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FieldsInstituteCommunicationsVolume00,0000(Nov.20,1995)Elementaryderivationsofsummationandtransformationformulasforq-seriesGeorgeGasperDepartmentofMathematicsNorthwesternUniversityEvanston,IL60208-2730Wepresentsomeelementaryderivationsofsummationandtransformationfor-mulasforq-series,whicharedi erentfrom,andinseveralcasessimplerorshorterthan,thosepresentedintheGasperandRahman[1990]\BasicHypergeometricSeriesbook(whichwewillrefertoasBHS),theBailey[1935]andSlater[1966]books,andinsomepapers;thusprovidingdeeperinsightsintothetheoryofq-series.Ourmainemphasisisonmethodsthatcanbeusedtoderiveformulas,ratherthantojustverifypreviouslyderivedorconjecturedformulas.Inx5thisapproachleadstothederivationofanewfamilyofsummationformulasforvery-well-poisedbasichypergeometricseries6+2kW5+2k;k=1;2;::::Severaloftheobservationsinthispaperwerepresented,alongwithrelatedexercises,intheauthor’sminicourseon\q-SeriesattheFieldsInstituteminiprogramon\SpecialFunctions,q-SeriesandRelatedTopics,June12{14,1995.Asiscustomary,weemploythenotationsusedinBHSfortheshiftedfactorial(a)0=1;(a)k=a(a+1) (a+k 1);k=1;2;:::;theq-shiftedfactorial(a;q)0=1;(a;q)k=(1 a)(1 aq) (1 aqk 1);k=1;2;:::;(a;q)1=limk!1(a;q)k=1Yk=0(1 aqk);jqj1;(a;q) =(a;q)1(aq ;q)1;0jqj1;therFshypergeometricseriesrFs(a1;a2;:::;ar;b1;:::;bs;z) rFs a1;a2;:::;arb1;:::;bs;z =1Xk=0(a1)k(a2)k (ar)kk!(b1)k (bs)kzk;1991MathematicsSubjectClassi cation.Primary33D15,33D20,33D65;Secondary33C05,33C20.ThisworkwassupportedinpartbytheNationalScienceFoundationundergrantDMS-9401452.c 0000AmericanMathematicalSociety0000-0000/00$1.00+$.25perpage12GeorgeGasperandther sbasichypergeometricseriesr s(a1;a2;:::;ar;b1;:::;bs;q;z) r s a1;a2;:::;arb1;:::;bs;q;z =1Xk=0(a1;a2;:::;ar;q)k(q;b1;:::;bs;q)kh( 1)kq(k2)i1+s rzk;where k2 =k(k 1)=2;(a1;a2;:::;ar;q)k=(a1;q)k(a2;q)k (ar;q)kandtheprincipalvalueofq istaken.Wealsoemploythecompactnotationr+1Wr(a1;a4;a5;:::;ar+1;q;z)forthevery-well-poisedr+1 rseriesr+1 ra1;qa121; qa121;a4;:::;ar+1a121; a121;qa1=a4;:::;qa1=ar+1;q;z#andde nethebilateralbasichypergeometricrsseriesbyrs(z) rs a1;a2;:::;arb1;b2;:::;bs;q;z =1Xk= 1(a1;a2;:::;ar;q)k(b1;b2;:::;bs;q)k( 1)(s r)kq(s r)(k2)zk:Forsimplicity,unlessstatedotherwiseweshallassumethatnisanonnegativeinteger,jqj1innonterminatingq-series,andthattheparametersandvariablesarecomplexnumberssuchthattheseriesconvergeabsolutelyandanysingulari-tiesareavoided(whichusuallyleadstoisolatedconditionsontheparametersandvariablessincethesingularitiesareusuallyatpolesandatlimitsofsequencesofpoles).Foradiscussionofwhentheaboveseriesconverge,seeSections1.2and5.1inBHS(inthethirdparagraphonp.5eachoftheratiosjb1b2 bsj=ja1a2 arjshouldbereplacedbyjb1b2 bsqj=ja1a2 arj).1.Theq-binomialtheoremThesummationformula1F0(a;|;z)=1Xk=0(a)kk!zk=(1 z) a;jzj1;(1:1)iscalledthebinomialtheorembecause,when a=nisanonnegativeintegerandz= x=y;itreducestothebinomialtheoremforthen-thpowerofthebinomialx+y:(x+y)n=nXk=0 nk xkyn k:(1:2)Since,byl’H^opital’srule,limq!11 qa1 q=aandhencelimq!1(qa;q)k(q;q)k=(a)kk!;Elementaryderivationsofsummationandtransformationformulasforq-series3itisnaturaltoconsiderwhathappenswhenthecoe cient(a)k=k!ofzkintheseriesin(1.1)isreplacedby(qa;q)k=(q;q)kor,moregenerally,by(a;q)k=(q;q)k:Hence,letussetf(a;z)=1Xk=0(a;q)k(q;q)kzk;jzj1;(1:3)withjqj1:Thecasewhenjqj1willbeconsideredlater.Notethat,bytheWeierstrassM-test,sincejqj1theseriesin(1.3)convergesuniformlyoncompactsubsetsoftheunitdiskfz:jzj1gtoafunctionf(a;z)thatisananalyticfunctionofz(andofa)whenjzj1.Onewayto ndaformulaforf(a;z)thatisageneralizationof(1.1)isto rstobservethat,since1 a=1 aqk+aqk a=(1 aqk) a(1 qk);f(a;z)=1+1Xk=1(a;q)k(q;q)kzk=1+1Xk=1(aq;q)k 1(q;q)k[(1 aqk) a(1 qk)]zk=1+1Xk=1(aq;q)k(q;q)kzk a1Xk=1(aq;q)k 1(q;q)k 1zk=f(aq;z) azf(aq;z)=(1 az)f(aq;z):(1:4)Byiteratingthisfunctionalequationn 1times,we ndthatf(a;z)=(az;q)nf(aqn;z);whichonlettingn!1andusingqn!0yieldsf(a;z)=(az;q)1f(0;z):(1:5)Nowseta=qin(1.5)togetf(0;z)=f(q;z)(qz;q)1=(1 z) 1(qz;q)1=1(z;q)1;which,combinedwith(1.3)and(1.5),givestheq-binomialtheorem1 0(a;|;q;z)=1Xk=0(a;q)k(q;q)kzk=(az;q)1(z;q)1;jzj1;jqj1:(1:6)ThissummationformulawasderivedbyCauchy[1843],Jacobi[1846],andHeine[1847].Heine’sproofof(1.6),whichisreproducedinthebooksHeine[1878],Bailey[1935,p.66],Slater[1966,p.92],andinx1.3ofBHSalongwithsomemo-tivationfromAskey[1980],consistsofusingseriesmanipulationstoderivethefunctionalequation(1 z)f(a;z)=(1 az)f(a;qz);(1:7)iterating(1.7)n 1times,andthenlettingn!1togetf(a;z)=(az;q)n(z;q)nf(a;qnz)=(az;q)1(z;q)1f(a;0)=(az;q)1(z;q)1;whichgives(1.6).4GeorgeGasperAnotherderivationoftheq-binomialtheoremcanbegivenbycalculatingthecoe cientsck=g(k)a(0)=k!;k=0;1;2;:::;intheTaylorseriesexpansionofthefunctionga(z)=(az;q)1(z;q)1=1Xk=0ckzk;(1:8)whichisananalyticfunctionofzwhenjzj1andjqj1:Clearlyc0=ga(0)=1:Onemayshowthatc1=g0a(0)=(1 a)=(1 q)bytakingthelogarithmicderivativeof(az;q)1=(z;q)1andthensettingz=0:But,unfortunately,thesucceedinghigherorderderivativesofga(z)becomemoreandmoredi culttocalculateforjzj1,andsooneisforcedtoabandonth