichisCurzon's integralforp"(x) , equation(4),pageIGI. 5.Letvn(x)denotetherightmemberofequation(A)ofEx.4.Prove(A)by showing that ro Lvn(x)Y"=0- 2xy+ y2)-i. n=O 6.Evaluate the integral onthe right in (B)Hn(x)=2n+1 exp(x2) J:':xP(dt by using n(:I:)_[n12Jn!(x2 =-f}.'j2x)n-2k (2t)Pn t-t;o(k!)2(n-2k)! derivedfromequation0),page164,andterm-by-termintegrationtoprove the validity of(B),whichisequation(5),page191. 7.Usethe Rodriguesformula anditeratedintegrationbyparts to showthat m=n. CHAPTER12 Laguerre Polynomials 112.ThepolynomialLn (Q)(x).Letusconsideranaturally terminatingIFIWe define,fornanon-negative integer, (1)L (a)()- (1+ a)nF(- 1+. ) nX- IIIn,a, X n. Thefactor(1+ a)n/n!isinsertedforconvenienceonly.The polynomials (1)are calledLaguerre, generalizedLaguerre, or Sonine polynomials.Thespecialcasea=receivesmuchindi\'idualat-tentionand isknown either astheLaguerre orthesimpleLaguerre polynomial.Whena=0,aisusuallyomittedfromthesymbol: (2) WeshallworkwithLn (a)(x),butforreferencepurposesalistof propertiesofLn(x)isincludedat theend ofthechapter. The notationin(1)isquite standardwith the oneexceptionthat some authors permit ato dependuponn; others donot.Weshall insistthatabeindependentofnbecauseforthepolynomials(1) somanypropertieswhicharevalidforaindependentofnfail (Shively[1])tobevalidforadependent upon n. It shouldbeapparenttothereaderbythetimehehasfinished readingthisbook,ifnotbefore,thatforapolynomial~ n ( X ) of hypergeometriccharacter,thewayinwhichthe indexnentersthe parameters of thepFqinvolwd has avital effect upon the properties ofthe polynomial.For amathematicianto usethe same name for 200 113]GENERATINGFUNCTIONS thetwopolynomials and (1 a)n.. -'--------;-,----'---'IF1( -n, 1 a, x) n. (1 C n)" --nr---1F1( -n; 1 c n; x), 201 inwhichaandc areindependent ofn,isroughlythe equivalent of alayman's usingthe samenameforaneagleand akitten. From(1)it followsat oncethat (3)L(al(x)- i: _( -I)k(ln-- k)!(1+ a): from whieh we see that the I'n (al(x)form a simple set of polynomials, the coefficientofxnbeing( -1) nln!. From(3)weobtain Lz(a)(x) = - (2+a)x+tx2, L3(a)(x)= i(1+a)(2+a)(3+a) - H2+a)(3+a)x+H3+a)x2- !X3. 113.Generating functions.Direetlyfrom(3)ofthepreceding section weobtain '"Ln(a)(x)tn'"n(-I)kxktn ?; (Ia)n= t=ok!(n- k)!(1 ah - (ff -n! n!(1+ a)n HencetheLaguerrepolynomialshavethegeneratingfunctionin-dicatedin (1) Sinceanyof!isaBesselfunction,weareledalsotowritetheleft member of(1)inthelesspretty form (2)r(I+ a)(xt)-a/ze t Ja(2yxt)=f (1 a)n Aset of other generating funetionsforthesepolynomials iseasily found.Let c bearbitrary andproeeed asfollows: 202LAGUERREPOLYNOMIALS[Ch.12 00(c)nLn(a)(x)tn00n(C)n( -x) ktn (1+ a)n=,?; {; h!(n- k)!(1+ ah (C)k(- xtp =LIFo(c+ k;-; t)-Iolro}+--)-k=(}(.ak '"(C)k( -xi)k = =-t)C+k' Wethusarrive at thegeneratingrelation(seealsopages134-135) (3) Equation(3)is aspeC'ialcase of aresult due to Chaundy [1].Note thecommonlyquotedf'pp(ialcasewithc=1+ a: (4) 1(-Xl)ro exp--- =LLn(a)(x)tn. (l-t)l+aI-t114.Recurrence relations.We han already seen in Chapter 8 thatthe\"eryformofthegeneratingfunctions(1)and(4)ofthe preceding sectionleads at onceto therelations(withD=d/dx) (1)xDLn (a)(x)=nLn (a\x)- (a+ (2)DLn (a)(x)=D- n-I (3) Eliminationofthederi\"ati\'esfrom(1)and(2)yieldsthepure recurrencerelation (4)nLn(a\x)=(2n- 1 + a- x)I';,".!.I(x)- (n- 1 + Wealreadyknowthree(2p+ q)contiguousfunctionrelations fortheIFIFromequations(15),(18),and(20)ofSection48, usingp=1,q=1,a]=-n, (3,=1+a,weobtain 115]THERODRIGUESFORMULA (5)(-n - a)IFI(-n; 1+ a; x) 203 =- nIFI ( - n+ 1;1+ a;x)- a IFI ( - n; a;x), (6)(-n + x)IF1(-n; 1+ a;x) (n+ a+l)x - nIFI ( - n+ 1; 1+ a;X)+1 + aIF1 (- n;2+ a;x), (7)IF1(-ni 1 + aiX)=IFI(-n - 1; 1 + a;X) X + I+ aIF1 (- n; 2+ a; x). Since n!L,,(a)(x)

equations(5),(6),(7)maybeconvertedintothemixedrecurrence relations (8)Ln(a)(x)= + L,,(a-l)(x), (9)(n- x)Ln(a)(x)=(a+ - xLn(a+l)(x), (10)(1+ a+ n)Ln (a) (x)=(n++ xLn (a+l)(x). Next ashift ofindex in(10)yields xLn_1 (a+l)(x)=(a+- nLn (a)(x) =-x DLn(a)(x), by(1)above.Hencewehave (11)D Ln(a)(x)=-L,,_I(a+1l(x). Comparisonof(3)and(11)showsthat (12) L,,(a+1l(x)=LLkia,(x). k_O 115.TheRodriguesformula.LetUI-'returntotheexpanded form (1) Since wemay write 204LAGUERREPOLYNOMIALS[Ch.12 involvingthebinomialcoefficientC",k.KowDke-z =(-l)ke-z; so,wemay concludethat eIx-a.no(2)L,,(a)(x)=-,- ~ C"ADn-kxn+a] [Dke-x]. n.k ~ OIn view of Leibnitz' rule for the nth derh'atiye of a product, equation (2)yields (3) thedesiredformulaofRodriguestype. 116.Thedifferentialequation.SincetheLaguerrepoly-nomial is aconstant multiple of aIF),wemay obtain the differential equation (1)xD2L,,(a)(x)+ (1+ a- x)DL,,(a)(x)+ nLn(a)(x)=0 fromthegeneraltheory.Equation(1)isalsoeasytoderiyeby eliminatingL ~ a ~ 1 (x)fromthetwodifferentialrecurrencerelations (1)and(2)ofSection114. Thethree-termpurerecurrencerelation(4),page202,suggests thatwelookforanorthogonalitypropertyoftheLaguerrepoly-nomials.Either the differential equation or the Rodriguesformula leadsusquicklyto the desiredresult. 117.Orthogonality.Theprecedingdifferentialequationfor L,,(a)(x)may beput intheform (1)D[xa+1e-zDLn(a)(x)]+ nxae-ZLn(a)(x)=0; asiseasilyverified.Equation(1)together with d D==dx' (2)D[xa+1e-zDLm(a)(x)]+ mxae-xLm(a)(x)=0 leads at onceto (m- n)xae-zL,,(a)(x)Lm(a)(x) =Lm(a)(x)D[xa+1e-z DL,,(a)(x)]- Ln(a)(x)D[xa+1e-z DLm(a)(x)] =D[xa+1e-z{Lm(a)(x)D L,,(a)(x)- /",,(al(x)J)L",(a)(x) l]. 117]ORTHOGONALITY205 Therefore we have (3)(m- n) fbxae-XLn(a) (x)Lm(a)(x)dx a Theproductofe-X andanypolynomialinx---40asx---400. Furthermore,xa+1 ---40asx---40ifRe(a)>-1, soequation(3) yieldstheorthogonality property (4)iooxae-XLn(Cl)(x)Lm(a)(x)dx=0, m n,Re(a)>-1. Equation(4)showsthatifRe(a)>-1, thepolynomialsLn(a)(x) form an orthogonal set over the interval (0,(0)with weight function xae-X.Wenowneedto e\aluatetheintegralontheleftin(4)for m=n.For the sakeofvarietyweusetheRodriguesformula d (5)Ln(a)(x)=-1z!Dn[e-xxn+a],D=dx' to evaluate the integral on the left in (4)both for m=nand m n. Becauseof(5)wemaywrite andthenintegrateby parts ntimestoobtain (6) forRe(a)>-1.Ateachintegrationbypartstheintegrated portion, o < k;n, vanishesbothat x=0andasx---400. Since Lm(a)(x)isof degree m,DnLm(a)(x)=0forn>m.There-foretheintegralontheleftin(6)\anishesforn> m.Sincethat integral issymmetric innand m,it alsovanishesforn< m,which completesour secondproofof(4). We knowthat DnLn(a)(x)=D{( + 1rn-1]=(-l)n. 206LAGUERREPOLYNOMIALS[Ch.12 Hence,for m=n, equation(6)yields or (7) L'"r(1+ a+ n) x"e-x[L,,(al(x)]2dx=-----,---, on. Re(a)>-1. In the notationofthetheory of orthogonalpolynomials,Chapter 9,weha\'e r(1+ a+ n) g"=,' n. ( -1)'1 hn=--,-' n. THEOREM68.Ifa>- 1,theLaguerrepolynomialshavethe followingproperties: (8)i"'xae-XLncal(X)Xkdx=0,k=01')",(n- 1)' ,, ... ", (9)Thezerosof Ln(a)(x)areposit'iveanddistinct; (10)k!Lk (al(x)Lk (al(y)= I)!(x)(al(y). (l+a)k(l+a)nx-y, (11)If i"'xae-XP(x)dxexists, Lim [Q_+, a)n]-lr;ac-xf(x)Ln(al(x)dx=O. n+ron.Jo' Thethree-termrecurrencerelationforLn(al(x)hasalreadybeen obtained; it is equation(4),page202. 118.Expansionofpolynomials.SincetheLn(al(x)forman orthogonalset,theclassicaltechniqueforexpandingapolynomial by the method indicatedinTheorem 56,page1;')1,isa\'ailable..-\s usual weprefer to treat the problem by obtaining firstthe expansion ofxnandthenusinggeneratingfunctiontechniqueswheneverwe can. Equation(1),page201,yields (1) Therefore 118]EXPANSIONOFPOLYNOMIALS 207 0>(-x)ntn 0>n(-l)n-kLk(a)(x)t" (1+ a)nn!= {; (n- k) !(1+ ah fromwhichit followsthat (2) Letusemploy(2)inexpandingtheHermitepolynomialina series ofLaguerrepolynomials.Consider the series 0>Hn(x)tn 00(-1)s(2x)nln+28 L --f- =exp(2xt- t2)=L,,._- n. 8.n. =f -Hn-I);-t] X ... k_O_Ha+n+k),- Ha+n+k-1); (- I)k2n+k(1+ a)n+kLk(a)(x)tnH

ron[-Hn-k),-Hn-k-l);] =LL2F2-!X I()I(1) -'2 a + n ,-'2 a + n - ; (-1)k2n(1+(n- k)!(l+ a)k From the abovewemay concludethat n[-Hn-k),-Hn-k-1);-41]X (3)Hn(x)=2"(1+ a)nL2F2 _Ha+n),- Ha+n-l); ((1+ a) k 208LAGUERREPOLYNOMIALS[Ch.12 N ext let us expand the Legendre polynomial in a series of Laguerre polynomials.Considertheseries =i: (-1)'( !)n+.(2x) ntn+2. s!n! =:_C-_1L"_+_82_n+_k_C,,_'(a_)C_x ) t n_+_k+_28 n.k . s!n!(l+ a)k 00n[-Hn-k),-Hn-k-l);] =LL2F3tX n=Ok=O!_n,- Ha+ n),- Ha+ n- 1); Wemaythereforewrite C-1)k2nCDn(1+ a)"Lk(a)(x)tn (n- k) !(1+ ;r:---' ( -n)kLkC"jx). (1+ a)k TheLaguerrepolynomialcanbeexpandedinseriesofeither LegendreorHermitepolynomialsby employingpreciselythetech-niqueusedabovewiththeaidofthepertinentexpansionsofxn 119]SPECIALPROPERTIES209 fromChapters10and11.Thisisleftforthereadertodo;the resultsmaybefoundinExs.2and3attheendofthischapter. 119.Specialproperties.ThegeneratingfunctionsofSection 113leadtocertainsimplefinitesumpropertiesoftheLaguerre polynomials.For instance,from (1) ( xt)ro (1- t)-J-aexp =LLnCal(X)tft, I- tn-O and (1-=(I- t)-Ca-/l)(1- it followsat oncethat (2)Ln(a)(x)=t(a- k. forarbitrary aand{3. Fromequation(1)andthefactthat (1- t)-J-aexp(-xt )(1-1-t1-t it followsthat =(1_t)-J-(a+/l+J)exp(=-(X+ y)t) 1- t n (3) Ln (a+Il+!)(x+ y)=LLk

The generating relation roLnCal(x)tn (4)et aF\( -; 1+ a;-xt)=L(1+) n::zOan together withthe factthat etoFJ( -; 1+ a;-xyt)=e(l-liltelit oFJ( -; 1+ a;-x(yt) yields roLn(al(xy)tn_(ro(1_y)ntn)(roLn(a)(x)yntn) L --- - LL --c--'--'--":- (1+ a)n n! (1+ a)n fromwhichweget (5)LnCa)(xy)=t (1+ a)n(1- y)n-kykLkCa)(x). (n- 1.:)!(1+ a)k For(5)seealsoEx.1,page145. 210LAGUERREPOLYNOMIALS[Ch.12 Weknow that forarbitrary c, (6) [C' (1- t)-c IFI' 1+ a; By Kummer's firstformula,page125,wehave (7)IFl[c; =exp(.i?t) IFl[1+ a- c; 1+ a;1+ a; xt] 1- t Using(6)and(7)wewrite i: ,,=0(1+ a)" ( t)[1+ a- c; =(1- t)-c exp tIF! 1+a; xt] 1- t ( t)[1+ a- c; =(1- t)-1-(2c-a-2)exp (I- t)-(I+a-c)\F\1+ a; xt] 1- t =[i: Ln(2C-a-2)(x)t n ][ i: - C)nLn(a)0x)tn], n=O,,=0(I+ O')n withtheaidof(1)and(6).Weconcludethat forarbitrary c(not zeroor anegativeinteger) (8)Ln(a)(x)=(1+ O')n(I+ a- C)kLk(a)((C)nk=O(1+ O'h Inequation(8)thetwospecialchoicesC=1+ !o'+ !mand c=1+ a+ m,fornon-negativeintegraltn,areparticularlyrec-ommended.SeeExs.6and7at theend ofthischapter. WenextseekforLaguerrepolynomialsarelationanalogousto equation(7)ofSection95onLegendrepolynomialsandequation (1)ofSection111onHermitepolynomials.Considerthe series '"'"(n+ '"nnltn-kvkLn(a)(x) {; Sklnl={; {;kl(n- k)l '"(-x(V + t) =LL,,(a)(x)(t+ v)n=(1- t- v)-\-aexp. n=O1- t- v 120]OTHERGENERATINGFUNCTIONS211 We wish to expand the right member above in powers ofv in another way.Now [VJ-I-a (1- t- v)-I-a=(1- t)-I-a1- 1_t and [-xCv + t)J( -xt)[-xvJ exp=exp exp 1 - t - v1- t(1- t) (1- t- v) [-X11] --0---xt1- t1- t =expC_t) exp11' 1---1- t Hence wemay write ff(n+ k_On_OkIn! =(1- t)-I-a fLIc(a)(_X )(__ 1- tk=O1- t1- t Wefind,by comparing coefficientsofVic,that (9) (n+ k) =(1_t)-I-a-k(-xt)L (a)(_X_) L..k IIexp1- tk1- t' """,0.n. arelationwhichisusefulindiscoveringgenerating functions. 120.Other generatingfunctions.Considertheseries fn!Ln(a)(x)Ln(a)(y)tn =fi: (-l)kn !ykLn(a)(x)tn n=O(1+ a)nn=O k!(n- k)!(1+ a)k Forthemomentlet n!Ln(a)(x)Ln(a)(y)tn. if;=L.. n=O(1+ a)n Wemay now useequation(9)aboveto concludethat 212LAGUERREPOLYNOMIALS[Ch.12 -xt1- t () co(1- t)-kLk(al(_X_) ( _yt)k if;=(1- t)-I-", exp1_t6(1+ a)k xyt] (1- t)2. THEOREMfi9.IfI t I q+ 1.Shealsoob-tainedafewresultsof interestforsomeofthesimplerofherpoly-nomials. Wequotethe easilyderivedresult which includes 1feo (5)P n(1- 2x)=.rIny-!e-lIfn( -; - ; XV)dy v7r and (6) Usingp=1,q=1,al=!, b1 =1,wefindthatSisterCeline's (4)in this sectionbecomes (7) 1feo fnC!;1; x)=--= Iny-ie-lIfn( - ; 1; xV)dy. V7r0 Asshe points out,forBateman's Zn, (8)fn(!;1; x)=Zn(X), and intermsofthe simpleLaguerrepolynomial, (9) 292OTHERPOLYNOMIALSETS[Ch.18 Equation(9)isthespecialcasea=-n,f3=1,ofRamanujan's theoremobtainedinEx.5,page106.By combining(7),(8),and (9),Sister Celine obtains (10) ThegeneralpolynomialofSister Celine,(2)inthissection,falls under the classificationof Theorem48,page137,withc=1.For the moment,denotethepolynomial of(2)byCn(x): (11) Then Theorem 48yields (12) (13) n-l (14)XC,,/(X)- nC,,(x)=- L[Ck(x)+ 2xC/(x)], k_O n-l (15)xCn/(x)- nC,,(x)=L(- I) ,,-k(2k+ I)Ck(x).

Nextletusturntothepolynomial(2)withnoa'sandnob's. Put (16) apolynomial whosepure recurrencerelationwederivedinChapter 14.AsSister Celinepoints out, (17) involving the Laguerre and Hermite polynomials with the symbolic notation ofChapter15. For thej,,(x)of(16)thegenerating function(1)becomes (18) [ -4xt]ro (1- t)-1exp(1_t)2= 150]BESSELPOLYNOMIALS293 Thereforethis fn(x)isapolynomial ofShefferA-typezero.Inthe notation ofChapter13, (19)A(t)=(1- t)-I, -4t H(t)=(1- t)2 Then the inverseofH Ct)is (20) 2