A First Order q-difference System for the BC 1 -Type Jackson Integral and Its Applications

Transcription

1 Symmetry Integrablty and Geometry: Methods and Applcatons SIGMA pages A Frst Order -Dfference System for the BC 1 -Type Jackson Integral and Its Applcatons Masahko ITO Department of Physcs and Mathematcs Aoyama Gakun Unversty Kanagawa Japan E-mal: mto@gemaoyamaacp Receved December n fnal form March ; Publshed onlne Aprl do:103842/sigma Abstract We present an explct expresson for the -dfference system whch the BC 1 - type Jackson ntegral -seres satsfes as frst order smultaneous -dfference euatons wth a concrete bass As an applcaton we gve a smple proof for the hypergeometrc summaton formula ntroduced by Gustafson and the product formula of the -ntegral ntroduced by Nassrallah Rahman and Gustafson Key words: -dfference euatons; Jackson ntegral of type BC 1 ; Gustafson s C n -type sum; Nassrallah Rahman ntegral 2000 Mathematcs Subect Classfcaton: 33D15; 33D67; 39A13 1 Introducton A lot of summaton and transformaton formulae for basc hypergeometrc seres have been found to date The BC 1 -type Jackson ntegral whch s the man subect of nterest n ths paper s a -seres whch can be wrtten as a basc hypergeometrc seres n a class of so called very-well-posed-balanced 2r ψ 2r A key reason to consder the BC 1 -type Jackson ntegrals s to gve an explanaton of these hypergeometrc seres from the vew ponts of the Weyl group symmetry and the -dfference euatons of the BC 1 -type Jackson ntegrals In [15] we showed that Slater s transformaton formula for a very-well-posed-balanced 2r ψ 2r seres can be regarded as a connecton formula for the solutons of -dfference euatons of the BC 1 - type Jackson ntegral e the Jackson ntegral as a general soluton of -dfference system s wrtten as a lnear combnaton of partcular solutons As a conseuence we gave a smple proof of Slater s transformaton formula See [15] for detals Also see [13] for a connecton formula for the BC n -type Jackson ntegral whch s a multsum generalaton of that of type BC 1 The am of ths paper s to present an explct form of the -dfference system as frst order smultaneous -dfference euatons for the BC 1 -type Jackson ntegral wth generc condton on the parameters We gve the Gauss decomposton of the coeffcent matrx of the system wth a concrete bass see Theorem 41 Each entry of the decomposed matrces s wrtten as a product of bnomals and as a conseuence the determnant of the coeffcent matrx s easy to calculate explctly As an applcaton we gve a smple proof of the product formula for Gustafson s multple C n -type sum [10] We also present an explct form of the -dfference system for the BC 1 -type Jackson ntegral wth a balancng condton on the parameters We fnally gve a smple proof of the product formula for the -ntegral of Nassrallah Rahman [16] and Gustafson [9] A recent work of Rans and Sprdonov [17] contans results for the ellptc Ths paper s a contrbuton to the Proceedngs of the Workshop Ellptc Integrable Systems Isomonodromy Problems and Hypergeometrc Functons July MPIM Bonn Germany The full collecton s avalable at

2 2 M Ito hypergeometrc ntegral of a smlar type to those contaned for the BC 1 -type Jackson ntegral obtaned here 2 BC 1 -type Jackson ntegral Throughout ths paper we assume 0 < < 1 and denote the -shfted factoral for all ntegers N by x := 1 x and x N := x; / N x; =0 Let OC be the set of holomorphc functons on the complex multplcatve group C A functon f on C s sad to be symmetrc or skew-symmetrc under the Weyl group acton f f satsfes f = f or f = f respectvely For ξ C and a functon f on C we defne the sum over the lattce Z ξ 0 f d := 1 ν= f ν ξ whch provded the ntegral converges we call the Jackson ntegral For an arbtrary postve nteger s we defne the functon Φ and the skew-symmetrc functon on C as follows: Φ := 1 2 αm /a m a m := 21 where a m = αm For a symmetrc functon ϕ on C and a pont ξ C we defne the followng sum over the lattce Z: ξ 0 ϕφ d whch we call the Jackson ntegral of type BC 1 and s smply denoted by ϕ ξ By defnton the sum ϕ ξ s nvarant under the shft ξ ν ξ for ν Z Let Θ be the functon on C defned by Θ := s α 1 α θ 2 θa m where θ denotes the functon / whch satsfes θ = θ/ and θ/ = θ 22 For a symmetrc functon ϕ OC we denote the functon ϕ /Θ by ϕ We call ϕ the regulared Jackson ntegral of type BC 1 whch satsfes the followng: Lemma 21 Assume α 1 + α α Z If ϕ OC s symmetrc then the functon ϕ s symmetrc and holomorphc on C Proof See [15 Proposton 22] For an arbtrary meromorphc functon ϕ on C we defne the functon ϕ on C by ϕ := ϕ Φ Φ ϕ

3 A Frst Order -Dfference System for the BC 1 -Type Jackson Integral 3 In partcular from 21 the functon Φ/Φ s the ratonal functon Φ Φ = m s+1 a m The followng proposton wll be used for the proof of the key euaton Theorem 31: Lemma 22 If ξ 0 Φϕ d Proof See [11 Lemma 51] for nstance s convergent for ϕ OC then ξ 0 Φ ϕ d = 0 3 Key euaton In ths secton we wll present a key euaton to construct the dfference euatons for the BC 1 -type Jackson ntegral Before we state t we ntroduce the functon ex; y defned by ex; y := x + x y + y whch s expressed by the product form ex; y = y x1 xy xy The basc propertes of ex; y are the followng: ex; = ex; y + ey; 31 ex; y = ey; x ex; y = e x ; y 32 ex; ye; w ex; ey; w + ex; wey; = 0 33 Remark 31 As we wll see later euaton 33 s gnorable n the case a 1 a 2 a 1 whle euaton 31 s gnorable n the case a 1 a 2 a = 1 For functons f g on C the functon fg on C s defned by fg := fg for C Set e := e; a and e 1 e 2 e s := e 1 e 2 e s The symbol e 1 ê k e s s eual to e 1 e k e k+1 e s The key euaton s the followng: Theorem 31 Suppose a a f If { 1 2 s } {1 2 2s + 2} then C 0 e 1 e 2 e s ξ + s C k e 1 ê k e s ξ = 0 where the coeffcents C 0 and C k 1 k s are gven by C 0 = 1 a 2 a and C k = k a m a s k 2 k ea k ; a l 1 l s l k

4 4 M Ito Proof Wthout loss of generalty t suffces to show that s C 0 e 1 e 2 e s ξ + C e 1 ê e s ξ = 0 34 =1 where the coeffcents C 0 and C are gven by C 0 = 1 a 2 a and C = a m a s 2 ea ; a k 35 1 k s k Set F = a m and G = m Then from Lemma 22 t follows that ξ 0 Φ F d s+1 = 0 where F F G s+1 = s+1 36 Snce F G/ s+1 s skew-symmetrc under the reflecton t s dvsble by and we can expand t as F G s+1 = C 0 e; a 1 e; a 2 e; a s + s C e; a 1 ê; a e; a s 37 where the coeffcents C wll be determned below We obtan C 0 = 1 a 2 a from the prncpal term of asymptotc behavor of 37 as + If we put = a 1 s then we have F a Ga a s 2 = C 1 k s k ea ; a k Snce F a = 0 and Ga = a m by defnton the above euaton mples 35 From 36 and 37 we obtan 34 whch completes the proof 4 The case a 1 a 2 a df ference euaton Set =1 v k := { e1 e 2 e s f k = 0 e 1 ê k e s f 1 k s 1 41 where the hat symbol denotes the term to be omtted Let T a be the dfference operator correspondng to the -shft a Theorem 41 Suppose a 1 a 2 a 1 For the BC 1 -type Jackson ntegrals f { 1 2 s } {1 2 2s + 2} and { 1 2 s } then the frst order vector-valued -dfference euaton wth respect to the bass {v 0 v 1 v s } defned by 41 s gven by T a v 0 ξ v s ξ = v 0 ξ v s ξ B 42

5 A Frst Order -Dfference System for the BC 1 -Type Jackson Integral 5 where B = UL Here U and L are the s s matrces defned by c c 1 d 1 1 U = c 2 L = d 2 1 cs d s 1 where c 0 = a m a s 1 a 2 a 2 s l=1 ea l ; a and c k = ea k ; a d k = k a m a k s 1 a 1 a 2 a 1 a 2 k ea ; a k 1 l s l k ea l ; a k 43 for k = 1 2 s 1 Moreover det B = a m a s 1 a 2 a 2 Proof Euaton 42 s rewrtten as T a v 0 ξ v s ξ L = v 0 ξ v s ξ U where 1 d 1 1 L = d 2 1 d s 1 Snce T a v ξ = e v ξ the above euaton s euvalent to and s e v 0 ξ d k e v k ξ = c 0 v 0 ξ 44 e v k ξ = v 0 ξ + c k v k ξ for k = 1 2 s 1 45 whch are to be proved Euaton 44 s a drect conseuence of 32 and Theorem 31 f a 1 a 2 a 1 Euaton 45 s trval usng e; a = e; a k + ea k ; a from 31 Lastly det B = det U det L = c 0 c 1 c s whch completes the proof Snce the functon Θ satsfes T a Θ = a Θ we mmedately have the followng from Theorem 41:

6 6 M Ito Corollary 41 Suppose a 1 a 2 a 1 For the regulared BC 1 -type Jackson ntegrals f { 1 2 s } {1 2 2s + 2} and { 1 2 s } then the frst order vector-valued -dfference euaton wth respect to the bass {v 0 v 1 v s } s gven by T a v 0 ξ v s ξ = v 0 ξ v s ξ B 46 where B = a c c 1 c 2 cs 1 d 1 1 d 2 1 d s 1 and c and d are gven by 43 In partcular the dagonal entres of the upper trangular part are wrtten as c 0 a = 1 a 2 a 2 s l=1 a m l a l a and c k = k a a k a for = 1 2 s 1 Moreover det B = 2 a m 1 a 2 a 47 Remark 41 The -dfference system for the BC n -type Jackson ntegral s dscussed n [3] for ts rank and n [4 5] for the explct expresson of the determnant of the coeffcent matrx of the system On the other hand though t s only for the BC 1 -type Jackson ntegral the coeffcent matrx n ts Gauss decomposton form s obtaned explctly only n the present paper 42 Applcaton The am of ths subsecton s to gve a smple proof of Gustafson s multple C n -type summaton formula Corollary 42 The pont of the proof s to obtan a recurrence relaton of Gustafson s multple seres of C n -type Before we state the recurrence relaton we frst gve the defnton of the multple seres of C n -type 1 x G For = 1 n C n we set Φ G := Cn := n =1 n =1 1/2 αm 1 2 m a m 1 <k n 1 / k 1 k

7 A Frst Order -Dfference System for the BC 1 -Type Jackson Integral 7 where αm = a m For an arbtrary ξ = ξ 1 ξ n C n we defne the -shft ξ ν ξ by a lattce pont ν = ν 1 ν n Z n where ν ξ := ν 1 ξ 1 νn ξ n C n For ξ = ξ 1 ξ n C n we defne the sum over the lattce Z n by 1 ξ G := 1 n Φ G ν ξ Cn ν ξ ν Z n whch we call the BC n -type Jackson ntegral Moreover we set 1 ξ G := 1 ξ G /Θ G ξ where Θ G ξ := n =1 ξ α 1 α 2 α θξ 2 θa m ξ 1 <k n θξ /ξ k θξ ξ k By defnton t can be confrmed that 1 ξ G s holomorphc on C n see [4 Proposton 37] and we call t the regulared BC n -type Jackson ntegral In partcular f we assume s = n we call 1 ξ G the regulared Jackson ntegral of Gustafson s C n -type whch s n partcular a constant not dependng on ξ C n Remark 42 For further results on BC n -type Jackson ntegrals see [ ] for nstance Now we state the recurrence relaton for Gustafson s sum 1 ξ G Proposton 41 Suppose s = n and x = x 1 x n C n The sum 1 x G satsfes T a 1 x G = 1 x G 2 2n+2 a m 1 a 2 a 2n+2 for = 1 2 2n + 2 Proof We assume s = n for the bass {v 0 v 1 v n } of the BC 1 -type Jackson ntegral Let P be the transton matrx from the bass {v 0 v 1 v n } to {χ n χ n 2 χ 0 }: χ n χ n 2 χ 0 = v 0 v 1 v n P where χ s the rreducble character of type C 1 defned by χ = +1 for = From 46 t follows that so that T a χ n ξ χ 0 ξ = χ n ξ χ 0 ξ P BP 49 T a det χ n x 1 n = det χ n x 1 n det B 410 for x = x 1 x n C n By defnton the relaton between the determnant of the BC 1 - type Jackson ntegrals and the Jackson ntegral of Gustafson s C n -type tself s gven as det χ n x 1 n = 1 x G 1 <k n θx /x k θx x k x 411 whch s also referred to n [14] From 410 and 411 we obtan T a 1 x G = 1 x G det B where det B has already been gven n 47

8 8 M Ito Remark 43 The explct form of the coeffcent matrx of the system 49 s gven n [1] or [3] Corollary 42 Gustafson [10] Suppose s = n and x = x 1 x n C n Then the sum 1 x G s wrtten as n a 1 x G = 1 n 1 < 2n+2 1 a 2 a 2n+2 Proof By repeated use of the recurrence relaton n Proposton 41 usng the asymptotc behavor of the Jackson ntegral as the boundary condton of the recurrence relaton we eventually obtan Corollary 42 See [12] for further detals about the proof Remark 44 From 411 and Corollary 42 we see det χ s x s 1 s = 1 s 1 < 1 a 2 a a 1 <k s θx /x k θx x k x whch s non-degenerate under generc condton Ths ndcates that the set {χ s χ s 2 χ 0 } s lnearly ndependent And we eventually know the rank of the -dfference system wth respect to ths bass s s so are the ranks of the systems 42 and 46 5 The case a 1 a 2 a = 1 51 Ref lecton euaton Theorem 51 Suppose a 1 a 2 a = 1 Let v k k = 1 2 s be the functons defned by 41 for the fxed ndces 1 2 s {1 2 2s + 2} If 1 2 { 1 2 s } then e 1 v 1 ξ e 1 v s ξ = e 2 v 1 ξ e 2 v s ξ M where M = M 2 NM 1 Here M and N are the matrces defned by γ 1 1 σ 2 σ 3 σ s γ 2 1 τ 2 M = γ 3 1 N = τ 3 γ s 1 τs where the entres of the above matrces are gven by Moreover σ k = ea 1 ; a 2 ea k ; a 2 τ k = ea 1 ; a k ea 2 ; a k γ k = as 2 k a m a s k 2 k a m 1 l s l k ea ; a l ea k ; a l det M = as a m a s a m

9 A Frst Order -Dfference System for the BC 1 -Type Jackson Integral 9 Proof Frst we wll prove the followng: e 1 v 1 ξ e 1 v 2 ξ e 1 v s ξ M 1 = v 0 ξ e 2 v 2 ξ e 2 v s ξ N 52 e 2 v 1 ξ e 2 v 2 ξ e 2 v s ξ M 2 = v 0 ξ e 2 v 2 ξ e 2 v s ξ 53 whch are euvalent to and s γ k e v k ξ = v 0 ξ 54 e 1 v k ξ = σ k v 0 ξ + τ k e 2 v k ξ k = 2 s 1 55 Under the condton a 1 a 2 a = 1 Euaton 54 s a drect conseuence of Theorem 31 Euaton 55 s trval from the euaton e; a = e; a ea ; a k ea ; a k + e; a k ea ; a ea k ; a whch was gven n 33 From 52 and 53 t follows M = M 2 NM 1 Moreover we obtan 2 2 det M = det M 1 det N det M 2 = γ 1 1 τ 2 τ s = as 2 γ 12 a s a m 2 a m whch completes the proof Corollary 51 Suppose s = 2 and the condton a 6 = the BC 1 -type Jackson ntegral 1 ξ s T a 1 ξ = 1 ξ a a 6 1 l 5 l a l 1 6 a l Proof Wthout loss of generalty t suffces to show that T a1 1 ξ = 1 ξ a 1 a a l 1 l=2 6 a l a 1 a 2 a 3 a 4 a 5 The recurrence relaton for for = Set Ja 1 a 2 a 3 a 4 a 5 a 6 ; ξ := 1 ξ Under the condton a 1 a 2 a 3 a 4 a 5 a 6 = 1 we have J 1 a 2 a 3 a 4 a 5 a 6 ; ξ = Ja 1 a 2 a 3 a 4 a 5 6 ; ξ a2 6 a l=2 1 a l 6 a l from Theorem 51 by settng 1 = 1 and 2 = 6 We now replace a 6 by a 6 n the above euaton Then under the condton a 1 a 2 a 3 a 4 a 5 a 6 = 1 we have J 1 a 2 a 3 a 4 a 5 a 6 ; ξ = Ja 1 a 2 a 3 a 4 a 5 a 6 ; ξ a 1 a a l 1 l=2 6 a l Snce T a1 1 ξ = J 1 a 2 a 3 a 4 a 5 a 6 ; ξ under ths condton a 6 = 1 a 2 a 3 a 4 a 5 we obtan 56 whch completes the proof

10 10 M Ito Corollary 52 Suppose s = n + 1 and the condton a 2n+4 = a 1 a 2 a 2n+3 Then the recurrence relaton for Gustafson s sum 1 x G where x = x 1 x n C n s gven by T a 1 x G = 1 x G a a 2n+4 1 l 2n+3 l a l 1 l a 2n+4 Proof Fx s = n + 1 For the BC 1 -type Jackson ntegral we frst set Ja 1 a 2 a 2n+4 ; x := det χ n x 1 n for = 1 2 2n + 3 where χ s defned n 48 under no condton on a 1 a 2 a 2n+4 By the defnton of Φ we have J 1 a 2 a 2n+4 ; x = det e 1 χ n x 1 n Let Q be the transton matrx from the bass {v 1 v 2 v n } to {χ n χ n 2 χ 0 } e χ n χ n 2 χ 0 = v 1 v 2 v n Q Under the condton a 1 a 2 a 2n+4 = 1 from Theorem 51 wth 1 = 1 and 2 = 2n+4 t follows that so that e 1 v 1 ξ e 1 v n ξ = e 2n+4 v 1 ξ e 2n+4 v n ξ M e 1 χ n ξ e 1 χ 0 ξ = e 2n+4 χ n ξ e 2n+4 χ 0 ξ Q MQ Ths ndcates that det e 1 χ n x 1 n = det e 2n+4 χ n x det M 1 n From 51 and the above euaton we have J 1 a 2 a 2n+4 ; x = Ja 1 a 2 2n+4 ; x a2n+4 a 1 n+1 2n+3 l=2 l a 1 l a 2n+4 under the condton a 1 a 2 a 2n+4 = 1 We now replace a 2n+4 by a 2n+4 n the above euaton Then we have J 1 a 2 a 2n+4 ; x = Ja 1 a 2 a 2n+4 ; x a 1 a 2n+4 under the condton a 1 a 2 a 2n+4 = 1 Snce T a1 det χ n x 1 n = J 1 a 2 a 2n+4 ; x f a 1 a 2 a 2n+4 = we have T a1 det χ n x 1 n = det χ n x 1 n a 1 a 2n+4 2n+3 l=2 2n+3 l=2 l a 1 1 l a 2n+4 l a 1 1 l a 2n+4

11 A Frst Order -Dfference System for the BC 1 -Type Jackson Integral 11 On the other hand f x = x 1 x n C n then by defnton we have det χ n x 1 n = 1 x G whch s also referred to n [14] Therefore under the condton a 2n+4 = 1 a 2 a 2n+3 we obtan T a1 1 x G = 1 x G a 1 a 2n+4 2n+3 l=2 l a 1 1 l a 2n+4 Snce the same argument holds for parameters a 2 a 2n+3 we can conclude Corollary 52 Remark 51 If we take ξ = a = 1 6 and add the termnatng condton a 1 a 2 = N N = 1 2 to the assumptons of Corollary 51 then the fnte product expresson of 1 ξ whch s euvalent to Jackson s formula for termnatng 8 φ 7 seres [8 p 43 euaton 262] s obtaned from fnte repeated use of Corollary 51 In the same way f we take a sutable x and add the termnatng condton to the assumptons of Corollary 52 then the fnte product expresson of 1 x G whch s euvalent to the Jackson type formula for termnatng multple 8 φ 7 seres see [7 Theorem 4] or [6 p 231 euaton 44] for nstance s obtaned from fnte repeated use of Corollary Applcaton The am of ths subsecton s to gve a smple proof of the followng propostons proved by Nassrallah and Rahman [16] and Gustafson [9] Proposton 51 Nassrallah Rahman Assume a < 1 for 1 5 If a 6 = then 1 2π T a a =1 d = where T s the unt crcle taken n the postve drecton a 1 a 2 a 3 a 4 a a k 57 a a 1 < 5 Proof We denote the left-hand sde of 57 by Ia 1 a 2 a 3 a 4 a 5 By resdue calculaton 5 θ 6 Ia 1 a 2 a 3 a 4 a 5 = Res θ 2 6 =a k ν ν=0 5 θ 1 2 m d 58 a m a m 5 = Res θ 6 θ 2 d ak 6 =a k 5 θ 1 2 m d a m 0 a m = 5 R k 1 a k 59 where θ 6 R k := Res θ 2 =a k 5 θ a m d = θ k 2 a k 6 a 1 m 5 m k θ a 2 k θ a m a k

12 12 M Ito whose recurrence relaton s T a R k = a a 6 Rk 510 for 1 k 5 whch s obtaned usng 22 From and Corollary 51 we obtan the recurrence relaton for Ia 1 a 2 a 3 a 4 a 5 as T a Ia 1 a 2 a 3 a 4 a 5 = Ia 1 a 2 a 3 a 4 a 5 By repeated use of the above relaton we obtan and Ia 1 a 2 a 3 a 4 a 5 = = 5 1 l 5 l a l 1 6 a l 6 a k 2N I N a 1 N a 2 N a 3 N a 4 N a 5 a a 2N 1 < a k lm a a I N a 1 N a 2 N a 3 N a 4 N a 5 N 1 < 5 lm I N a 1 N a 2 N a 3 N a 4 N 1 a 5 = N 2π 2 2 d T = 2 Ths completes the proof Remark 52 Strctly speakng the resdue calculaton 58 reures that 1 I ε := 2π d 0 f ε =ε 5 a a =1 whch can be shown n the followng way We frst take ε = N ε for ε > 0 and postve nteger N If we put 6 F := a a =1 then we have F = G 1 G 2 where G 1 = θ 6 θ 2 5 θ G 2 = a = =1 a Snce G 1 s a contnuous functon on the compact set = ε and s nvarant under the -shft under the condton a 6 = 1 a 2 a 3 a 4 a 5 G 1 s bounded on = N ε G 2 s also bounded because G 2 1 f 0 Thus there exsts C > 0 such that F < C If we put = εe 2π τ then I ε < 1 whch proves F εe 2π τ dτ < C 1 0 εe 2π τ dτ = Cε 0 ε 0

13 A Frst Order -Dfference System for the BC 1 -Type Jackson Integral 13 Proposton 52 Gustafson [9] Assume a < 1 for 1 2n + 3 If a 2n+4 = then 1 n n 2π 2n+4 2n T n 2n+3 = n 1 < n 2 n n! 2n+3 =1 2s+4 a k 1 < 2s+3 a k ak d 1 d n 1 n a 1 a 2 a 2n+3 a a 512 where T n s the n-fold drect product of the unt crcle traversed n the postve drecton The proof below s based on an dea usng resdue computaton due to Gustafson [10] whch s done for the case of the hypergeometrc ntegral under no balancng condton Here we wll show that hs resdue method s stll effectve even for the ntegral under the balancng condton a 1 a 2 a 2n+4 = In partcular ths s dfferent from hs proof n [9] Proof Let L be the set of ndces defned by L := {λ = λ 1 λ n ; 1 λ 1 < λ 2 < < λ n 2n + 3} Set a µ := a µ1 a µn C n for µ = µ 1 µ n L We denote the left-hand sde of 512 by Ia 1 a 2 a 2n+3 By resdue calculaton we have Ia 1 a 2 a 2n+3 = µ L R µ 1 a µ G 513 where the coeffcents R µ µ L are n θ 2n+4 R µ := Res θ 2 1 =a µ1 2n+3 =1 n=a µn θ a m 1 <k n θ k θ k d 1 1 d n n The recurrence relaton for R µ s T a R µ = a a 2n+4 Rµ 514 From and Corollary 52 we obtan the recurrence relaton for Ia 1 a 2 a 2n+3 as T a Ia 1 a 2 a 2n+3 = Ia 1 a 2 a 2n+3 By repeated use of the above relaton we obtan 1 l 2n+3 l a l 1 2n+4 a l Ia 1 a 2 a 2n+3 = 2n+3 2n+4 a k 2N 1 < 2n+3 a a 2N I N a 1 N a 2 N a 2n+3

14 14 M Ito = 2n+3 2n+4 a k 1 < 2n+3 lm a a I N a 1 N a 2 N a 2n+3 N and lm I N a 1 N a 2 N 1 n a 2n+3 = N 2π T n 1 < n n 2 =1 d d n = 2n n! n n Ths completes the proof References [1] Aomoto K A normal form of a holonomc -dfference system and ts applcaton to BC 1-type Int J Pure Appl Math [2] Aomoto K Ito M On the structure of Jackson ntegrals of BC n type and holonomc -dfference euatons Proc Japan Acad Ser A Math Sc [3] Aomoto K Ito M Structure of Jackson ntegrals of BC n type Tokyo J Math [4] Aomoto K Ito M BC n-type Jackson ntegral generaled from Gustafson s C n-type sum J Dfference Eu Appl [5] Aomoto K Ito M A determnant formula for a holonomc -dfference system assocated wth Jackson ntegrals of type BC n Adv Math to appear do:101016/am [6] van Deen JF Sprdonov VP Modular hypergeometrc resdue sums of ellptc Selberg ntegrals Lett Math Phys [7] Dens RY Gustafson RA An SUn -beta ntegral transformaton and multple hypergeometrc seres denttes SIAM J Math Anal [8] Gasper G Rahman M Basc hypergeometrc seres 2nd ed Encyclopeda of Mathematcs and ts Applcatons Vol 96 Cambrdge Unversty Press Cambrdge 2004 [9] Gustafson RA Some -beta and Melln Barnes ntegrals wth many parameters assocated to the classcal groups SIAM J Math Anal [10] Gustafson RA Some -beta and Melln Barnes ntegrals on compact Le groups and Le algebras Trans Amer Math Soc [11] Ito M -dfference shft for a BC n type Jackson ntegral arsng from elementary symmetrc polynomals Adv Math [12] Ito M Another proof of Gustafson s C n-type summaton formula va elementary symmetrc polynomals Publ Res Inst Math Sc [13] Ito M A multple generalaton of Slater s transformaton formula for a very-well-posed-balanced 2rψ 2r seres Q J Math [14] Ito M Okada S An applcaton of Cauchy Sylvester s theorem on compound determnants to a BC n-type Jackson ntegral n Proceedngs of the Conference on Parttons -Seres and Modular Forms Unversty of Florda March to appear [15] Ito M Sanada Y On the Sears Slater basc hypergeometrc transformatons Ramanuan J [16] Nassrallah B Rahman M Proecton formulas a reproducng kernel and a generatng functon for -Wlson polynomals SIAM J Math Anal [17] Rans EM Sprdonov VP Determnants of ellptc hypergeometrc ntegrals Funct Anal Appl to appear arxv: