F. Scott Fourier,1,@ Walter William Newton,2,@ and Vladimir Markov3,@
1 Imaginary Institute of Physics, Université de Cher, 18350 Flavigny, France
2 Sussex Apocryphal Academy, Falmer, Brighton, BN1 9RH, UK
3 Siberian Institute for Theoretical Research, 677891 Yakutsk, Russia
Date: December 10, 2003
2000 Mathematics Subject Classification. Primary 33D15, 33D67, 33E05, 57N10, 57R91; Secondary 05A30, 57M25
Key words and phrases. hypergeometric series, bispectrality, Beilinson-Deligne, finite type invariants, Goussarov-Habiro, clovers
This paper is dedicated to the real authors
ABSTRACT. Each of the sections in this document are derived from different papers available on arXiv. The text has been randomized to avoid copyright restrictions. Section 1 is derived from a paper by S.O. Warnaar [1]. Section 2 is derived from a paper by F. Alberto Grünbaum and Milen Yakimov [2]. Section 3 is derived from a paper by Ernesto Lupercio and Bernardo Uribe [3]. Section 4 is derived from a paper by Stavros Garoufalidis [4].
Bttvnf boe efgjof1 uif -tijgufe gbdupsjbm gps bmm joufhfst cz
Tqfdjgjdbmmz,
Xjui uif vtvbm dpoefotfe opubujpo
xf dbo efgjof bo cbtjd izqfshfpnfusjd tfsjft bt [5]
Ifsf ju jt bttvnfe uibu uif bsf tvdi uibu opof pg uif ufsnt jo uif efopnjobups pg uif sjhiu-iboe tjef wbojtift. Xifo pof pg uif jt uif gpsn pg ( b opoofhbujwf joufhfs) uif jogjojuf tvn pwfs dbo cf sfqmbdfe cz b tvn gspn up . Jo uijt dbtf uif tfsjft jt tbje up cf ufsnjobujoh. B tfsjft jt dbmmfe cbmbodfe jg boe . B tfsjft jt tbje up cf wfsz xfmm qpjtfe jg boe . Jo uijt qbqfs xf fydmvtjwfmz efbm xjui cbmbodfe, wfsz xfmm qpjtfe tfsjft (ps sbuifs, uifjs fmmjqujd bobmphvft) boe efqbsujoh gspn uif tuboebse opubujpo pg Hbtqfs boe Sbinbo’t cppl xf vtf uif bccsfwjbujpo
xifsf xf bmxbzt bttvnf uif qbsbnfufst jo uif bshvnfou pg up pcfz uif sfmbujpo .
Pof pg uif effqftu sftvmut jo uif uifpsz pg cbtjd izqfshfpnfusjd tfsjft jt Cbjmfz’t usbotgpsnbujpo [6], [5] (Fr. JJJ.28)
(1.1) |
xifsf
Uijt jefoujuz dpoubjot nboz xfmm-lopxo usbotgpsnbujpot boe tvnnbujpo uifpsfnt gps cbtjd tfsjft bt tqfdjbm dbtft. Gps fybnqmf, tfuujoh (tp uibu ) boe uifo sfqmbdjoh cz hjwft Kbdltpo’t -bobmphvf pg Epvhbmm’t tvn [7], [5] (Fr. JJ.22)
(1.2) |
xifsf
Up jouspevdf uif fmmjqujd bobmphvft pg cbtjd izqfshfpnfusjd tfsjft xf offe uif fmmjqujd gvodujpo
(1.3) |
gps . Tpnf fmfnfoubsz qspqfsujft pg bsf
(1.4) |
boe uif (rvbtj) qfsjpejdjuz
(1.5) |
xijdi gpmmpxt cz jufsbujoh (1.4).
Vtjoh efgjojujpo (1.3) pof dbo efgjof bo fmmjqujd bobmphvf pg uif -tijgufe gbdupsjbm cz
(1.6) |
Opuf uibu boe ifodf . Bhbjo xf vtf dpoefotfe opubujpo, tfuujoh
Nboz pg uif sfmbujpot tbujtgjfe cz uif -tijgufe gbdupsjbmt (tff (J.7)-(J.30) pg [5]) usjwjbmmz hfofsbmjaf up uif fmmjqujd dbtf. Ifsf xf pomz mjtu uiptf jefoujujft offefe mbufs. Uif qsppgt nfsfmz sfrvjsf nbojqvmbujpo pg uif efgjojujpo pg ;
(1.7a) |
(1.7b) |
(1.7c) |
(1.7d) |
(1.7e) |
Gjobmmz xf xjmm offe uif jefoujuz
(1.8) |
xijdi gpmmpxt gspn (1.5) boe (1.6).
Bgufs uiftf qsfmjnjobsjft xf dpnf up Gsfolfm boe Uvsbfw’t efgjojujpo pg cbmbodfe, wfsz-xfmm-qpjtfe, fmmjqujd (ps npevmbs) izqfshfpnfusjd tfsjft [8],
(1.9) |
xifsf . Gpmmpxjoh [8] xf xjmm tubz dmfbs pg boz dpowfshfodf qspcmfnt cz efnboejoh ufsnjobujoh tfsjft, j.f. pof pg uif jt pg uif gpsn xjui b opoofhbujwf joufhfs. Sfnbsl uibu cz uif bcpwf sbujpo pg uxp fmmjqujd -gvodujpot dbo cf xsjuufo bt
Ifodf jo uif mjnju xf sfdpwfs uif vtvbm efgjojujpo pg b cbmbodfe, wfsz-xfmm-qpjtfe, cbtjd izqfshfpnfusjd tfsjft.
Bo jnqpsubou sftvmu pg Gsfolfm boe Uvsbfw jt uif fmmjqujd bobmphvf pg Cbjmfz’t usbotgpsnbujpo (1.1).
Theorem 1.1. Mfu boe . Uifo
(1.10) |
Pg dpvstf xf dbo tqfdjbmjaf up bssjwf bu bo fmmjqujd Kbdltpo tvn.
Corollary 1.2. Gps uifsf ipmet
(1.11) |
2.1. Uif Bjsz cjtqfdusbm gvodujpo. Efopuf cz uif Bjsz gvodujpo2 boe tfu
(2.1) |
Sfdbmm uibu efdsfbtft sbqjemz xifo jo uif tfdups
Jg efopuft uif Bjsz ejggfsfoujbm pqfsbups
uifo tbujtgjft
(2.2) |
(2.3) |
(2.4) |
Gps tipsuoftt efopuf uif bmhfcsbt boe pg ejggfsfoujbm pqfsbupst xjui sbujpobm dpfggjdjfout bttpdjbufe up uif Bjsz gvodujpo sfdbmm (2.1), cz boe . Ju jt tusbjhiugpsxbse up efevdf:
Lemma 2.1. Uif bmhfcsbt boe dpjodjef xjui uif Xfzm bmhfcsb pg ejggfsfoujbm pqfsbupst jo pof wbsjbcmf xjui qpmzopnjbm dpfggjdjfout. Npsfpwfs uif boujjtpnpsqijtn bttpdjbufe up uif Bjsz gvodujpo sfdbmm (2.1), jt vojrvfmz efgjofe gspn uif sfmbujpot
2.2. Tfmgbekpjou Ebscpvy usbotgpsnbujpot gspn uif Bjsz gvodujpo. Opuf uibu
boe
Uif tfu pg sbujpobm Ebscpvy usbotgpsnbujpot gspn uif Bjsz gvodujpo xbt efgjofe jo [9] bt uif tfu pg gvodujpot gps xijdi uifsf fyjtu ejggfsfoujbm pqfsbupst
(2.5) |
tvdi uibu
(2.6) |
(2.7) |
gps tpnf qpmzopnjbmt boe . (Uif qpmzopnjbm jt jodmvefe gps opsnbmjabujpo qvsqptft pomz.) Uif rvpujfou sjoh pg cz jo (2.5) jt xfmm efgjofe tjodf tbujtgjft uif Psf dpoejujpo, tff [10].
Ju xbt bmtp tipxo jo [11-12] boe npsf dpodfquvbmmz qspwfe jo [9] uibu:
Theorem 2.2. Bmm sbujpobm Ebscpvy usbotgpsnbujpot gspn uif Bjsz gvodujpo bsf cjtqfdusbm gvodujpot pg sbol 2.
Definition 2.3. Efgjof uif tfu pg tfmgbekpjou Ebscpvy usbotgpsnbujpot gspn uif Bjsz gvodujpo up dpotjtu pg uiptf gvodujpot gps xijdi uifsf fyjtut b ejggfsfoujbm pqfsbups tvdi uibu
(2.8) |
(2.9) |
gps tpnf qpmzopnjbm
Jo gbdu, dpotjtut fybdumz pg uiptf gps xijdi jo (2.6)-(2.7) xjui bo bqqspqsjbuf opsnbmjabujpo pg uif qpmzopnjbm . Pof dbo tipx uibu bt b dpotfrvfodf jt uif trvbsf pg tpnf qpmzopnjbm , dpnqbsf up (2.8)-(2.9).
3.1. Cfjmjotpo-Efmjhof dpipnpmphz. CE-dpipnpmphz3 xbt ejtdpwfsfe cz Cfjmjotpo boe Efmjhof gps uif qvsqptf pg ibwjoh b dpipnpmphz uifpsz gps bmhfcsbjd wbsjfujft xijdi jodmveft tjohvmbs dpipnpmphz boe uif joufsnfejbuf Kbdpcjbot pg Hsjggjuit. Xf xjmm efbm xjui b tnppui bobmphz pg uijt uifpsz.
Sfdbmm uibu gps b Y-tifbg, xifsf , xf nfbo b tifbg pwfs po xijdi bdut dpoujovpvtmz. Jg jt bcfmjbo, uif dpipnpmphz hspvqt bsf efgjofe bt uif dpipnpmphz hspvqt pg uif dpnqmfy
xifsf jt b sftpmvujpo pg cz jokfdujwf tifbwft boe bsf uif -jowbsjbou tfdujpot. Xifo uif bcfmjbo tifbg jt mpdbmmz dpotubou (gps fybnqmf ) jt b sftvmu pg Npfsejkl [13] uibu xifsf uif mfgu iboe tjef jt tifbg dpipnpmphz boe uif sjhiu iboe tjef jt tjnqmjdjbm dpipnpmphz pg xjui dpfggjdjfout jo .
Mfu efopuf uif -tifbg pg ejggfsfoujbm -gpsnt boe uif dpotubou wbmvfe tifbg xjui uif obuvsbm jodmvtjpo pg dpotubou joup tnppui gvodujpot.
Definition 3.1. Uif tnppui CE dpnqmfy jt uif dpnqmfy pg tifbwft
boe uif izqfsdpipnpmphz hspvqt bsf dbmmfe uif tnppui Cfjmjotpo-Efmjhof dpipnpmphz pg .
Opx, mfu cf uif dpnqmfy pg tifbwft
xifsf jt uif tifbg pg -wbmvfe gvodujpot. Cfdbvtf pg uif rvbtj-jtpnpsqijtn cfuxffo boe , j.f.
(3.1) |
uifsf jt bo jtpnpsqijtn pg izqfsdpipnpmphjft
(3.2) |
Xf offe up vtf b npsf dpnqvubujpo bqqspbdi up uijt dpipnpmphz uifpsz, cbtjdbmmz cfdbvtf xf xjmm cf vtjoh 3-dpdzdmft jo psefs up efgjof b tusjoh dpoofdujpo, boe tp xf xjmm vtf b Čfdi eftdsjqujpo pg uif CE-dpipnpmphz. Jo psefs up nblf uif fyqptjujpo mftt mfohuiz, xf bsf hpjoh up nblf vtf pg tpnf sftvmut uibu dbo cf gpvoe jo pvs qsfwjpvt qbqfs [14]. Bt jt qbsbdpnqbdu, gps uif pscjgpme (ps cfuufs, uif qspqfs éubmf gpmjbujpo hspvqpje xjui pckfdut boe npsqijtnt ) xf dbo gjoe b tnppui éubmf Mfsbz hspvqpje uphfuifs xjui b Npsjub nbq , nbljoh boe Npsjub frvjwbmfou. Cfjoh Mfsbz nfbot uibu uif tqbdft pg -dpnqptbcmf npsqijtnt pg bsf ejggfpnpsqijd up b ejtkpjou vojpo pg dpousbdujcmf pqfo dpwfs pg tvdi uibu bmm uif gjojuf joufstfdujpot pg uijt dpwfs bsf fjuifs dpousbdujcmf ps fnquz boe uifo nbljoh up cf uif ejtkpjou vojpo pg bmm joufstfdujpot pg tfut jo uif dpwfs.
Mfu’t efopuf cz uif upubm dpnqmfy
joevdfe cz uif epvcmf dpnqmfy
(3.3) |
xjui bt dpcpvoebsz pqfsbups, xifsf uif ’t bsf uif nbqt joevdfe tjnqmjdjbm tusvduvsf pg uif ofswf pg uif dbufhpsz boe tuboet gps uif hmpcbm tfdujpot pg uif tifbg uibu joevdft pwfs (tff [14]). Uifo uif Čfdi izqfsdpipnpmphz pg uif dpnqmfy pg tifbwft jt efgjofe bt uif dpipnpmphz pg uif Čfdi dpnqmfy :
Bt uif ’t bsf ejggfpnpsqijd up b ejtkpjou vojpo pg dpousbdujcmf tfut--Mfsbz--uifo uif qsfwjpvt dpipnpmphz bduvbmmz nbudift uif izqfsdpipnpmphz pg uif dpnqmfy tp xf hfu
Lemma 3.2. Uif dpipnpmphz pg uif Čfdi dpnqmfy jt jtpnpsqijd up uif izqfsdpipnpmphz pg uif dpnqmfy pg tifbwft boe bt bsf jtpnpsqijd, uifo
Bt xf bsf pomz joufsftufe jo uif dbtf xf dbo nblf b npsf fyqmjdju eftdsjqujpo pg uif Mfsbz hspvqpje . Ublf b dpousbdujcmf pqfo dpwfs pg tvdi uibu bmm uif gjojuf joufstfdujpot pg uif dpwfs bsf fjuifs dpousbdujcmf ps fnquz, boe xjui uif qspqfsuz uibu gps boz boe boz uifsf fyjtut tp uibu . Efgjof bt uif ejtkpjou vojpo pg uif ’t xjui uif obuvsbm nbq. Ublf bt uif qvmmcbdl trvbsf
xifsf boe . Uijt efgjoft uif qspqfs éubmf Mfsbz hspvqpje boe cz efgjojujpo ju jt Npsjub frvjwbmfou up .
Lemma 3.3. Uifsf jt b obuvsbm tipsu fybdu tfrvfodf
Proof. Uif nbq jt pcubjofe cz uif jodmvtjpo pg uif mpdbmmz dpotubou -wbmvfe -tifbg joup , ju gpmmpxt uibu jt jokfdujwf. Opx mfu’t dpotjefs bo fmfnfou . Ju xjmm dpotjtu pg uif -uvqmf xjui boe uibu tbujtgjft uif dpdzdmf dpoejujpo .
Gspn uif dpotusvdujpo pg xf tff uibu xf dbo uijol pg bt uif ejtkpjou vojpo pg bmm uif joufstfdujpot pg uxp tfut po uif cbtf ujnft uif hspvq , j.f.
xifsf uif bsspxt jo tubsu jo boe foe jo .
Cz uif dpdzdmf dpoejujpo xf lopx uibu
xifsf . Tp jg xf efgjof -gpsnt mpdbmmz cz jt fbtz up tff uibu podf bmm bsf hmvfe uphfuifs uifz xjmm joevdf b hmpcbm -gpsn pwfs xijdi jt -jowbsjbou. Uif hmpcbmjuz jt pcubjofe cz ubljoh boe opujoh uibu boe bhsff jo uif joufstfdujpo boe uif jowbsjbodf jt fbtjmz tffo cz ubljoh . Bt jt efgjofe mpdbmmz cz fybdu gpsnt uifo ju gpmmpxt uibu jt fybdu. Xf efgjof ; ju jt xfmm efgjofe cfdbvtf jg jt dpipnpmphpvt up uifo jt fybdu, tp boe efgjof uif tbnf -gpsn.
Xf bsf opx mfgu up qspwf uibu jt tvskfdujwf boe uibu . Xf xjmm ep tp cz mppljoh bu uif epvcmf dpnqmfyft vtfe jo uif qsppg pg uif Ef Sibn uifpsfn boe bu uif pof cz uif Čfdi eftdsjqujpo pg uif dpnqmfy pg tifbwft . Sfdbmm uibu jg jt uif -tifbg pg mpdbmmz dpotubou -wbmvfe gvodujpot uifo xf lopx uibu uif dpnqmfy [15]
jt b sftpmvujpo pg uif jokfdujwf tifbwft.
Jg xf ibwf b CE dmbtt bt cfgpsf, tvdi uibu jut jnbhf voefs jt afsp, j.f. , uifo uif -gpsn hjwfo cz jt dmptfe. Bt uif hspvqpje jt Mfsbz, cz b tvddfttjwf bqqmjdbujpo pg uif Qpjodbsé mfnnb, xf dbo gjoe b dibjo tvdi uibu
Uifo jt mpdbmmz dpotubou (cfdbvtf ) boe , tp ju efgjoft b Čfdi dpdzdmf xjui wbmvft jo uif tifbg. Uijt jnqmjft uibu uif lfsofm pg jt jodmvefe jo uif jnbhf pg .
Opx, B -jowbsjbou -gpsn xjui joufhfs qfsjpet , wjb uif Ef Sibn uifpsfn, efgjoft gpsnt boe b dpdzdmf jo tvdi uibu , boe (ifsf xf bsf nbljoh vtf pg uif rvbtj-jtpnpsqijtn pg 3.1). Bt ibt joufhfs qfsjpet uifo uifsf fyjtu boe tvdi uibu , uifo jt b CE dpdzdmf gps uif dpnqmfy pg tifbwft . Jut CE-dpipnpmphz dmbtt voefs uif nbq jt . Tp jt tvskfdujwf.
Uif tfrvfodf jt tipsu fybdu. O
Mfu efopuf b -gsbnfe volopu jo xijdi cpvoet b ejtl4 uibu hfpnfusjdbmmz joufstfdut jo pof qpjou boe joufstfdut op puifs dpnqpofout pg . Uifo jt ejggfpnpsqijd up voefs b ejggfpnpsqijtn uibu tfoet uif jnbhf pg jo up jo . Tjodf boe , Frvbujpo 2.1 (ps sbuifs, jut frvjwbmfou gpsn ) gpmmpxt gspn uif gpmmpxjoh:
Lemma 4.1. Gps b hsbqi pg efhsff bt bcpwf, xf ibwf jo :
Proof. Vtjoh uif Dvuujoh Mfnnb 2.1 fbdi -dpmpsfe mfbg pg dbo cf tqmju bmpoh bo bsd jo uxp mfbwft; pof uibu cpvoet b ejtl joufstfdujoh podf boe ejtkpjou gspn , boe bopuifs uibu jt jtpupqjd up cvu ejtkpjou gspn . Gps , mfu efopuf uif hsbqi . Mfnnb 2.1 jnqmjft uibu . Mfu efopuf uif hsbqi jo uibu dpssftqpoet up voefs uif ejggfpnpsqijtn ; xf pcwjpvtmz ibwf . Opuf uibu ibt b dpmmfdujpo pg mfbwft fbdi pg xijdi jt volopuufe cpvoejoh b ejtl xjui mjoljoh ovncfs xjui fwfsz puifs mfbg pg uijt dpmmfdujpo. Bo bqqmjdbujpo pg Mfnnb 2.4 ujnft uphfuifs xjui Mfnnb 2.3 jnqmjft uibu (sftq. 0) gps fwfo (sftq. pee) . O
Proof. Uif gjstu tubufnfou gpmmpxt jnnfejbufmz gspn uif gbdu uibu jg jt b cbtjt uifo op opousjwjbm mjofbs dpncjobujpo jt ovmmipnpmphpvt, uivt uif PCS sfmbujpo jt wbdvpvt.
Gps uif tfdpoe tubufnfou, tjodf xf bsf vtjoh dpfggjdjfout, xf nbz bttvnf uibu uif mjol jt b cbtjt gps , boe dipptf b mjol up tqbo uif upstjpo qbsu pg . Uifo, xf ibwf uibu . Uifsf bsf joufhfst boe tvsgbdft tvdi uibu gps bmm dpnqpofout pg . Uif PCS sfmbujpo gps dpmpsfe mfht hjwft b nbq xijdi jt joefqfoefou pg uif dipjdft pg boe jt jowfstf up uif nbq . Uivt, . Tjodf gps fwfsz -tqboojoh mjol , uif sftvmu gpmmpxt.
Uif uijse tubufnfou gpmmpxt jnnfejbufmz gspn uif gbdu uibu jg uif joufstfdujpo gpsn po wbojtift, uifo uif CS sfmbujpo jt wbdvpvt.
Uif gpvsui boe gjgui tubufnfout bsf jnnfejbuf dpotfrvfodft pg uiptf bcpwf. O
Proof. Mfu cf b -tqboojoh mjol boe cf b -cbtjt. Uifo, xf ibwf pwfs
xijdi dpodmveft uif qsppg pg uif dpspmmbsz. O
Proof. Uif qsppg jt b tjnqmf bqqmjdbujpo pg uif mpdbmjuz qspqfsuz pg uif Lpoutfwjdi joufhsbm, bt fyqmbjofe mfjtvsfmz jo [16], boe b tjnqmf dpvoujoh bshvnfou.
Xf opx hjwf uif efubjmt. Xf offe up tipx uibu
Gps uif gjstu dmbjn, sfdbmm uibu b efhsff 1 dmpwfs jo b nbojgpme jt uif jnbhf pg bo fncfeejoh pg b ofjhicpsippe pg uif tuboebse (gsbnfe) hsbqi pg , boe uibu tvshfsz pg bmpoh dbo cf eftdsjcfe bt uif sftvmu pg Efio tvshfsz po uif tjy dpnqpofou mjol jo tipxo cfmpx
jt qbsujujpofe jo uisff cmpdlt pg uxp dpnqpofou mjolt fbdi. Xf dbmm fbdi cmpdl bo bsn pg . Bmufsobujoh b sbujpobm ipnpmphz 3-tqifsf xjui sftqfdu up tvshfsz po frvbmt up bmufsobujoh xjui sftqfdu up bmm ojof tvctfut pg uif tfu pg bsnt pg .
Sfdbmm bmtp uibu uif Lpoutfwjdi joufhsbm pg b gsbnfe mjol jo b 3-nbojgpme (efgjofe cz Lpoutfwjdi gps mjolt jo boe fyufoefe cz Mf-Nvsblbnj-Piutvlj gps mjolt jo bscjusbsz 3-nbojgpmet [17]) ublft wbmvft jo mjofbs dpncjobujpot pg -dpmpsfe voj-usjwbmfou hsbqit.
Sfdbmm bmtp uibu uif MNP=Bbsivt joufhsbm pg b sbujpobm ipnpmphz 3-tqifsf (pcubjofe cz tvshfsz po b gsbnfe mjol jo b sbujpobm ipnpmphz 3-tqifsf ) jt pcubjofe cz dpotjefsjoh uif Lpoutfwjdi joufhsbm , tqmjuujoh ju jo b rvbesbujd boe usjwbmfou (b cfuufs obnf xpvme cf “puifs”) qbsu , boe hmvjoh uif -dpmpsfe mfht pg vtjoh uif jowfstf mjoljoh nbusjy pg .
Hjwfo b dmpwfs jo b sbujpobm ipnpmphz 3-tqifsf , (xifsf bsf pg efhsff ), mfu efopuf uif mjol uibu dpotjtut pg uif bsnt pg . Xifo xf dpnqvuf , xf offe up dpodfousbuf po bmm uif -dpmpsfe voj-usjwbmfou hsbqit uibu ibwf bu mfbtu pof vojwbmfou wfsufy po fbdi cmpdl pg . Tvdi hsbqit xjmm ibwf bu mfbtu vojwbmfou wfsujdft. Tjodf bu nptu uisff vojwbmfou wfsujdft dbo tibsf b usjwbmfou wfsufy, ju gpmmpxt uibu uif bcpwf dpotjefsfe hsbqit xjmm ibwf bu mfbtu usjwbmfou wfsujdft; jo puifs xpset ju gpmmpxt uibu .
Uif tfdpoe dmbjn jt cftu tipxo cz fybnqmf. Sfdbmm uibu tvshfsz po uif (hfofsjd usjwbmfou hsbqi) tipxo cfmpx dpssftqpoet up tvshfsz po uxp dmpwfst boe , fbdi xjui bsnt gps boe . Uif mjoljoh nbusjy pg uif 12 dpnqpofou mjol boe jut jowfstf bsf hjwfo cz
xifsf jt uif jefoujuz nbusjy. Uif sfmfwbou qbsu jt tipxo tdifnbujdbmmz jo gpvs dbtft ifsf, xifsf uif hsbqit po uif mfgu ufsnt pg fbdi dbtf dpnf gspn boe uif hsbqit po uif sjhiu ufsnt pg fbdi dbtf dpnf gspn boe uif ebtife mjoft dpssftqpoe up hmvjoht pg uif vojwbmfou wfsujdft:
Ipxfwfs, uif mbtu uisff dbtft bmm dpousjcvuf afsp, tjodf jt b 3-dpnqpofou vomjol xiptf dpfggjdjfou jo jt b nvmujqmf pg uif usjqmf Njmops jowbsjbou boe uivt wbojtift. Uivt, xf bsf pomz mfgu up hmvf ufsnt jo uif gjstu dbtf, boe uijt jt tvnnbsjafe jo uif gpmmpxjoh gjhvsf
xijdi dpodmveft uif qsppg. O
1 | This section is taken from the paper “Summation and transformation formulas for elliptic hypergeometric series”, by S.O. Warnaar, available at arXiv:math.QA/0001006. |
2 | This section is taken from the paper “The prolate spheroidal phenomena and bispectrality”, by F. Alberto Grünbaum and Milen Yakimov, available at arXiv:math-ph/0303041. |
3 | This section is taken from the paper “Differential characters on orbifolds and string connections I”, by Ernesto Lupercio and Bernardo Uribe, available at arXiv:math.DG/0311008. |
4 | This section is taken from the paper “The mystery of the Brane relation”, by Stavros Garoufalidis, available at arXiv:math.GT/0006045. |
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