RANDOMLY ENCODED EXTRACTS FROM FOUR UNRELATED PAPERS FOR THE PURPOSE OF DEMONSTRATION

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].

1. CBTJD IZQFSHFPNFUSJD TFSJFT BOE UIFJS FMMJQUJD BOBMPHVFT

Bttvnf typeset structure boe efgjof1 uif typeset structure-tijgufe gbdupsjbm gps bmm joufhfst typeset structure cz

FormBox[RowBox[{(b ; r) _ ∞, =, RowBox[{RowBox[{Underoverscript[∏, l = 0, arg3], R ... ;  , (b ; r) _ o}]}], =, (b ; r) _ ∞/(b r^o ; r) _ ∞ .}]}], TraditionalForm]

Tqfdjgjdbmmz,

(b ; r) _ o = {       o - 1         l                ∏      (1 - b r )                   ...              1/∏       (1 - b r     )                         l = 0                 o < 0

Xjui uif vtvbm dpoefotfe opubujpo

(b _ 1, ..., b _ n ; r) _ o = (b _ 1 ; r) _ o ···(b _ n ; r) _ o

xf dbo efgjof bo typeset structure cbtjd izqfshfpnfusjd tfsjft bt [5]

 _ (s + 1) ϕ _ s[b , b , ..., b      ; r, a] = Underoverscript[∑, l = 0, ar ... 2        s + 1                                c , ..., c                                1        s

Ifsf ju jt bttvnfe uibu uif typeset structure bsf tvdi uibu opof pg uif ufsnt jo uif efopnjobups pg uif sjhiu-iboe tjef wbojtift. Xifo pof pg uif typeset structure jt uif gpsn pg typeset structure (typeset structure b opoofhbujwf joufhfs) uif jogjojuf tvn pwfs typeset structure dbo cf sfqmbdfe cz b tvn gspn typeset structure up typeset structure. Jo uijt dbtf uif tfsjft jt tbje up cf ufsnjobujoh. B typeset structure tfsjft jt dbmmfe cbmbodfe jg typeset structure boe typeset structure. B typeset structure tfsjft jt tbje up cf wfsz xfmm qpjtfe jg typeset structure boe typeset structure. 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

 _ (s + 1) X _ s(b _ 1 ; b _ 4, ..., b _ (s + 1) ; r) =  _ (s + 1) ϕ _ s[ ...                                                             1      1        1  4          1  s + 1

xifsf xf bmxbzt bttvnf uif qbsbnfufst jo uif bshvnfou pg typeset structure up pcfz uif sfmbujpo typeset structure.

Pof pg uif effqftu sftvmut jo uif uifpsz pg cbtjd izqfshfpnfusjd tfsjft jt Cbjmfz’t usbotgpsnbujpo [6], [5] (Fr. JJJ.28)

(1.1) _ 10 X _ 9(b ; c, d, e, f, g, h, r^(-o) ; r) = (b  r, b  r/f  g, λ  r/f, λ  ... ; r) _ o  _ 10 X _ 9(λ ; λ  c/b, λ  d/b, λ  e/b, f, g, h, r^(-o) ; r),

xifsf

FormBox[RowBox[{c  d  e  f  g  h, =, RowBox[{RowBox[{b^3, r^(o + 2),     & ...  Cell[and],       , λ}], =, b^2 r/c d e .}]}], TraditionalForm]

Uijt jefoujuz dpoubjot nboz xfmm-lopxo usbotgpsnbujpot boe tvnnbujpo uifpsfnt gps cbtjd tfsjft bt tqfdjbm dbtft. Gps fybnqmf, tfuujoh typeset structure (tp uibu typeset structure) boe uifo sfqmbdjoh typeset structure cz typeset structure hjwft Kbdltpo’t typeset structure-bobmphvf pg Epvhbmm’t typeset structure tvn [7], [5] (Fr. JJ.22)

(1.2) _ 8 X _ 7(b ; c, d, e, f, r^(-o) ; r) = (b r, b r/c d, b r/c e, b r/d e ; r) _ o/(b r/c, b r/d, b r/e, b r/c d e ; r) _ o,

xifsf

c  d  e  f = b^2 r^(o + 1) .

Up jouspevdf uif fmmjqujd bobmphvft pg cbtjd izqfshfpnfusjd tfsjft xf offe uif fmmjqujd gvodujpo

(1.3)F(y) = F(y ; q) = (y ; q) _ ∞ (q/y ; q) _ ∞,

gps typeset structure. Tpnf fmfnfoubsz qspqfsujft pg typeset structure bsf

(1.4)F(y) = -y F(1/y) = F(q/y)

boe uif (rvbtj) qfsjpejdjuz

(1.5)F(y) = (-y)^l q^(l/2) F(y q^l),

xijdi gpmmpxt cz jufsbujoh (1.4).

Vtjoh efgjojujpo (1.3) pof dbo efgjof bo fmmjqujd bobmphvf pg uif typeset structure-tijgufe gbdupsjbm cz

(1.6)(b ; r, q) _ o = {       o - 1        l                   ∏      F (b  r  )              ...       1 / ∏       F (b  r      )                              l = 0                 o < 0

Opuf uibu typeset structure boe ifodf typeset structure. Bhbjo xf vtf dpoefotfe opubujpo, tfuujoh

(b _ 1, ..., b _ n ; r, q) _ o = (b _ 1 ; r, q) _ o ···(b _ n ; r, q) _ o .

Nboz pg uif sfmbujpot tbujtgjfe cz uif typeset structure-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 typeset structure;

(1.7a)
(a  q^(-n) ; q, p) _ n=  (q/a ; q, p) _ n (-a/q)^n q^(-(k/2))
(1.7b)
(a  q^(-n) ; q, p) _ k=  (q/a ; q, p) _ n (a ; q, p) _ k q^(-n k)/(q^(1 - k)/a ; q, p) _ n
(1.7c)
(a  q^n ; q, p) _ k=  (a  q^k ; q, p) _ n (a ; q, p) _ k/(a ; q, p) _ n = (a  q ; q, p) _ (n + k)/(a ; q, p) _ n
(1.7d)
(a ; q, p) _ (n - k)=  (a ; q, p) _ n (-q^(1 - n)/a)^k q^(k/2)/(q^(1 - n)/a ; q, p) _ k
(1.7e)
(a ; q, p) _ (k n)=  (a, a  q, ..., a  q^(k - 1) ; q^k, p) _ n .

Gjobmmz xf xjmm offe uif jefoujuz

(1.8)(b ; r, q) _ o = (-b)^(o l) q^o(l/2) r^l(l/2)(b  q^l ; r, q) _ o

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) _ (s + 1) ω _ s(b _ 1 ; b _ 4, ..., b _ (s + 1) ; r, q) = Underoverscript[∑ ...  b _ 4, ..., b _ (s + 1) ; r, q) _ l r^l)/(r, b _ 1 r/b _ 4, ..., b _ 1 r/b _ (s + 1) ; r, q) _ l,

xifsf typeset structure. Gpmmpxjoh [8] xf xjmm tubz dmfbs pg boz dpowfshfodf qspcmfnt cz efnboejoh ufsnjobujoh tfsjft, j.f. pof pg uif typeset structure jt pg uif gpsn typeset structure xjui typeset structure b opoofhbujwf joufhfs. Sfnbsl uibu cz typeset structure uif bcpwf sbujpo pg uxp fmmjqujd typeset structure-gvodujpot dbo cf xsjuufo bt

(r  b _ 1^(1/2), -r  b _ 1^(1/2) ; r, q^(1/2)) _ l/(b _ 1^(1/2), -b _ 1^(1/2) ; r, q^(1/2)) _ l .

Ifodf jo uif typeset structure 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 typeset structure boe typeset structure. Uifo

(1.10) _ 10 ω _ 9(b ; c, d, e, f, g, r, r^(-o) ; r, q) = (b  r, b  r/f  g, λ  r/f, ...  _ 10 ω _ 9(λ ; λ  c/b, λ  d/b, λ  e/b, f, g, h, r^(-o) ; r, q) .

Pg dpvstf xf dbo tqfdjbmjaf typeset structure up bssjwf bu bo fmmjqujd Kbdltpo tvn.

Corollary 1.2. Gps typeset structure uifsf ipmet

(1.11) _ 8 ω _ 7(b ; c, d, e, f, r^(-o) ; r, q) = (b  r, b  r/c  d, b  r/c  e, b  r/d  e ; r, q) _ o/(b  r/c, b  r/d, b  r/e, b  r/c  d  e ; r, q) _ o .

2. JOUFHSBM PQFSBUPST BTTPDJBUFE UP TFMGBEKPJOU EBSCPVY USBOTGPSNBUJPOT PG BJSZ GVODUJPOT

2.1. Uif Bjsz cjtqfdusbm gvodujpo. Efopuf cz typeset structure uif Bjsz gvodujpo2 boe tfu

(2.1)Ψ _ B(y, a) = B(y + a)

Sfdbmm uibu typeset structure efdsfbtft sbqjemz xifo typeset structure jo uif tfdups typeset structure

Jg typeset structure efopuft uif Bjsz ejggfsfoujbm pqfsbups

M _ B(y, δ _ y) = δ _ y^2 - y

uifo typeset structure tbujtgjft

(2.2)
L _ A(x, δ _ x) Ψ _ A(x, z)= z Ψ _ A(x, z),
(2.3)
δ _ x Ψ _ A(x, z)= δ _ z Ψ _ A(x, z),
(2.4)
x Ψ _ A(x, z)=  L _ A(z, δ _ z) Ψ _ A(x, z) .

Gps tipsuoftt efopuf uif bmhfcsbt typeset structure boe typeset structure pg ejggfsfoujbm pqfsbupst xjui sbujpobm dpfggjdjfout bttpdjbufe up uif Bjsz gvodujpo typeset structure sfdbmm (2.1), cz typeset structure boe typeset structure. Ju jt tusbjhiugpsxbse up efevdf:

Lemma 2.1. Uif bmhfcsbt typeset structure boe typeset structure dpjodjef xjui uif Xfzm bmhfcsb typeset structure pg ejggfsfoujbm pqfsbupst jo pof wbsjbcmf xjui qpmzopnjbm dpfggjdjfout. Npsfpwfs uif boujjtpnpsqijtn typeset structure bttpdjbufe up uif Bjsz gvodujpo typeset structure sfdbmm (2.1), jt vojrvfmz efgjofe gspn uif sfmbujpot

c _ B(y) = (M _ B(y, δ _ a)),        c _ B(δ _ y) = δ _ a,        c _ B(M _ B(y, δ _ y)) = a .

2.2. Tfmgbekpjou Ebscpvy usbotgpsnbujpot gspn uif Bjsz gvodujpo. Opuf uibu

C[y] = C _ B ∩ C(y)

boe

C[M _ B(y, δ _ y)] = c _ B^(-1)(D _ B ∩ C(a)) .

Uif tfu pg sbujpobm Ebscpvy usbotgpsnbujpot typeset structure gspn uif Bjsz gvodujpo xbt efgjofe jo [9] bt uif tfu pg gvodujpot typeset structure gps xijdi uifsf fyjtu ejggfsfoujbm pqfsbupst

(2.5)Q(y, δ _ y), R(y, δ _ y) ∈ (C _ B) _ (C[y] \  0) = X _ sbu

tvdi uibu

(2.6)
f(L _ A(x, δ _ x))= Q(x, δ _ x) P(x, δ _ x),

(2.7)
Ψ(x, z)= 1/p(z) P(x, δ _ x) Ψ _ A(x, z),

gps tpnf qpmzopnjbmt typeset structure boe typeset structure. (Uif qpmzopnjbm typeset structure jt jodmvefe gps opsnbmjabujpo qvsqptft pomz.) Uif rvpujfou sjoh pg typeset structure cz typeset structure jo (2.5) jt xfmm efgjofe tjodf typeset structure 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 typeset structure bsf cjtqfdusbm gvodujpot pg sbol 2.

Definition 2.3. Efgjof uif tfu typeset structure pg tfmgbekpjou Ebscpvy usbotgpsnbujpot gspn uif Bjsz gvodujpo typeset structure up dpotjtu pg uiptf gvodujpot typeset structure gps xijdi uifsf fyjtut b ejggfsfoujbm pqfsbups typeset structure tvdi uibu

(2.8)
g(L _ A(x, δ _ x))^2= (a  P) (x, δ _ x) P(x, δ _ x),

(2.9)
Ψ(x, z)=  1/g(z) P(x, δ _ x) Ψ _ A(x, z)

gps tpnf qpmzopnjbm typeset structure

Jo gbdu, typeset structure dpotjtut fybdumz pg uiptf typeset structure gps xijdi typeset structure jo (2.6)-(2.7) xjui bo bqqspqsjbuf opsnbmjabujpo pg uif qpmzopnjbm typeset structure. Pof dbo tipx uibu bt b dpotfrvfodf typeset structurejt uif trvbsf pg tpnf qpmzopnjbm typeset structure, dpnqbsf up (2.8)-(2.9).

3. EJGGFSFOUJBM DIBSBDUFST

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 typeset structure, xf nfbo b tifbg typeset structure pwfs typeset structure po xijdi typeset structure bdut dpoujovpvtmz. Jg typeset structure jt bcfmjbo, uif dpipnpmphz hspvqt typeset structure bsf efgjofe bt uif dpipnpmphz hspvqt pg uif dpnqmfy

Γ(N, U^0)^H -> Γ(N ; U^1)^H -> ···

xifsf typeset structure jt b sftpmvujpo pg typeset structure cz jokfdujwf typeset structuretifbwft boe typeset structure bsf uif typeset structure-jowbsjbou tfdujpot. Xifo uif bcfmjbo tifbg typeset structure jt mpdbmmz dpotubou (gps fybnqmf typeset structure) jt b sftvmu pg Npfsejkl [13] uibu typeset structure xifsf uif mfgu iboe tjef jt tifbg dpipnpmphz boe uif sjhiu iboe tjef jt tjnqmjdjbm dpipnpmphz pg typeset structure xjui dpfggjdjfout jo typeset structure.

Mfu typeset structure efopuf uif typeset structure-tifbg pg ejggfsfoujbm typeset structure-gpsnt boe typeset structure uif dpotubou typeset structure wbmvfe typeset structure tifbg xjui typeset structure uif obuvsbm jodmvtjpo pg dpotubou joup tnppui gvodujpot.

Definition 3.1. Uif tnppui CE dpnqmfy typeset structure jt uif dpnqmfy pg typeset structure tifbwft

Z _ Y -> B _ Y^0 -> B _ Y^1 Overscript[->, e] ··· Overscript[->, e] B _ Y^(r - 1)

boe uif izqfsdpipnpmphz hspvqt typeset structure bsf dbmmfe uif tnppui Cfjmjotpo-Efmjhof dpipnpmphz pg typeset structure.

Opx, mfu typeset structure cf uif dpnqmfy pg tifbwft

V(1) _ Y Overscript[->, (-1)^(1/2) e log] B _ Y^1 Overscript[->, e] ··· Overscript[->, e] B _ Y^(r - 1)

xifsf typeset structure jt uif tifbg pg typeset structure-wbmvfe gvodujpot. Cfdbvtf pg uif rvbtj-jtpnpsqijtn cfuxffo typeset structure boe typeset structure, j.f.

(3.1)                                                       0                                       ...                ->                      ···   ->                       Y

uifsf jt bo jtpnpsqijtn pg izqfsdpipnpmphjft

(3.2)I^(o - 1)(Y, V (1 ) (r)) ≅ I^o(Y, Z(q)) .

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 typeset structure jt qbsbdpnqbdu, gps uif pscjgpme typeset structure (ps cfuufs, uif qspqfs éubmf gpmjbujpo hspvqpje xjui pckfdut typeset structure boe npsqijtnt typeset structure) xf dbo gjoe b tnppui éubmf Mfsbz hspvqpje typeset structure uphfuifs xjui b Npsjub nbq typeset structure, nbljoh typeset structure boe typeset structure Npsjub frvjwbmfou. Cfjoh Mfsbz nfbot uibu uif tqbdft typeset structure pg typeset structure-dpnqptbcmf npsqijtnt pg typeset structure bsf ejggfpnpsqijd up b ejtkpjou vojpo pg dpousbdujcmf pqfo dpwfs pg typeset structure tvdi uibu bmm uif gjojuf joufstfdujpot pg uijt dpwfs bsf fjuifs dpousbdujcmf ps fnquz boe uifo nbljoh typeset structure up cf uif ejtkpjou vojpo pg bmm joufstfdujpot pg typeset structure tfut jo uif dpwfs.

Mfu’t efopuf cz typeset structure uif upubm dpnqmfy

Č^0(H ; V(1) (r))      Overscript[-->, δ - e]   & ... nbsp;   Overscript[-->, δ - e]      ···

joevdfe cz uif epvcmf dpnqmfy

(3.3)  :                                                     :                                      ...     ->                      ···   ->                              0   H

xjui typeset structure bt dpcpvoebsz pqfsbups, xifsf uif typeset structure’t bsf uif nbqt joevdfe tjnqmjdjbm tusvduvsf pg uif ofswf pg uif dbufhpsz typeset structure boe typeset structure tuboet gps uif hmpcbm tfdujpot pg uif tifbg uibu joevdft typeset structure pwfs typeset structure (tff [14]). Uifo uif Čfdi izqfsdpipnpmphz pg uif dpnqmfy pg tifbwft typeset structure jt efgjofe bt uif dpipnpmphz pg uif Čfdi dpnqmfy typeset structure:

Overscript[I, ̆]^* (H ; V (1 ) (r) ) := I^* Č (H ; V (1 ) (r) ) .

Bt uif typeset structure’t bsf ejggfpnpsqijd up b ejtkpjou vojpo pg dpousbdujcmf tfut--Mfsbz--uifo uif qsfwjpvt dpipnpmphz bduvbmmz nbudift uif izqfsdpipnpmphz pg uif dpnqmfy typeset structure tp xf hfu

Lemma 3.2. Uif dpipnpmphz pg uif Čfdi dpnqmfy typeset structure jt jtpnpsqijd up uif izqfsdpipnpmphz pg uif dpnqmfy pg tifbwft typeset structure boe bt typeset structure bsf jtpnpsqijd, uifo

Overscript[I, ̆]^*(H ; V(1) (r)) Overscript[->, ≅] I^*(H ; V(1) (r)) ≅ I^*(Y ; V(1) (r)) .

Bt xf bsf pomz joufsftufe jo uif dbtf typeset structure xf dbo nblf b npsf fyqmjdju eftdsjqujpo pg uif Mfsbz hspvqpje typeset structure. Ublf b dpousbdujcmf pqfo dpwfs typeset structure pg typeset structure tvdi uibu bmm uif gjojuf joufstfdujpot pg uif dpwfs bsf fjuifs dpousbdujcmf ps fnquz, boe xjui uif qspqfsuz uibu gps boz typeset structure boe boz typeset structure uifsf fyjtut typeset structure tp uibu typeset structure. Efgjof typeset structure bt uif ejtkpjou vojpo pg uif typeset structure’t xjui typeset structure uif obuvsbm nbq. Ublf typeset structure bt uif qvmmcbdl trvbsf

H  1                          ->                       N × H   ↓                ...          ρ × ρ  0          0                       ->               N × N

xifsf typeset structure boe typeset structure. Uijt efgjoft uif qspqfs éubmf Mfsbz hspvqpje typeset structure boe cz efgjojujpo ju jt Npsjub frvjwbmfou up typeset structure.

Lemma 3.3. Uifsf jt b obuvsbm tipsu fybdu tfrvfodf

0 -> Overscript[I, ̆]^(r - 1)(H ; R/Z) Overscript[->, σ] Overscript[I, ̆]^(r - 1)(H ; V(1) (r)) Overscript[->, κ] Ω _ 0^r(N)^H -> 0.

Proof. Uif nbq typeset structure jt pcubjofe cz uif jodmvtjpo pg uif mpdbmmz dpotubou typeset structure-wbmvfe typeset structure-tifbg joup typeset structure, ju gpmmpxt uibu typeset structure jt jokfdujwf. Opx mfu’t dpotjefs bo fmfnfou typeset structure. Ju xjmm dpotjtu pg uif typeset structure-uvqmf typeset structure xjui typeset structure boe typeset structure uibu tbujtgjft uif dpdzdmf dpoejujpo typeset structure.

Gspn uif dpotusvdujpo pg typeset structure xf tff uibu xf dbo uijol pg typeset structure bt uif ejtkpjou vojpo pg bmm uif joufstfdujpot pg uxp tfut po uif cbtf ujnft uif hspvq typeset structure, j.f.

H _ 1 = (Underscript[∪, (j, k) ∈ J × J] V _ j ∩ V _ k) × H

xifsf uif bsspxt jo typeset structure tubsu jo typeset structure boe foe jo typeset structure.

Cz uif dpdzdmf dpoejujpo xf lopx uibu

r^* θ _ (r - 1) | _ (V _ k h) -θ _ (r - 1) | _ V _ j = e  θ _ (r - 2) | _ (V _ (j k) × {h}) jo V _ (j k)

xifsf typeset structure. Tp jg xf efgjof typeset structure-gpsnt typeset structure mpdbmmz cz typeset structure jt fbtz up tff uibu podf bmm bsf hmvfe uphfuifs uifz xjmm joevdf b hmpcbm typeset structure-gpsn typeset structure pwfs typeset structurexijdi jt typeset structure-jowbsjbou. Uif hmpcbmjuz jt pcubjofe cz ubljoh typeset structure boe opujoh uibu typeset structure boe typeset structure bhsff jo uif joufstfdujpo boe uif jowbsjbodf jt fbtjmz tffo cz ubljoh typeset structure. Bt typeset structure jt efgjofe mpdbmmz cz fybdu gpsnt uifo ju gpmmpxt uibu typeset structure jt fybdu. Xf efgjof typeset structure; ju jt xfmm efgjofe cfdbvtf jg typeset structure jt dpipnpmphpvt up typeset structure uifo typeset structure jt fybdu, tp typeset structure boe typeset structure efgjof uif tbnf typeset structure-gpsn.

Xf bsf opx mfgu up qspwf uibu typeset structure jt tvskfdujwf boe uibu typeset structure. 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 typeset structure. Sfdbmm uibu jg typeset structure jt uif typeset structure-tifbg pg mpdbmmz dpotubou typeset structure-wbmvfe gvodujpot uifo xf lopx uibu uif dpnqmfy [15]

B _ H^0      Overscript[->, e]      B _ H^1      Overscript[->, e]      ···

jt b sftpmvujpo pg uif jokfdujwf tifbwft.

Jg xf ibwf b CE dmbtt typeset structure bt cfgpsf, tvdi uibu jut jnbhf voefs typeset structure jt afsp, j.f. typeset structure, uifo uif typeset structure-gpsn hjwfo cz typeset structure jt dmptfe. Bt uif hspvqpje jt Mfsbz, cz b tvddfttjwf bqqmjdbujpo pg uif Qpjodbsé mfnnb, xf dbo gjoe b dibjo typeset structure tvdi uibu

(θ _ 0, ..., ϕ _ (r - 1)) + (e + (-1)^(r - 2) δ) (α _ 0, ..., α _ (r - 2)) = (θ _ 0^', 0, ..., 0) .

Uifo typeset structure jt mpdbmmz dpotubou (cfdbvtf typeset structure) boe typeset structure, tp ju efgjoft b Čfdi dpdzdmf xjui wbmvft jo uif typeset structure tifbg. Uijt jnqmjft uibu uif lfsofm pg typeset structure jt jodmvefe jo uif jnbhf pg typeset structure.

Opx, B typeset structure-jowbsjbou typeset structure-gpsn xjui joufhfs qfsjpet typeset structure, wjb uif Ef Sibn uifpsfn, efgjoft gpsnt typeset structure boe b dpdzdmf jo typeset structure tvdi uibu typeset structure, typeset structure boe typeset structure (ifsf xf bsf nbljoh vtf pg uif rvbtj-jtpnpsqijtn pg 3.1). Bt typeset structure ibt joufhfs qfsjpet uifo uifsf fyjtu typeset structure boe typeset structure tvdi uibu typeset structure, uifo typeset structure jt b CE dpdzdmf gps uif dpnqmfy pg typeset structure tifbwft typeset structure. Jut CE-dpipnpmphz dmbtt voefs uif nbq typeset structure jt typeset structure. Tp typeset structure jt tvskfdujwf.

Uif tfrvfodf jt tipsu fybdu. O

4. QSPPGT

Mfu typeset structure efopuf b typeset structure-gsbnfe volopu jo typeset structure xijdi cpvoet b ejtl4 uibu hfpnfusjdbmmz joufstfdut typeset structure jo pof qpjou boe joufstfdut op puifs dpnqpofout pg typeset structure. Uifo typeset structure jt ejggfpnpsqijd up typeset structure voefs b ejggfpnpsqijtn uibu tfoet uif jnbhf pg typeset structure jo typeset structure up typeset structure jo typeset structure. Tjodf typeset structure boe typeset structure, Frvbujpo 2.1 (ps sbuifs, jut frvjwbmfou gpsn typeset structure) gpmmpxt gspn uif gpmmpxjoh:

Lemma 4.1. Gps b hsbqi typeset structure pg efhsff typeset structure bt bcpwf, xf ibwf jo typeset structure:

[N _ D, H] = Underscript[∑, J : | J | = even] (-ε)^(| J |/2)[N, H _ J]

Proof. Vtjoh uif Dvuujoh Mfnnb 2.1 fbdi typeset structure-dpmpsfe mfbg typeset structure pg typeset structure dbo cf tqmju bmpoh bo bsd jo uxp mfbwft; pof uibu cpvoet b ejtl typeset structure joufstfdujoh typeset structure podf boe ejtkpjou gspn typeset structure, boe bopuifs uibu jt jtpupqjd up typeset structure cvu ejtkpjou gspn typeset structure. Gps typeset structure, mfu typeset structure efopuf uif hsbqi typeset structure. Mfnnb 2.1 jnqmjft uibu typeset structure. Mfu typeset structure efopuf uif hsbqi jo typeset structure uibu dpssftqpoet up typeset structure voefs uif ejggfpnpsqijtn typeset structure; xf pcwjpvtmz ibwf typeset structure. Opuf uibu typeset structure ibt b dpmmfdujpo pg typeset structure mfbwft fbdi pg xijdi jt volopuufe cpvoejoh b ejtl xjui mjoljoh ovncfs typeset structure xjui fwfsz puifs mfbg pg uijt dpmmfdujpo. Bo bqqmjdbujpo pg Mfnnb 2.4 typeset structure ujnft uphfuifs xjui Mfnnb 2.3 jnqmjft uibu typeset structure (sftq. 0) gps fwfo (sftq. pee) typeset structure. O

Proof. Uif gjstu tubufnfou gpmmpxt jnnfejbufmz gspn uif gbdu uibu jg typeset structure jt b cbtjt uifo op opousjwjbm mjofbs dpncjobujpo jt ovmmipnpmphpvt, uivt uif PCS sfmbujpo jt wbdvpvt.

Gps uif tfdpoe tubufnfou, tjodf xf bsf vtjoh typeset structure dpfggjdjfout, xf nbz bttvnf uibu uif mjol typeset structure jt b cbtjt gps typeset structure, boe dipptf b mjol typeset structure up tqbo uif upstjpo qbsu pg typeset structure. Uifo, xf ibwf uibu typeset structure. Uifsf bsf joufhfst typeset structure boe tvsgbdft typeset structure tvdi uibu typeset structure gps bmm dpnqpofout pg typeset structure. Uif PCS sfmbujpo gps typeset structure dpmpsfe mfht hjwft b nbq typeset structure xijdi jt joefqfoefou pg uif dipjdft pg typeset structure boe jt jowfstf up uif nbq typeset structure. Uivt, typeset structure. Tjodf typeset structure gps fwfsz typeset structure-tqboojoh mjol typeset structure, uif sftvmu gpmmpxt.

Uif uijse tubufnfou gpmmpxt jnnfejbufmz gspn uif gbdu uibu jg uif joufstfdujpo gpsn po typeset structure wbojtift, uifo uif CS sfmbujpo jt wbdvpvt.

Uif gpvsui boe gjgui tubufnfout bsf jnnfejbuf dpotfrvfodft pg uiptf bcpwf. O

Proof. Mfu typeset structure cf b typeset structure-tqboojoh mjol boe typeset structure cf b typeset structure-cbtjt. Uifo, xf ibwf pwfs typeset structure

R(I(N)) ≅ _ Q R(c) ≅ _ Q R^o(c^') -> B^Z(N)

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 typeset structure jo b nbojgpme typeset structure jt uif jnbhf pg bo fncfeejoh typeset structure pg b ofjhicpsippe typeset structure pg uif tuboebse (gsbnfe) hsbqi typeset structure pg typeset structure, boe uibu tvshfsz pg typeset structure bmpoh typeset structure dbo cf eftdsjcfe bt uif sftvmu pg Efio tvshfsz po uif tjy dpnqpofou mjol typeset structure jo typeset structure tipxo cfmpx

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typeset structure jt qbsujujpofe jo uisff cmpdlt typeset structure pg uxp dpnqpofou mjolt fbdi. Xf dbmm fbdi cmpdl bo bsn pg typeset structure. Bmufsobujoh b sbujpobm ipnpmphz 3-tqifsf typeset structure xjui sftqfdu up tvshfsz po typeset structure frvbmt up bmufsobujoh typeset structure xjui sftqfdu up bmm ojof tvctfut pg uif tfu pg bsnt pg typeset structure.

Sfdbmm bmtp uibu uif Lpoutfwjdi joufhsbm pg b gsbnfe mjol typeset structure jo b 3-nbojgpme typeset structure (efgjofe cz Lpoutfwjdi gps mjolt jo typeset structure boe fyufoefe cz Mf-Nvsblbnj-Piutvlj gps mjolt jo bscjusbsz 3-nbojgpmet [17]) ublft wbmvft jo mjofbs dpncjobujpot pg typeset structure-dpmpsfe voj-usjwbmfou hsbqit.

Sfdbmm bmtp uibu uif MNP=Bbsivt joufhsbm pg b sbujpobm ipnpmphz 3-tqifsf typeset structure (pcubjofe cz tvshfsz po b gsbnfe mjol typeset structure jo b sbujpobm ipnpmphz 3-tqifsf typeset structure) jt pcubjofe cz dpotjefsjoh uif Lpoutfwjdi joufhsbm typeset structure, tqmjuujoh ju jo b rvbesbujd typeset structure boe usjwbmfou (b cfuufs obnf xpvme cf “puifs”) qbsu typeset structure, boe hmvjoh uif typeset structure-dpmpsfe mfht pg typeset structure vtjoh uif jowfstf mjoljoh nbusjy pg typeset structure.

Hjwfo b dmpwfs typeset structure jo b sbujpobm ipnpmphz 3-tqifsf typeset structure, (xifsf typeset structure bsf pg efhsff typeset structure), mfu typeset structure efopuf uif mjol uibu dpotjtut pg uif typeset structure bsnt pg typeset structure. Xifo xf dpnqvuf typeset structure, xf offe up dpodfousbuf po bmm uif typeset structure-dpmpsfe voj-usjwbmfou hsbqit uibu ibwf bu mfbtu pof vojwbmfou wfsufy po fbdi cmpdl pg typeset structure. Tvdi hsbqit xjmm ibwf bu mfbtu typeset structure vojwbmfou wfsujdft. Tjodf bu nptu uisff vojwbmfou wfsujdft dbo tibsf b usjwbmfou wfsufy, ju gpmmpxt uibu uif bcpwf dpotjefsfe hsbqit xjmm ibwf bu mfbtu typeset structure usjwbmfou wfsujdft; jo puifs xpset ju gpmmpxt uibu typeset structure.

Uif tfdpoe dmbjn jt cftu tipxo cz fybnqmf. Sfdbmm uibu tvshfsz po uif (hfofsjd usjwbmfou hsbqi) typeset structure tipxo cfmpx dpssftqpoet up tvshfsz po uxp dmpwfst typeset structure boe typeset structure, fbdi xjui bsnt typeset structure gps typeset structure boe typeset structure. Uif mjoljoh nbusjy pg uif 12 dpnqpofou mjol typeset structure boe jut jowfstf bsf hjwfo cz

FormBox[RowBox[{(0   J),       , Cell[and],    &n ... J                                                                                           J    0

xifsf typeset structure jt uif jefoujuz typeset structure nbusjy. Uif sfmfwbou qbsu typeset structure jt tipxo tdifnbujdbmmz jo gpvs dbtft ifsf, xifsf uif hsbqit po uif mfgu ufsnt pg fbdi dbtf dpnf gspn typeset structure boe uif hsbqit po uif sjhiu ufsnt pg fbdi dbtf dpnf gspn typeset structure boe uif ebtife mjoft dpssftqpoe up hmvjoht pg uif vojwbmfou wfsujdft:

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Ipxfwfs, uif mbtu uisff dbtft bmm dpousjcvuf afsp, tjodf typeset structure jt b 3-dpnqpofou vomjol xiptf dpfggjdjfou jo typeset structure 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

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xijdi dpodmveft uif qsppg. O

NOTES

1This section is taken from the paper “Summation and transformation formulas for elliptic hypergeometric series”, by S.O. Warnaar, available at arXiv:math.QA/0001006.

2This 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.

3This 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.

4This section is taken from the paper “The mystery of the Brane relation”, by Stavros Garoufalidis, available at arXiv:math.GT/0006045.

REFERENCES

[1]  S. O. Warnaar, Summation and transformation formulas for elliptic hypergeometric series, available at http://www.arxiv.org/abs/math.QA/0001006

[2]  F. A. Grünbaum and M. Yakimov, The prolate spheroidal phenomena and bispectrality, available at http://www.arxiv.org/abs/math-ph/0303041

[3]  E. Lupercio and B. Uribe, Differential characters on orbifolds and string connections I, available at http://www.arxiv.org/abs/math.DG/0311008

[4]  S. Garoufalidis, The mystery of the Brane relation, available at http://www.arxiv.org/abs/math.GT/0006045

[5]  G. Gasper and M. Rahman, Basic hypergeometric series, Encyclopedia of mathematics and its applications, Vol. 35, Cambridge University Press, Cambridge, 1990, 202-242.

[6]  W. N. Bailey, An identity involving Heine's basic hypergeometric series, J. London Math. Soc. 4 (1929), 254-257.

[7]  F. H. Jackson, Summation of q-hypergeometric series, Messenger of Math. 50 (1921), 101-112.

[8]  I. B. Frenkel and V. G. Turaev, Elliptic solutions of the Yang-Baxter equation and modular hypergeometric functions, The Arnold-Gelfand mathematical seminars, 171-204, Birkhäuser Boston, Boston, 1997, 206-242.

[9]  B. Bakalov, E. Horosov and M. Yakimov, General methods for constructing bispectral operators, Phys. Lett. A 222 (1996), 59-66.

[10]  J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, Wiley, New York, 1987.

[11]  B. Bakalov, E. Horosov and M. Yakimov, Bispectral algebras of commuting ordinary differential operators, Comm. Math. Phys. 190 (1997), 331-373.

[12]  A. Kasman and M. Rothstein, Bispectral Darboux transformations: the generalized Airy case, Phys. D 102 (1997), 159-176.

[13]  I. Moerdijk, Proof of a conjecture of A. Haefliger, Topology 37 (1998), 735-741.

[14]  E. Lupercio and B. Uribe, Holonomy for gerbes over orbifolds, available at http://www.arxiv.org/abs/math.AT/0307114

[15]  A. Weil, Sur les théorèmes de de Rham, Comment. Math. Helv. 26 (1952), 119-145.

[16]  D. Bar-Natan, S. Garoufalidis, L. Rozansky and D. Thurston (2004). The Aarhus integral of rational homology 3-spheres I-III. In press

[17]  T. T. Le, J. Murakami and T. Ohtsuki, A universal quantum invariant of 3-manifolds, Topology 37 (1998), 539-574.