An Isomonodromy Interpretation of the Hypergeometric Solution of the Elliptic Painleve Equation (and Generalizations)

Symmetry , Integrabilit y and Geometry: Metho ds and Applications SIGMA 7 (2011), 088, 24 pages An Isomono drom y In terp r e tation of the Hyp ergeometri c Solut ion of the Elliptic P ain lev ´ e Equation (and Gener alizations) ⋆ Eric M. RAINS Dep artment of M athematics, California Institute of T e c hnolo gy, 1200 E. California Boulevar d, Pasadena, CA 91125, USA E-mail: r ains@c al te ch.e du Received April 25, 2011, in f inal f orm Septem b e r 06, 2 011; P ublished online September 09, 20 11 ht tp://dx.doi.or g/10 .3842/ SIGMA.2011.088 Abstract. W e construct a family of second-or der linear dif ference eq ua tions parametrized by the h ypergeo metric so lution of the elliptic Painlev´ e equation (or higher-or der analog ues), and admitting a large family of monodr omy-preserving defor ma tions. The solutions are certain semiclass ic al biorthogonal functions (and their Cauch y transforms), biorthog o nal with r esp ect to higher-or der analogues of Spirido nov’s elliptic b eta integral. Key wor ds: is omono dromy; hyper geometric; Painlev ´ e; biorthog onal functions 2010 Mathematics Subje ct Classiﬁc ation: 3 3E17 ; 3 4M55; 3 9A13 1 In tro duction In [21], Sak ai introd uced an elliptic analogue of th e P ainlev ´ e equati ons, includ ing all of the kno wn discrete (and con tin uous) Pa inlev ´ e equations as sp ecial cases. Unf ortunately , although Sak ai’s construction is quite n atural and geometric, it do es not ref lect the most imp ortan t role of the ordin ary Painlev ´ e trans cend en ts, namely as parameters con trolling mon o drom y-preserving deformations. As with the ord inary Pa inlev ´ e equ ations, the elliptic P ainlev ´ e equation adm its a s p ecial class of “h yp ergeometric” solutions [12, 18] that in th e most general case can b e expr essed via n - dimensional con tour integrals with integrands expressed in terms of elliptic Gamma fu nctions. It is th us natural, as a f irst step in constructing an isomono dromy int erpretation of the elliptic P ainlev ´ e equ ation, to attempt to und er s tand that interpretatio n in the hypergeometric case (and th us gain in sigh t to the general case). Note that we wan t to unders tand the hyp ergeometric case for all n ≥ 1, to a v oid the p ossibility that the small n cases might dif fer from th e general P ainlev ´ e case in some qualitativ ely signif icant w a y . (F or instance, [13] considers the isomono drom y in terpretation of the usual 2 F 1 (corresp onding to n = 1 in our setting), but this is simplif ied greatly from the general Pa inlev ´ e VI case b y the fact that not only the mono dr om y but th e equation itself can b e tak en to b e triangular.) In th e present work, we do pr ecisely that: associated to eac h elliptic h yp ergeometric solution of the elliptic P ainlev ´ e equation, w e construct a corresp onding second-order linear dif f erence equation that admits a f amily of d iscrete “mono dromy-preserving” deformations. (In fact, th e construction wo rks equally we ll for higher-ord er analogues of th e relev an t elliptic hyp ergeometric in tegrals, wh ic h should corresp ond to s p ecial solutions of “elliptic Garnier equations”.) Th e construction is b ased on an analog ue of the approac h in [15, 11]. There, a linear dif f eren tial ⋆ This pap er is a contribution to the Sp ecial Issue “Relationship of Orthogonal Polynomials and Sp e- cial F unctions with Quantum Groups and Integrable S ystems”. The full collection is a v aila ble at http://w ww.emis.de/j ournals/SIGMA/OPSF.html 2 E.M. Rains equation deformed by the hyp ergeometric case of the Painlev ´ e VI equation is constructed as a dif feren tial equ ation s atisf ied by a family of “semiclassical” (bi-)orthogonal p olynomials. Our construction is m uc h the same, although there are sev eral tec hnical issues to o v ercome. The f irst suc h issue is, simply put, to u nderstand precisely wh at it means for a deformation of an elliptic d if feren ce equation to p reserv e mono dromy , or ev en what the mono d rom y of an elliptic d if feren ce e quation is. While we giv e only a partial answer to this question, w e do def in e (in S ection 2 b elow; note that many of the considerations there tu rn out to ha v e b een an ticipated b y Etingof in [8]) a weak ened form of mono dromy that, while somewhat w eak er than th e analogous notions at the q -dif ference [9] and lo w er [4] leve ls, is still strong enough to give a reasonably rigid notion of isomono dromy deformation. Ind eed, tw o elliptic dif ference equations ha v e the same wea k mono d rom y if f th e corresp onding d if feren ce m o dules (see [17]) are isomorph ic; the same holds for ordinary dif f erence equations, ev en relativ e to the s tronger notion of mono dromy [5]. The k ey observ atio n is th at a fundamenta l matrix for a p -elliptic q -dif ference equation is also a fu ndamenta l matrix for a q -elliptic p -d if feren ce equation; this latter equ ation (up to a certa in equiv alence relation) pla ys the role of the mono dromy . (The result is similar to the n otion of mono dr omy int ro du ced b y Krichev er in [14]; while our notion is we ak ened b y an equiv al ence relation, it a v oids any assu mptions of genericit y .) In S ection 3, we deve lop th e theory of semiclassical elliptic biorth ogonal fun ctions, functions biorthogonal w ith r esp ect to a dens ity generalizing Spiridonov’s elliptic b eta in tegral [23] b y adding m additional pairs of p arameters. The key observ ation is that such functions can b e constructed as higher-order elliptic Selb erg int egrals of a sp ecia l form; in addition, th eir “Cauc h y transforms” can also be so written. This giv es rise to sev eral nice rela tions b et w een these functions, whic h we describ e. Most imp ortan t for our purp oses is their b eha vio r u nder p -shifts; the biorthogonal functions themselv es are p -elliptic, but if we include the Cauc h y transforms, the o v er all action is triangular. W e can th us constru ct from these functions a 2 × 2 matrix whic h s atisf ies a triangular q -elliptic p -dif ference equation, analogous to the Riemann –Hilb ert problem asso ciated to orthogonal p olynomial s ([10, § 3.4]; see also [6] for a general exp osition). By the theo ry of Section 2, this im m ediately give s rise to a p -elliptic q -dif ference equation, and symmetries of th e p -dif ference equation ind uce m on o drom y-preserving deformations of the q -dif ference equation. Finally , in S ection 4, w e compute this dif ference equation and the asso ciated deformations. Although w e cannot give a close d form expression for the dif ference equation, w e are able at least to d etermine precisely where the dif ference equation is singular, and at eac h suc h p oint , compute the v alue (or residue, as app r opriate) of the s hift matrix. T ogether w ith the fact that the co ef f icients are meromorphic p -theta fu nctions, this d ata suf f ices to (o v er)d etermine the shift matrix. In a follo wup pap er [3], with Ar inkin and Boro din, w e w ill complete the isomono dr om y in ter- pretation of the elliptic Pa inlev ´ e equation by applying the ideas of [2] to sho w that any d if f erence equation ha ving the same structure as the ones constructed b elo w admits a corresp onding fa- mily of mono dromy-preserving deformations, and moreo v er that (wh en m = 1) Sak ai’s rational surface can b e reco vered as a mo duli space of suc h dif feren ce equations. A rather dif ferent geometric app roac h to such an int erpretation (via a L ax pair) for the m = 1 case h as b een given in [26]. Notation The el liptic Gamma function [20] is def ined for complex num ber s p , q , z with | p | , | q | < 1, z 6 = 0, b y Γ p,q ( z ) := Y 0 ≤ i,j 1 − p i +1 q j +1 /z 1 − p i q j z , Isomono dromy Int erpretation of Hyp ergeome tric S olution of E lliptic Painlev ´ e 3 and satisf ies th e ref lection relation Γ p,q ( pq /z ) = Γ p,q ( z ) − 1 as w ell as the shift relations Γ p,q ( pz ) = θ q ( z )Γ p,q ( z ) , Γ p,q ( q z ) = θ p ( z )Γ p,q ( z ) , where the fun ction θ p ( z ) := Y 0 ≤ i  1 − p i +1 /z  1 − p i z  satisf ies θ p ( z ) = − z θ p (1 /z ) = θ p ( p/z ) , so that Γ p,q ( pq z )Γ p,q ( z ) = − z − 1 Γ p,q ( pz )Γ p,q ( q z ) . By con v en tion, multiple argumen ts to a Gamma or th eta function represent a pro du ct; th us, for instance Γ p,q ( u 0 z ± 1 ) = Γ p,q ( u 0 z )Γ p,q ( u 0 /z ) . W e w ill also mak e b rief use of the third-order elliptic Gamma f unction Γ + p,q ,t ( x ) := Y 0 ≤ i,j,k  1 − p i q j t k x  1 − p i +1 q j +1 t k +1 /x  , whic h satisf ies Γ + p,q ,t ( tx ) = Γ p,q ( x )Γ + p,q ,t ( x ) , Γ + p,q ,t ( pq t/x ) = Γ + p,q ,t ( x ); for our purp oses, this app ears only as a normalization f actor relating the order 1 elliptic Selb erg in tegral to the hypergeometric tau fu nction f or elliptic P ainlev ´ e. 2 Elliptic dif ference equations Let p b e a complex num b er w ith | p | < 1. A (meromorphic) p -theta fu nction of multiplier αz k is a meromorph ic fu nction f ( z ) on C ∗ := C \ { 0 } with the p eriodicit y p rop erty f ( pz ) = αz k f ( z ). (T o ju stify this def inition, observe that the comp osition f (exp(2 π √ − 1 t )) is meromorphic on C , p erio d ic with p eriod 1, an d quasi-p erio d ic w ith p eriod log ( p ) / 2 π √ − 1; in other w ords, it is a theta function in the usu al sense.) The canonical example of such a fu nction is θ p ( z ), a h olomorphic p - theta f u nction with m ultiplier − z − 1 ; indeed, any holomorphic p -theta function can b e w ritten as a pr o duct of functions θ p ( uz ), and an y meromorph ic p -theta function as a ratio of s uc h pro ducts. In th e s p ecial case of multiplier 1, the function is called p -elliptic, for similar reasons. By stand ard con v en tio n, a p -theta function, if not explicitly allo w ed to b e meromorph ic, is holomorphic; ho w ev er, p -elliptic functions are alw a ys allo we d to b e meromorp hic (sin ce a holomorphic p - elliptic function is constant) . Let q b e another complex num b er w ith | q | < 1, such that p Z ∩ q Z = ∅ . 4 E.M. Rains Deﬁnition 2.1. A p - theta q - diﬀer enc e e quation of multiplier µ ( z ) = αz k is an equation of th e form v ( q z ) = A ( z ) v ( z ) , where A ( z ) is a nonsingular mer omorphic matrix (a square matrix, eac h co ef f icien t of whic h is meromorphic on C ∗ , and the determinan t of wh ic h is not iden tically 0), called the shift matrix of the equation, such that A ( pz ) = µ ( z ) A ( z ) , so in p articular the coef f icien ts of A are meromorp hic p -theta f unctions of m u ltiplier µ ( z ). Similarly , a p -elliptic q -dif ference equation is a p -theta q -dif feren ce equation of multiplier 1. W e w ill r efer to the d imension of the matrix A as the or der of the corresp onding dif ference equation. W e note the follo wing fact ab out nonsingular meromorph ic matrices. Prop osition 2.2. L et M ( z ) b e a nonsingular mer omo rphic matrix. Then M ( z ) − 1 is also a non- singular mer om orphic matrix, and if the c o eﬃcients of M ( z ) ar e mer omorphic p -theta functions of multiplier µ ( z ) , then those of M ( z ) − 1 ar e mer omor phic p - theta fu nc tions of multiplier µ ( z ) − 1 . Pro of . Indeed, the co ef f icien ts of the adjoin t matrix det( M ( z )) M ( z ) − 1 are minors of M ( z ), and th us, as p olynomials in meromorph ic functions, are meromorphic; this con tin ues to hold after m ultiplying by the meromorph ic fu nction d et( M ( z )) − 1 . F or the second claim, if M ( pz ) = µ ( z ) M ( z ) , then M ( pz ) − 1 = µ ( z ) − 1 M ( z ) − 1 .  Deﬁnition 2.3. Let v ( q z ) = A ( z ) v ( z ) b e a p -theta q -dif ference equatio n. A mer omo rphic fundamental matrix for this equation is a non s ingular meromorphic matrix M ( z ) satisfying M ( q z ) = A ( z ) M ( z ) . It follo ws from a theorem of Pr aagman [16, Theorem 3] that for any nons in gular m eromorphic matrix A ( z ), there exists a nonsin gular meromorphic matrix M ( z ) satisfying M ( q z ) = A ( z ) M ( z ) (this is the sp ecial case of the theorem in whic h the discon tinuous group ac ting on CP 1 is that generated b y multiplicatio n by q ). In particular, an y p -theta q -d if feren ce equ ation admits a meromorphic fun d amen tal matrix. In the case of a f irst order equation, we can explicitly construct such a matrix. Prop osition 2.4. Any ﬁrst or der p -theta q -diﬀer e nc e e quation admits a mer omorphic funda- mental matrix. Pro of . F or any nonzero meromorphic p -theta fu nction a ( z ), w e n eed to construct a nonzero meromorphic function f ( z ) such that f ( q z ) = a ( z ) f ( z ) . Since a ( z ) can b e f actored into fun ctions θ p ( uz ), it suf f ices to consider the case a ( z ) = θ p ( uz ), with meromorphic solution f ( z ) = Γ p,q ( uz ); this includes the case a ( z ) = bz k b y w riting bz k = θ p ( − bz ) θ p ( − pz ) k − 1 θ p ( − bpz ) θ p ( − z ) k − 1 .  Isomono dromy Int erpretation of Hyp ergeome tric S olution of E lliptic Painlev ´ e 5 W e note in particular that, since the elliptic Gamma function is symmetrical in p and q , the solution th us obtained f or a f ir s t ord er p -theta q -dif ference equation also satisf ies a q -theta p -dif ference equ ation. This is quite typical , and in fact we h a v e the f ollo wing result. Lemma 2.5. L et v ( q z ) = A ( z ) v ( z ) b e a p -theta q -diﬀer enc e e quation of multiplier µ ( z ) , and let M ( z ) b e a mer omorphic fundamental matrix for this e quation. Then ther e exists a ( uni q ue ) q -theta p - diﬀe r enc e e q uation of multiplier µ ( z ) for which M ( z ) t is a fu ndamental matrix. Pro of . An equation w ( pz ) = C ( z ) w ( z ) with fu n dament al matrix M ( z ) t satisf ies M ( pz ) t = C ( z ) M ( z ) t , and th us, since M ( z ) is nonsin gular, we can compute C ( z ) = M ( pz ) t M ( z ) − t . (Here M − t denotes the in v erse of the transp ose of M .) T h is matrix is meromorp hic, and s atisf ies C ( q z ) = M ( pq z ) t M ( q z ) − t = M ( pz ) A ( q z ) t A ( z ) − t M ( z ) − t = µ ( z ) C ( z ) .  By symmetry , we obtain the f ollo w ing r esult. Theorem 2.6. L et M ( z ) b e a nonsingular mer omorphic matrix. Then the fol lowing ar e e qui v - alent: (1) M ( z ) is a mer omorphic fundamental matrix for some p - theta q -diﬀer enc e e quation; (2) M ( z ) t is a mer omo rphic fundamental matrix for some q -theta p -diﬀer enc e e quation; (1 ′ ) M ( z ) − t is a mer omo rphic fundamental matrix for some p -theta q -diﬀer enc e e quation; (2 ′ ) M ( z ) − 1 is a mer omorphic fundamental matrix for some q -theta p -diﬀer enc e e quation, as ar e the c o rr esp onding statements with “some” r e plac e d by “a unique”. F urthermor e, if the ab ove c onditions hold, the multipliers of the diﬀer e nc e e quations of (1) and (2) agr e e, and ar e inverse to those of (1 ′ ) and (2 ′ ) . Remark 2.7. In th e elliptic case, the ab o v e observ at ions w ere made by Etingof [8], who also noted that the asso ciated q -elliptic p -dif f erence equation can b e thought of as the mono dr om y of M . Giv en a p -theta q -dif ference equation, the corresp ondin g meromorphic fund amen tal matrix is b y no means un iqu e, and thus w e obtain a w h ole family of r elated q -theta p -dif ference equations. There is, how ever, a natural equiv alence relation on q -theta p -dif ference equations suc h th at an y p -theta q -dif ference equation giv es rise to a w ell-def ined equiv ale nce class. F irst, we need to understand the exten t to which the fu ndament al matrix fails to b e unique. Lemma 2.8. L et M ( z ) and M ′ ( z ) b e fundamental matric es for the same p - theta q -diﬀer enc e e quation v ( q z ) = A ( z ) v ( z ) . Then M ′ ( z ) = M ( z ) D ( z ) t for some nonsingular mer om orphic matrix D ( z ) with q -el liptic c o eﬃcients. 6 E.M. Rains Pro of . Certainly , ther e is a uniqu e meromorphic matrix D ( z ) with M ′ ( z ) = M ( z ) D ( z ) t , and comparing d etermin ants shows it to b e nonsingular. It thus remains to sho w that D ( z ) has q -elliptic co ef f icients, or equiv alen tly that D ( q z ) = D ( z ). As in th e p r o of of Lemma 2.5, we can write A ( z ) = M ( q z ) M ( z ) − 1 = M ′ ( q z ) M ′ ( z ) − 1 , and th us D ( q z ) t D ( z ) − t = M ( q z ) − 1 M ′ ( q z ) M ′ ( z ) − 1 M ( z ) = 1 , as required.  Theorem 2.9. Deﬁne an e qu ivalenc e r elation on q -theta p -diﬀe r enc e e quations by saying  v ( pz ) = C ( z ) v ( z )  ∼ =  v ( pz ) = C ′ ( z ) v ( z )  iﬀ ther e exists a nonsingular q - el liptic matrix D ( z ) such that C ′ ( z ) D ( z ) = D ( pz ) C ( z ) . Then the set of q - theta p -diﬀer enc e e qu ations asso ciate d to a given p -theta q -diﬀe r enc e e quation is an e q uivalenc e class. Pro of . Let M ( z ) b e a meromorph ic fu ndament al matrix for the p -theta q -dif ference equation v ( q z ) = A ( z ) v ( z ), and asso cia ted q -theta p -dif ference equation w ( pz ) = C ( z ) w ( z ). If M ′ ( z ) is another meromorphic f undamental matrix for the q -dif ference equation, with asso ciated p - dif ference equ ation w ( pz ) = C ′ ( z ) w ( z ), then M ′ ( z ) = M ( z ) D ( z ) t , and th us C ′ ( z ) = M ′ ( pz ) t M ′ ( z ) − t = D ( pz ) M ( pz ) t M ( z ) − t D ( z ) − 1 = D ( pz ) C ( z ) D ( z ) − 1 . Con v ersely , if C ′ ( z ) D ( z ) = D ( pz ) C ( z ) , then M ( z ) D ( z ) t is a fu ndamenta l matrix for a q -dif ference equation with asso ciated p -d if feren ce equation w ( pz ) = C ′ ( z ) w ( z ).  Deﬁnition 2.10. The we ak mono dr om y of a p -theta q -dif ference equation is th e asso ciated equiv alence class of q -theta p -d if ference equations. Tw o p -theta q -dif ference equations are iso- mono dr omic if they ha v e the same w eak mono dromy . Theorem 2.11. The p -theta q -diﬀer enc e e quations v ( q z ) = A ( z ) v ( z ) , v ( q z ) = A ′ ( z ) v ( z ) ar e isomono dr omic iﬀ ther e exists a nonsingular p -el liptic matrix B ( z ) su c h that A ′ ( z ) B ( z ) = B ( q z ) A ( z ) . Pro of . Cho ose a q -theta p -dif ference equation w ( q z ) = C ( z ) w ( z ) rep r esen ting the weak mono- drom y of th e f irst equation. The equ ations are isomono dromic if f w ( q z ) = C ( z ) w ( z ) represent s the we ak mono dr omy of the second equation, if f the t w o equations hav e fund amen tal matri- ces satisfying M ( q z ) = M ( z ) C ( z ). But by T heorem 2.9 (swapping p and q ), this holds if f A ′ ( z ) B ( z ) = B ( q z ) A ( z ) for some p -elliptic matrix B ( z ).  Isomono dromy Int erpretation of Hyp ergeome tric S olution of E lliptic Painlev ´ e 7 Remark 2.12. C ompare [5], w here the analogous r esu lt is pro v ed for d if f erence equations, relativ e to Birkhof f ’s [4] notion of m ono dromy . Corollary 2.13. The map fr om isomono dr omy c lasses of p -theta q -diﬀer enc e e quations to their we ak mono dr om ies is wel l-deﬁne d, and inverse to the map fr om isomono dr omy classes of q -theta p -diﬀer e nc e e quations to their we ak mono dr omies. Remark 2.14. The isomono dr om y equiv alence relation is also quite n atural fr om th e p ersp ec- tiv e of th e general theory of dif ference equations (see, e.g., [17]); to b e precise, t wo p -theta q -dif ference equations are isomono dromic if f they ind uce isomorphic dif ference mo d ules. The latter fact in duces a natural isomorp hism b et ween their dif ference Galois grou p s (at least when the latter are d ef ined , i.e., when the equations are elliptic), as can b e seen directly from the in terpretation of d if fer en ce Galois groups via T annakian categories. This preserv atio n of Galois groups seems to b e what is tru ly in tended by the w ord “isomono dromy”, ev en in the dif feren tial setting. F or ins tance, for non-F uc hsian equations, where the mono d rom y group con v eys rela- tiv ely little inform ation, one only obtains the r elev an t Painlev ´ e equ ations by insisting that the corresp ondin g deformations shou ld p reserv e S tok es d ata as w ell. It will b e con v enien t in the sequel to introd uce a sligh tly we ak er equiv ale nce relation. Deﬁnition 2.15. Tw o p -theta q -dif ference equations are theta-isomono dr omic if there exists a nons ingular meromorphic p -theta matrix B ( z ) suc h that the shift matrices A ( z ), A ′ ( z ) of the equations satisfy A ′ ( z ) B ( z ) = B ( q z ) A ( z ). Theorem 2.16. Two p - theta q -diﬀer enc e e quations ar e theta-isomono dr omic i ﬀ their we ak mo- no dr omies agr e e up to multiplic ation of the shift matrix by a factor of the form az k . Note that v ( q z ) = A ( z ) v ( z ) and v ( q z ) = A ( q z ) v ( z ) are theta-isomono dromic with B ( z ) = A ( z ). Remark 2.17. Though this equiv alence r elation no longer preserves Galois groups, ev en if b oth equations are elliptic, it comes qu ite close to doing so. Indeed, the Galois group of an n -th order equation is n atur ally a subgroup of GL n , and we ma y th us consider its image in PGL n , wh ic h one migh t call the pr oje ctive Galois group. Sin ce P GL n is the image of GL n under the adj oin t represent ation, one f inds that the pr o jectiv e Galois group of the equation with shift matrix A ( z ) can b e id en tif ied with the ordin ary Galois group of the equation w ith shift matrix A ( z ) ⊗ A ( z ) − t . If t w o equations are theta-isomono dromic, their images under the adjoint repr esen tation are thus fully isomono dromic, and thus the original equ ations had the same pro jectiv e Galois groups. This even extends to p -theta q -d if feren ce equations once we observe that the image of such an equation un der the adj oin t repr esen tation is elliptic, and th us the pr o jectiv e Galois group s of suc h equations are still we ll-def ined. Remark 2.18. The relation of isomono drom y to Galois groups suggests some further qu estions, whic h are in the main outside the scop e of the cu rrent pap er, b ut seem to merit a b rief mention nonetheless. First, since (theta-)isomonodr omic equations ha v e isomorph ic (pro jectiv e) Galois groups, it is natural to ask whether one can reco v er the (pr o jectiv e) Galois group more directly from the w eak mono dromy . Since the weak mono dr om y is itself a isomono drom y class, any t w o repr esen tativ es of the we ak mono d r om y hav e the same Galois group, and one would exp ect th at group to b e related to the original Galois group. It can b e shown (Etingof, p ersonal communicati on) that in f act the groups are naturally isomorph ic, with dual asso ciated r epresen tations. Th us, for instance, the fact that the dif ference equ ations w e will b e considering ha v e triangular w eak 8 E.M. Rains mono dromy imp lies that they h a v e solv ab le Galois group. (This also follo ws immediately f rom the fact that, by construction, they hav e theta function solutions.) Another natural question is wh ether ther e exists a str onger notion of mono dr omy; for ratio- nal q -dif ference equations with suf f icient ly nice singularities, there is a w ell -def in ed notion of mono dromy , an asso cia ted n onsingular q -elliptic matrix the nonsin gu lar v alues of whic h generate a Zariski den se su b group of the Galois group ([9]; see also Chapter 12 of [17]). Kric hev er [14] def in es an analogous matrix for generic dif ference equations with theta function coef f icien ts (although the relation to the Galois group is again unclear); although Kric hev er’s genericit y assumptions exp licitly exclude the s itu ation we consider ab o v e (raising the question of whether there is an analogue in our setting) , his m ono dromy is again a d if ference equ ation with theta function co ef f icient s. T his su ggests that the rational q -d if feren ce notion of mon o drom y should corresp ond at the elliptic leve l to a represen tativ e of our wea k mono dr omy , and th us suggests the qu estion of whether giv en a p -elliptic q -dif ference equ ation, there exists a representa tiv e of its w eak mono dr omy su c h that the nonsingular v alues of the corresp onding C matrix are Z ariski dense in its Galois group. 3 Semiclassical biorthogonal elliptic functions In [22], Spiridonov constructed a family of elliptic hyp ergeometric functions b iorth ogonal with resp ect to the density of the elliptic b eta int egral: ( p ; p )( q ; q ) 2 Z C Q 0 ≤ r< 6 Γ p,q ( u r z ± 1 ) Γ p,q ( z ± 2 ) dz 2 π √ − 1 z = Y 0 ≤ r< s< 6 Γ p,q ( u r u s ) , where the parameters satisfy the b alancing c onditio n Y 0 ≤ r< 6 u r = p q , and th e (p ossib ly disconn ected, but closed) con tour is chosen to b e symm etrical under z 7→ 1 /z , and to con tain all p oin ts of the form p i q j u r , i, j ≥ 0, 0 ≤ r < 6, or more p recisely , all p oles of the int egrand of that form. If we view this as the “classical” case , then this suggests, by analogy with [15, 11] that w e should s tudy biorthogonal fu nctions with resp ect to the more general densit y ∆ ( m ) ( z ; u 0 , . . . , u 2 m +5 ) = Q 0 ≤ r< 2 m +6 Γ p,q ( u r z ± 1 ) Γ p,q ( z ± 2 ) , with new balancing condition Y 0 ≤ r< 2 m +6 u r = ( pq ) m +1 , and the corresp ondin g cont our condition, integrat ed against the dif feren tial ( p ; p )( q ; q ) 2 dz 2 π √ − 1 z . Note that if u 2 m +4 u 2 m +5 = pq , then the corresp ondin g factors of the densit y cancel, an d thus w e reduce to the ord er m − 1 d ensit y . Also , it w ill b e con v enien t to m ultiply the integrands b y theta functions, not elliptic functions; suc h multiplicati on has the ef f ect of shifting th e b alan- cing condition. (Th e extent of the required shift can b e determined via th e observ ation that Isomono dromy Int erpretation of Hyp ergeome tric S olution of E lliptic Painlev ´ e 9 m ultiplying a parameter by q m ultiplies the in tegrand b y a p -theta function; in any ev en t, we will giv e the explicit balancing condition for eac h of the in tegrals app earing b elo w.) One n atur al m ultiv ariate analogue of the elliptic b eta int egral is the elliptic Selb erg in teg - ral [7, 19], the higher-order ve rsion of w hic h we def ine as follo ws I I ( m ) n ; t ; p,q ( u 0 , . . . , u 2 m +5 ) := ( p, p ) n ( q ; q ) n Γ p,q ( t ) − n 2 n n ! Z C n Y 1 ≤ i

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