Connection problem for an extension of q-hypergeometric systems
aa r X i v : . [ m a t h . C A ] F e b Connection problem for an extension of q -hypergeometric systems Takahiko NobukawaDepartmant of Mathematics, Graduate School of Science, Kobe University,1-1, Rokkodai, Nada-ku, Kobe 657-8501, JapanEmail: [email protected]: q -difference equations; Barnes-type integral;connection matrices; Yang-Baxter equationMSC2020: 33D70; 39A13 Abstract
We solve the connection problem of a certain system of linear q -difference equations re-cently introduced by K. Park. The result contains the connection formulas of the q -Lauricellahypergeometric function ϕ D and those of the q -generalized hypergeometric function N +1 ϕ N as special cases. Our result gives a simultaneous extension of them. In [1], a hypergeometric function F L,N , which is a certain extension of hypergeometric functions,was defined by T. Tsuda. He also obtained a Hamiltonian system H L,N , which describes anisomonodromic deformation of an L × L Fuchsian system on P with N + 3 regular singularities,and which has particular solutions in terms of the function F L,N . In [2, 3], a q -analog ofTsuda’s result was obtained by K. Park. Namely, she defined a q -hypergeometric function F N,M , which is given by (2.7) below, and a system P N,M as a q -analog of the function F N +1 ,M and the system H N +1 ,M , respectively. The function F N,M converges locally and satisfies linear q -difference equations, given by (2.14) and (2.15) below. Thus the function F N,M must becontinued analytically to C M .Our aim is to obtain connection formulas of the function F N,M , and to solve the connectionproblem of the q -difference equations (2.14), (2.15). The main result is Theorem 3.1 in subsection3.3, and Proposition 3.1, 3.2 and 3.4 are important steps for deriving Theorem 3.1.The function F N,M is an extension of the q -Lauricella function ϕ D and generalized q -hypergeometric function N +1 ϕ N (see (2.8), (2.9)). Thus our results contain the connectionformulas of ϕ D and also of N +1 ϕ N . Connection formulas of N +1 ϕ N were obtained by G. N.Watson [4], and connection formulas of the Lauricella function F D were obtained by P. O. M.Olsson [5] and by S. I. Bezrodnykh [6]. Our results include Watson’s formula and a q -analog ofsome of Olsson’s formulas and Bezrodnykh’s formulas.The contents of this paper are as follows. In section 2, we give notations and the propeties ofthe function F N,M and the system of q -difference equations satisfied by F N,M . In section 3, wesolve the connection problem of the system. In subsection 3.1, we show a Barnes-type integralrepresentation of the function F N,M . In subsection 3.2, we show fundamental solutions of the q -difference system. In subsection 3.3, we give the matrices which connect the fundamentalsolution with the other fundamental solution. In section 4, we obtain an elliptic solution of theYang-Baxter equation as an application of subsection 3.3.1 Preliminaries
In this paper, we fix q ∈ C so that 0 < | q | <
1. We use the following notations throughout thepaper: ( a ) ∞ = ∞ Y k =0 (1 − aq k ) , (2.1)( a ) m = ( a ) ∞ ( aq m ) ∞ , (2.2)( a , . . . , a n ) m = ( a ) m · · · ( a n ) m , (2.3) θ ( x ) = ( x, q/x ) ∞ , (2.4)and S n is the symmetric group of degree n . In addition, t A is the transpose of a matrix A .We also use the notation abc/def g for the fraction ( abc ) / ( def g ). Moreover, for a multi-index m = ( m , . . . , m M ), we often use the notations | m | = M X i =1 m i , (2.5) m ( l ) = l X i =1 m i − M X i = l +1 m i , (2.6)where 0 ≤ l ≤ M . Here, the empty sum is equal to 0, and the empty product is equal to 1. Definition 2.1 ([2], Definition 2.1) . We define the function F N,M as F N,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19) = X m i ≥ N Y j =1 ( a j ) | m | ( c j ) | m | M Y i =1 ( b i ) m i ( q ) m i M Y i =1 t m i i , (2.7)where 1 ≤ j ≤ N and 1 ≤ i ≤ M . This series F N,M converges in the region | t i | < N = 1 or M = 1, the function (2.7) is equal to the q -Appell Lauricella function ϕ D or the generalized q -hypergeometric function N +1 ϕ N , respectively: F ,M (cid:18) a, { b i } c ; { t i } (cid:19) = X m i ≥ ( a ) | m | ( c ) | m | M Y i =1 ( b i ) m i ( q ) m i M Y i =1 t m i i = ϕ D (cid:18) a, { b i } c ; { t i } (cid:19) , (2.8) F N, (cid:18) { a j } , b { c j } ; t (cid:19) = X m ≥ N Y j =1 ( a j ) m ( c j ) m ( b ) m ( q ) m t m = N +1 ϕ N (cid:18) { a j } , b { c j } ; t (cid:19) . (2.9) Proposition 2.1 ([2], Proposition 2.1) . The series F N,M satisfies the relation F N,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19) = N Y j =1 ( a j ) ∞ ( c j ) ∞ M Y i =1 ( b i t i ) ∞ ( t i ) ∞ F M,N (cid:18) { t i } , { c j /a j }{ b i t i } ; { a j } (cid:19) . (2.10) Remark 2.1.
When N = 1 or M = 1, the relation (2.10) was obtained in [7], pp. 621 (4.1): ϕ D (cid:18) a, { b i } c ; { t i } (cid:19) = ( a ) ∞ ( c ) ∞ M Y i =1 ( b i t i ) ∞ ( t i ) ∞ M +1 ϕ M (cid:18) { t i } , c/a { b i t i } ; a (cid:19) . (2.11)2e can interpret the relation (2.10) as a Jackson integral representation of the series F N,M as follows.
Corollary 2.1 ([2], Corollary 2.1) . With a j = q α j , the relation (2.10) can be rewritten as F N,M (cid:18) { q α j } , { b i }{ c j } ; { t i } (cid:19) = N Y j =1 ( q α j , c j /q α j ) ∞ ( c j , q ) ∞ Z · · · Z N Y j =1 ( z α j − j − q ( qz j ) ∞ ( c j z j /q α j ) ∞ ) M Y i =1 ( b i t i z · · · z N ) ∞ ( t i z · · · z N ) ∞ d q z · · · d q z N , (2.12)where the Jackson integral is defined as Z c f ( z ) d q z = c (1 − q ) X m ≥ f ( cq m ) q m , (2.13)for c ∈ C . Proposition 2.2 ([2], Proposition 2.2) . The series F = F N,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19) satisfies the q -difference equations t s N Y j =1 (1 − a j T )(1 − b s T s ) − N Y j =1 (1 − c j q − T )(1 − T s ) F = 0 (1 ≤ s ≤ M ) , (2.14) { t r (1 − b r T r )(1 − T s ) − t s (1 − b s T s )(1 − T r ) }F = 0 (1 ≤ r < s ≤ M ) , (2.15)where T s is the q -shift operator for the variable t s and T = Q Ms =1 T s .In this paper, we use the notation E N,M for the system of q -difference equations (2.14) and(2.15) Theorem 2.1 ([3], Theorem 4.1) . The rank of E N,M is N M + 1. q -difference equations In this section, we consider the connection problem of the system E N,M . First, we show anotherintegral representation: a Barnes-type integral representation of the series F N,M . Second, bycalculating this Barnes-type integral, we find solutions of the system E N,M , which convergelocally. Finally, we calculate connection matrices. F N,M
In this subsection, we show a Barnes-type integral representation of the series F N,M and wecalculate the analytic continuation of F N,M . We assume | a j | < ≤ j ≤ N in thissubsection. Definition 3.1.
Take an
R > | a j | R M < ≤ j ≤ N . Take paths C i (1 ≤ i ≤ M ) which satisfy the following conditions • C i is a positively oriented simple closed curve. • z i = 0 is in the interior of C i . 3 For a non-negative integer m , z i = q m is in the interior of C i . • For a non-negative integer m , z i = q − m /b i is in the exterior of C i . • C i ⊂ { z i ; 0 < | z i | < R } .We define the function ϕ N,M as a Barnes-type integral: ϕ N,M = ϕ N,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19) = 1(2 π √− M Z C · · · Z C M ψ N,M dz z · · · dz M z M , (3.1)where ψ N,M = N Y j =1 ( c j z · · · z M ) ∞ ( a j z · · · z M ) ∞ M Y i =1 ( qz i /t i , t i /z i ) ∞ ( b i z i , /z i ) ∞ . (3.2)If | t i | are small, then we can calculate ϕ N,M by applying the residue theorem.
Lemma 3.1 ([8], pp.125-126) . Let P ( z ) = ( a z, . . . , a n z, b /z, . . . , b m /z ) ∞ ( c z, . . . , c n z, d /z, . . . , d m /z ) ∞ (3.3)and let C be a positively oriented simple closed curve such that z = 0 is in the interior of C and C does not pass through any of the poles of P ( z ).1. If (cid:12)(cid:12)(cid:12)(cid:12) b · · · b m d · · · d m (cid:12)(cid:12)(cid:12)(cid:12) < , (3.4)then 12 π √− Z C P ( z ) dzz = X w ∈ D Res (cid:18) P ( z ) z , z = w (cid:19) , (3.5)where D = { w ∈ C \{ } ; w is the pole of P ( z ) and is in the interior of C } .2. If (cid:12)(cid:12)(cid:12)(cid:12) a · · · a n c · · · c n (cid:12)(cid:12)(cid:12)(cid:12) < , (3.6)then 12 π √− Z C P ( z ) dzz = − X w ∈ D ′ Res (cid:18) P ( z ) z , z = w (cid:19) , (3.7)where D ′ = { w ∈ C \{ } ; w is the pole of P ( z ) and is in the exterior of C } . Proof.
1. Let δ be a positive number such that δ = | d j q l | for 1 ≤ j ≤ m and δ = | q − l /c j | for1 ≤ j ≤ n and l = 0 , , , . . . . Also let C ( l ) be the circle | z | = δ | q | l , where l = 0 , , . . . . Then C ( l ) does not pass through any of the poles of P ( z ). By setting z = δq l e √− θ we obtain that Z C ( l ) P ( z ) dzz = √− Z π P ( δq l e √− θ ) dθ, (3.8)4nd we have that (cid:12)(cid:12)(cid:12) P ( δq l e √− θ ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( a δq l e √− θ , . . . , a n δq l e √− θ , b /δq l e √− θ , . . . , b m /δq l e √− θ ) ∞ ( c δq l e √− θ , . . . , c n δq l e √− θ , d /δq l e √− θ , . . . , d m /δq l e √− θ ) ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( a δe √− θ , . . . , a n δe √− θ , b /δe √− θ , . . . , b m /δe √− θ ) ∞ ( c δe √− θ , . . . , c n δe √− θ , d /δe √− θ , . . . , d m /δe √− θ ) ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( c δe √− θ , . . . , c n δe √− θ , qδe √− θ /b , . . . , qδe √− θ /b m ) l ( a δe √− θ , . . . , a n δe √− θ , qδe √− θ /d , . . . , qδe √− θ /d m ) l (cid:18) b · · · b m d · · · d m (cid:19) l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O (cid:12)(cid:12)(cid:12)(cid:12) b · · · b m d · · · d m (cid:12)(cid:12)(cid:12)(cid:12) l ! ( l → ∞ ) . (3.9)So if (cid:12)(cid:12)(cid:12)(cid:12) b · · · b m d · · · d m (cid:12)(cid:12)(cid:12)(cid:12) < , (3.10)then lim l →∞ Z C ( l ) P ( z ) dzz = 0 . (3.11)Hence, by applying the residue theorem to the region between C and C ( l ) for large l and letting l → ∞ , we find 12 π √− Z C P ( z ) dzz = X w ∈ D Res (cid:18) P ( z ) z , z = w (cid:19) . (3.12)By using the inversion z ′ = z − , we obtain part 2 of Lemma 3.1 similarly. Proposition 3.1. If | t i | < ≤ i ≤ M ), then ϕ N,M = N Y j =1 ( c j ) ∞ ( a j ) ∞ M Y i =1 θ ( t i )( b i , q ) ∞ F N,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19) . (3.13) Proof.
First, we consider 12 π √− Z C ψ N,M dz z . (3.14)By the conditions of C , the poles of ψ N,M in the interior of C are only z = q m ( m =0 , , , . . . ). So, by Lemma 3.1, we obtain12 π √− Z C ψ N,M dz z = X m ≥ Res (cid:18) ψ N,M z , z = q m (cid:19) , (3.15)if | t | <
1. Since Res (cid:18) z (1 /z ) ∞ , z = q m (cid:19) = ( − m q ··· + m ( q ) m ( q ) ∞ , ( m = 0 , , , . . . ) , (3.16)5t follows that12 π √− Z C ψ N,M dz z = θ ( t )( b , q ) ∞ X m ≥ N Y j =1 ( c j q m z · · · z M ) ∞ ( a j q m z · · · z M ) ∞ M Y i =2 ( qz i /t i , t i /z i ) ∞ ( b i z i , /z i ) ∞ ( b ) m ( q ) m t m . (3.17)Second, we consider 1(2 π √− Z C Z C ψ N,M dz z dz z . (3.18)If | t | <
1, then we have1(2 π √− Z C Z C ψ N,M dz z dz z = 12 π √− Z C θ ( t )( b , q ) ∞ X m ≥ N Y j =1 ( c j q m z · · · z M ) ∞ ( a j q m z · · · z M ) ∞ M Y i =2 ( qz i /t i , t i /z i ) ∞ ( b i z i , /z i ) ∞ ( b ) m ( q ) m t m dz z = θ ( t )( b , q ) ∞ X m ≥ π √− Z C N Y j =1 ( c j q m z · · · z M ) ∞ ( a j q m z · · · z M ) ∞ M Y i =2 ( qz i /t i , t i /z i ) ∞ ( b i z i , /z i ) ∞ dz z ( b ) m ( q ) m t m . (3.19)Similar to the above calculation, we can calculate12 π √− Z C N Y j =1 ( c j q m z · · · z M ) ∞ ( a j q m z · · · z M ) ∞ M Y i =2 ( qz i /t i , t i /z i ) ∞ ( b i z i , /z i ) ∞ dz z = θ ( t )( b , q ) ∞ X m ≥ N Y j =1 ( c j q m + m z · · · z M ) ∞ ( a j q m + m z · · · z M ) ∞ M Y i =3 ( qz i /t i , t i /z i ) ∞ ( b i z i , /z i ) ∞ ( b ) m ( q ) m t m (3.20)if | t | <
1. Thus we have1(2 π √− Z C Z C ψ N,M dz z dz z = Y i =1 θ ( t i )( b i , q ) ∞ X m ,m ≥ N Y j =1 ( c j q m + m z · · · z M ) ∞ ( a j q m + m z · · · z M ) ∞ M Y i =3 ( qz i /t i , t i /z i ) ∞ ( b i z i , /z i ) ∞ Y i =1 ( b i ) m i ( q ) m i t m i i (3.21)if | t | , | t | <
1. Similarly, we can calculate ϕ N,M = M Y i =1 θ ( t i )( b i , q ) ∞ X m i ≥ N Y j =1 ( c j q m + ··· + m M ) ∞ ( a j q m + ··· + m M ) ∞ M Y i =1 ( b i ) m i ( q ) m i t m i i = N Y j =1 ( c j ) ∞ ( a j ) ∞ M Y i =1 θ ( t i )( b i , q ) ∞ F N,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19) (3.22)inductively if | t i | < ≤ i ≤ M ).So we can obtain the analytic continuations of F N,M by calculating ϕ N,M .6 efinition 3.2. We define series F LN,M , F L ; k,lN,M as F LN,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19) = X m i ≥ N Y j =1 ( a j /b L +1 · · · b M ) m ( L ) ( c j /b L +1 · · · b M ) m ( L ) M Y i =1 ( b i ) m i ( q ) m i L Y i =1 t m i i M Y i = L +1 (cid:18) qb i t i (cid:19) m i , (3.23) F L ; k,lN,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19) = X m i ≥ ( N Y j =1 ( qa k /c j ) m L +1 ( qa k /a j ) m L +1 L Y i =1 ( b i ) m i ( q ) m i l − Y i = L +1 ( b i ) m i +1 ( q ) m i +1 M Y i = l +1 ( b i ) m i ( q ) m i × ( a k /b l +1 · · · b M ) m ( l ) ( qa k /b l · · · b M ) m ( l ) L Y i =1 (cid:18) qt i b l t l (cid:19) m i l − Y i = L +1 (cid:18) qt i b l t l (cid:19) m i +1 M Y i = l +1 (cid:18) b l t l b i t i (cid:19) m i N Y j =1 c j a j qb l t l m L +1 ) , (3.24)where 0 ≤ L ≤ M , 1 ≤ k ≤ N and L + 1 ≤ l ≤ M . Here, as mentioned in Preliminaries (2.6), m ( l ) = l X i =1 m i − M X i = l +1 m i . The series (3.23) converges in n | t i | < ≤ i ≤ L ) , (cid:12)(cid:12)(cid:12) c ··· c N qa ··· a N b i t i (cid:12)(cid:12)(cid:12) < L + 1 ≤ i ≤ M ) o andthe series (3.24) converges in n(cid:12)(cid:12)(cid:12) c ··· c N qa ··· a N b l t l (cid:12)(cid:12)(cid:12) < , (cid:12)(cid:12)(cid:12) qt i b l t l (cid:12)(cid:12)(cid:12) < ≤ i ≤ l − , (cid:12)(cid:12)(cid:12) qt l b i t i (cid:12)(cid:12)(cid:12) < l + 1 ≤ i ≤ M ) o . Lemma 3.2.
Let f I ( { m i } ) = I − Y i =1 θ ( t i b i )( b i ) m i ( q, b i ) ∞ ( q ) m i I − Y i =1 (cid:18) qb i t i (cid:19) m i N Y j =1 ( c j z I · · · z M /b · · · b I − q m + ··· + m I − ) ∞ ( a j z I · · · z M /b · · · b I − q m + ··· + m I − ) ∞ + N X k =1 I − X l =1 (Y j = k ( c j /a k ) ∞ ( qa k /c j ) m ( a j /a k ) ∞ ( qa k /a j ) m ( c k /a k ) ∞ ( qa k /c k ) m ( q ) ∞ ( q ) m N Y j =1 (cid:18) c j a j (cid:19) m × l − Y i =1 θ ( t i )( b i ) m i +1 ( q, b i ) ∞ ( q ) m i +1 l − Y i =1 t m i +1 i I − Y i = l +1 θ ( t i b i )( b i ) m i ( q, b i ) ∞ ( q ) m i I − Y i = l +1 (cid:18) qb i t i (cid:19) m i × θ ( t l a k z I · · · z M q m l,I − /b l +1 · · · b I − )( b l · · · b I − /a k z I · · · z M q m l,I − , a k z I · · · z M q m l,I − /b l +1 · · · b I − ) ∞ ) , (3.25)for I = 1 , , . . . , M + 1. Here, m l,I − = l X i =1 m i − I − X i = l +1 m i . (3.26)Then we have 12 π √− Z C ( qz /t , t /z ) ∞ ( b z , /z ) ∞ f dz z = X m ≥ f ( m ) (3.27)if (cid:12)(cid:12)(cid:12) c ··· c N qa ··· a N b t (cid:12)(cid:12)(cid:12) <
1, and for I = 2 , . . . , M , we have X m ,...,m I − ≥ π √− Z C I ( qz I /t I , t I /z I ) ∞ ( b I z I , /z I ) ∞ f I ( { m i } ) dz I z I = X m ,...,m I ≥ f I +1 ( { m i } ) (3.28)if (cid:12)(cid:12)(cid:12) c ··· c N qa ··· a N b i t i (cid:12)(cid:12)(cid:12) < ≤ i ≤ I ) and (cid:12)(cid:12)(cid:12) qt i b j t j (cid:12)(cid:12)(cid:12) < ≤ i < j ≤ I ).7 roof. By using Lemma 3.1.2, we have12 π √− Z C ( qz /t , t /z ) ∞ ( b z , /z ) ∞ f dz z = 12 π √− Z C ( qz /t , t /z ) ∞ ( b z , /z ) ∞ N Y j =1 ( c j z · · · z M ) ∞ ( a j z · · · z M ) ∞ dz z = − X m ≥ Res ( qz /t , t /z ) ∞ ( b z , /z ) ∞ N Y j =1 ( c j z · · · z M ) ∞ ( a j z · · · z M ) ∞ z , z = q − m b − X m ≥ N X k =1 Res ( qz /t , t /z ) ∞ ( b z , /z ) ∞ N Y j =1 ( c j z · · · z M ) ∞ ( a j z · · · z M ) ∞ z , z = q − m a k z · · · z M = X m ≥ θ ( t b )( b ) m ( q, b ) ∞ ( q ) m (cid:18) qb t (cid:19) m N Y j =1 ( c j z · · · z M /b q m ) ∞ ( a j z · · · z M /b q m ) ∞ + N X k =1 X m ≥ Y j = k ( c j /a k ) ∞ ( qa k /c j ) m ( a j /a k ) ∞ ( qa k /a j ) m ( c k /a k ) ∞ ( qa k /c k ) m ( q ) ∞ ( q ) m N Y j =1 (cid:18) c j a j (cid:19) m × θ ( t a k z · · · z M q m )( b /a k z · · · z M q m , a k z · · · z M q m ) ∞ = X m ≥ f ( m ) (3.29)if (cid:12)(cid:12)(cid:12) c ··· c N qa ··· a N b t (cid:12)(cid:12)(cid:12) <
1. Next, let f I ;0 ( { m i } ) = I − Y i =1 ( q/t i b i q m i , t i b i q m i ) ∞ ( q, b i q m i ) ∞ ( q − m i ) m i N Y j =1 ( c j z I · · · z M /b · · · b I − q m + ··· + m I − ) ∞ ( a j z I · · · z M /b · · · b I − q m + ··· + m I − ) ∞ , (3.30)and let f I ; k,l ( { m i } ) = Y j = k ( c j /a k q m ) ∞ ( a j /a k q m ) ∞ ( c k /a k q m ) ∞ ( q ) ∞ ( q − m ) m l − Y i =1 ( qq m i +1 /t i , t i /q m i +1 ) ∞ ( q, b i q m i +1 ) ∞ ( q − m i +1 ) m i +1 I − Y i = l +1 ( q/t i b i q m i , t i b i q m i ) ∞ ( q, b i q m i ) ∞ ( q − m i ) m i × θ ( t l a k z I · · · z M q m l,I − /b l +1 · · · b I − )( b l · · · b I − /a k z I · · · z M q m l,I − , a k z I · · · z M q m l,I − /b l +1 · · · b I − ) ∞ , (3.31)for 1 ≤ k ≤ N , 1 ≤ l ≤ I − I = 2 , . . . , M . Here, as mentioned above in (3.26), m l,I − = P li =1 m i − P I − i = l +1 m i . By using Lemma 3.1.2, we have12 π √− Z C I ( qz I /t I , t I /z I ) ∞ ( b I z I , /z I ) ∞ f I ;0 ( { m i } ) dz I z I = − X m I ≥ Res (cid:18) ( qz I /t I , t I /z I ) ∞ ( b I z I , /z I ) ∞ f I ;0 ( { m i } ) z I , z I = q − m I b I (cid:19) − N X k =1 X w ∈ D I ; k, ( { m i } ) Res (cid:18) ( qz I /t I , t I /z I ) ∞ ( b I z I , /z I ) ∞ f I ;0 ( { m i } ) z I , z I = w (cid:19) (3.32)8f (cid:12)(cid:12)(cid:12) c ··· c N qa ··· a N b I t I (cid:12)(cid:12)(cid:12) <
1. Here, D I ; k, ( { m i } ) = (cid:26) w = b · · · b I − q m + ··· + m I − a k z I +1 · · · z M q m I ; w is in the exterior of C I , m I = 0 , , , . . . (cid:27) . (3.33)Also, by using Lemma 3.1.2, we obtain12 π √− Z C I ( qz I /t I , t I /z I ) ∞ ( b I z I , /z I ) ∞ f I ; k,l ( { m i } ) dz I z I = − X m I ≥ Res (cid:18) ( qz I /t I , t I /z I ) ∞ ( b I z I , /z I ) ∞ f I ; k,l ( { m i } ) z I , z I = q − m I b I (cid:19) − X w ∈ D I ; k,l ( { m i } ) Res (cid:18) ( qz I /t I , t I /z I ) ∞ ( b I z I , /z I ) ∞ f I ; k,l ( { m i } ) z I , z I = w (cid:19) − X w ∈ D ′ I ; k,l ( { m i } ) Res (cid:18) ( qz I /t I , t I /z I ) ∞ ( b I z I , /z I ) ∞ f I ; k,l ( { m i } ) z I , z I = w (cid:19) (3.34)if (cid:12)(cid:12)(cid:12) qt l b I t I (cid:12)(cid:12)(cid:12) <
1. Here, D I ; k,l ( { m i } ) = (cid:26) w = b l +1 · · · b I − q m l +1 + ··· + m I − a k z I +1 · · · z M q m + ··· + m l + m I ; w is in the exterior of C I , m I = 0 , , , . . . (cid:27) (3.35)and D ′ I ; k,l ( { m i } ) = (cid:26) w = b l · · · b I − q m l +1 + ··· + m I − + m I a k z I +1 · · · z M q m + ··· + m l ; w is in the exterior of C I , m I = 0 , , , . . . (cid:27) , (3.36)for 1 ≤ k ≤ N and 1 ≤ l ≤ I −
1. We can calculate the following residues easily:Res (cid:18) ( qz I /t I , t I /z I ) ∞ ( b I z I , /z I ) ∞ f I ;0 ( { m i } ) z I , z I = q − m I b I (cid:19) = − f I +1;0 ( { m i } ) , (3.37)Res (cid:18) ( qz I /t I , t I /z I ) ∞ ( b I z I , /z I ) ∞ f I ; k,l ( { m i } ) z I , z I = q − m I b I (cid:19) = − f I +1; k,l ( { m i } ) , (3.38)Res (cid:18) ( qz I /t I , t I /z I ) ∞ ( b I z I , /z I ) ∞ f I ; k,I − ( { m i } ) z I , z I = 1 a k z I +1 · · · z M q m + ··· + m I − + m I (cid:19) = − f I +1; k,I ( { m i } ) , (3.39)for 1 ≤ k ≤ N and 1 ≤ l ≤ I −
1. Also, let A k,l ( { m i } ) = Res (cid:18) ( qz I /t I , t I /z I ) ∞ ( b I z I , /z I ) ∞ f I ; k,l ( { m i } ) z I , z I = b l · · · b I − q m l +1 + ··· + m I − + m I a k z I +1 · · · z M q m + ··· + m l (cid:19) , (3.40)for 1 ≤ k ≤ N and 1 ≤ l ≤ I −
1, then we have easilyRes (cid:18) ( qz I /t I , t I /z I ) ∞ ( b I z I , /z I ) ∞ f I ;0 ( { m i } ) z I , z I = b · · · b I − q m + ··· + m I − a k z I +1 · · · z M q m I (cid:19) = − A k, ( { m σ ( i ) } ) , (3.41)Res (cid:18) ( qz I /t I , t I /z I ) ∞ ( b I z I , /z I ) ∞ f I ; k,l ( { m i } ) z I , z I = b l +1 · · · b I − q m l +1 + ··· + m I − a k z I +1 · · · z M q m + ··· + m l + m I (cid:19) = − A k,l +1 ( { m σ l +1 ( i ) } ) , (3.42)9here 1 ≤ k ≤ N , 1 ≤ l ≤ I − σ j = ( j, I ) ∈ S I . Thus, letting˜ D I ; k, = (cid:26) m ∈ Z ; b · · · b I − q m a k z I +1 · · · z M is in the exterior of C I (cid:27) , (3.43)we have X m ,...,m I − ≥ X w ∈ D I ; k, ( { m i } ) Res (cid:18) ( qz I /t I , t I /z I ) ∞ ( b I z I , /z I ) ∞ f I ;0 ( { m i } ) z I , z I = w (cid:19) = X m ,...,m I ≥ m + ··· + m I − − m I ∈ ˜ D I ; k, − A k, ( { m σ ( i ) } )= − X m ,...,m I ≥ m + ··· + m I − m ∈ ˜ D I ; k, A k, ( { m i } )= − X m ,...,m I − ≥ X w ∈ D ′ I ; k, ( { m i } ) Res (cid:18) ( qz I /t I , t I /z I ) ∞ ( b I z I , /z I ) ∞ f I ; k, ( { m i } ) z I , z I = w (cid:19) , (3.44)for 1 ≤ k ≤ N . Similarly, we obtain X m ,...,m I − ≥ X w ∈ D I ; k,l ( { m i } ) Res (cid:18) ( qz I /t I , t I /z I ) ∞ ( b I z I , /z I ) ∞ f I ; k,l ( { m i } ) z I , z I = w (cid:19) = − X m ,...,m I − ≥ X w ∈ D ′ I ; k,l +1 ( { m i } ) Res (cid:18) ( qz I /t I , t I /z I ) ∞ ( b I z I , /z I ) ∞ f I ; k,l +1 ( { m i } ) z I , z I = w (cid:19) , (3.45)for 1 ≤ k ≤ N , 1 ≤ l ≤ I −
2. By definitions (3.30), (3.31), we have X m ,...,m I − ≥ π √− Z C I ( qz I /t I , t I /z I ) ∞ ( b I z I , /z I ) ∞ f I ( { m i } ) dz I z I = X m ,...,m I − ≥ π √− Z C I ( qz I /t I , t I /z I ) ∞ ( b I z I , /z I ) ∞ f I ;0 ( { m i } ) + N X k =1 I − X l =1 f I ; k,l ( { m i } ) ! dz I z I , (3.46)and by using the equations (3.32), (3.34), (3.37), (3.38), we have X m ,...,m I − ≥ π √− Z C I ( qz I /t I , t I /z I ) ∞ ( b I z I , /z I ) ∞ f I ;0 ( { m i } ) + N X k =1 I − X l =1 f I ; k,l ( { m i } ) ! dz I z I = X m ,...,m I − ≥ X m I ≥ f I +1;0 ( { m i } ) + N X k =1 I − X l =1 X m I ≥ f I +1; k,l ( { m i } ) − N X k =1 X w ∈ D I ; k,I − ( { m i } ) Res (cid:18) ( qz I /t I , t I /z I ) ∞ ( b I z I , /z I ) ∞ f I ; k,I − ( { m i } ) z I , z I = w (cid:19)! . (3.47)According to conditions of the paths C i ( I ≤ i ≤ M ), z I = 1 /a k z I +1 · · · z M q m is in the exteriorof C I for any non-negative integer m . Therefore, by using the equation (3.39), we have X w ∈ D I ; k,I − ( { m i } ) Res (cid:18) ( qz I /t I , t I /z I ) ∞ ( b I z I , /z I ) ∞ f I ; k,I − ( { m i } ) z I , z I = w (cid:19) = − X m I ≥ f I +1; k,I ( { m i } ) . (3.48)This completes the proof. 10 roposition 3.2. By calculating the integral ϕ N,M , we have F N,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19) = N Y j =1 ( a j , c j /b L +1 · · · b M ) ∞ ( c j , a j /b L +1 · · · b M ) ∞ M Y i = L +1 θ ( t i b i ) θ ( t i ) F LN,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19) + N X k =1 M X l = L +1 (Y j = k ( c j /a k , a j ) ∞ ( a j /a k , c j ) ∞ ( c k /a k , a k , b l ) ∞ ( c k , b l · · · b M /a k , a k /b l +1 · · · b M ) ∞ × θ ( t l a k /b l +1 · · · b M ) θ ( t l ) M Y i = l +1 θ ( t i b i ) θ ( t i ) F L ; k,lN,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19)) . (3.49) Proof.
We already obtained ϕ N,M = N Y j =1 ( c j ) ∞ ( a j ) ∞ M Y i =1 θ ( t i )( b i , q ) ∞ F N,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19) , (3.50)if | t i | <
1. Hence, we should have ϕ N,M = N Y j =1 ( c j /b L +1 · · · b M ) ∞ ( a j /b L +1 · · · b M ) ∞ L Y i =1 θ ( t i )( b i , q ) ∞ M Y i = L +1 θ ( t i b i )( b i , q ) ∞ F LN,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19) + N X k =1 M X l = L +1 (Y j = k ( c j /a k ) ∞ ( a j /a k ) ∞ ( c k /a k , b l ) ∞ ( b l · · · b M /a k , a k /b l +1 · · · b M ) ∞ × l − Y i =1 θ ( t i )( b i , q ) ∞ θ ( t l a k /b l +1 · · · b M )( b l , q ) ∞ M Y i = l +1 θ ( t i b i )( b i , q ) ∞ F L ; k,lN,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19)) (3.51)if | t i | < ≤ i ≤ L ), (cid:12)(cid:12)(cid:12) c ··· c N qa ··· a N b i t i (cid:12)(cid:12)(cid:12) < L + 1 ≤ i ≤ M ) and (cid:12)(cid:12)(cid:12) qt i b j t j (cid:12)(cid:12)(cid:12) < ≤ i ≤ M − , L + 1 ≤ j ≤ M, i < j ). Similar to the proof of Proposition 3.1, we have ϕ N,M = L Y i =1 θ ( t i )( b i , q ) ∞ X m ,...,m L ≥ ( M Y i =1 ( b i ) m i ( q ) m i t m i i × π √− M − L Z C L +1 · · · Z C M N Y j =1 ( c j q m + ··· + m L z L +1 · · · z M ) ∞ ( a j q m + ··· + m L z L +1 · · · z M ) ∞ M Y i = L +1 ( qz i /t i , t i /z i ) ∞ ( b i z i , /z i ) ∞ dz L +1 z L +1 · · · dz M z M ) (3.52)if | t i | < ≤ i ≤ L ). We consider the integral1(2 π √− M − L Z C L +1 · · · Z C M N Y j =1 ( c ′ j z L +1 · · · z M ) ∞ ( a ′ j z L +1 · · · z M ) ∞ M Y i = L +1 ( qz i /t i , t i /z i ) ∞ ( b i z i , /z i ) ∞ dz L +1 z L +1 · · · dz M z M , (3.53)where a ′ j = a j q m + ··· + m L and c ′ j = c j q m + ··· + m L . By using Lemma 3.2 inductively, we obtain1(2 π √− M − L Z C L +1 · · · Z C M N Y j =1 ( c ′ j z L +1 · · · z M ) ∞ ( a ′ j z L +1 · · · z M ) ∞ M Y i = L +1 ( qz i /t i , t i /z i ) ∞ ( b i z i , /z i ) ∞ dz L +1 z L +1 · · · dz M z M X m L +1 ,...,m M ≥ ( M Y i = L +1 θ ( t i b i )( b i ) m i ( q, b i ) ∞ ( q ) m i M Y i = L +1 (cid:18) qb i t i (cid:19) m i N Y j =1 ( c ′ j /b L +1 · · · b M q m L +1 + ··· + m M ) ∞ ( a ′ j /b L +1 · · · b M q m L +1 + ··· + m M ) ∞ + N X k =1 M X l = L +1 Y j = k ( c ′ j /a ′ k ) ∞ ( qa ′ k /c ′ j ) m L +1 ( a ′ j /a ′ k ) ∞ ( qa ′ k /a ′ j ) m L +1 ( c ′ k /a ′ k ) ∞ ( qa ′ k /c ′ k ) m L +1 ( q ) ∞ ( q ) m L +1 N Y j =1 c ′ j a ′ j ! m L +1 × l − Y i = L +1 θ ( t i )( b i ) m i +1 ( q, b i ) ∞ ( q ) m i +1 l − Y i = L +1 t m i +1 i M Y i = l +1 θ ( t i b i )( b i ) m i ( q, b i ) ∞ ( q ) m i M Y i = l +1 (cid:18) qb i t i (cid:19) m i × θ ( t l a ′ k q m L +1 + ··· + m l − m l +1 −···− m M /b l +1 · · · b M )( b l · · · b M /a ′ k q m L +1 + ··· + m l − m l +1 −···− m M , a ′ k q m L +1 + ··· + m l − m l +1 −···− m M /b l +1 · · · b M ) ∞ ) , (3.54)if (cid:12)(cid:12)(cid:12) c ··· c N qa ··· a N b i t i (cid:12)(cid:12)(cid:12) < L + 1 ≤ i ≤ M ) and (cid:12)(cid:12)(cid:12) qt i b j t j (cid:12)(cid:12)(cid:12) < L + 1 ≤ i < j ≤ M ). Since a ′ j = a j q m + ··· + m L and c ′ j = c j q m + ··· + m L , we have ϕ N,M = L Y i =1 θ ( t i )( b i , q ) ∞ X m ,...,m L ≥ M Y i =1 ( b i ) m i ( q ) m i t m i i × X m L +1 ,...,m M ≥ ( M Y i = L +1 θ ( t i b i )( b i ) m i ( q, b i ) ∞ ( q ) m i M Y i = L +1 (cid:18) qb i t i (cid:19) m i N Y j =1 ( c j q m ( L ) /b L +1 · · · b M ) ∞ ( a j q m ( L ) /b L +1 · · · b M ) ∞ + N X k =1 M X l = L +1 Y j = k ( c j /a k ) ∞ ( qa k /c j ) m L +1 ( a j /a k ) ∞ ( qa k /a j ) m L +1 ( c k /a k ) ∞ ( qa k /c k ) m L +1 ( q ) ∞ ( q ) m L +1 N Y j =1 (cid:18) c j a j (cid:19) m L +1 l − Y i = L +1 θ ( t i )( b i ) m i +1 ( q, b i ) ∞ ( q ) m i +1 × l − Y i = L +1 t m i +1 i M Y i = l +1 θ ( t i b i )( b i ) m i ( q, b i ) ∞ ( q ) m i M Y i = l +1 (cid:18) qb i t i (cid:19) m i θ ( t l a k q m ( l ) /b l +1 · · · b M )( b l · · · b M /a k q m ( l ) , a k q m ( l ) /b l +1 · · · b M ) ∞ ) = N Y j =1 ( c j /b L +1 · · · b M ) ∞ ( a j /b L +1 · · · b M ) ∞ L Y i =1 θ ( t i )( b i , q ) ∞ M Y i = L +1 θ ( t i b i )( b i , q ) ∞ F LN,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19) + N X k =1 M X l = L +1 (Y j = k ( c j /a k ) ∞ ( a j /a k ) ∞ ( c k /a k , b l ) ∞ ( b l · · · b M /a k , a k /b l +1 · · · b M ) ∞ × l − Y i =1 θ ( t i )( b i , q ) ∞ θ ( t l a k /b l +1 · · · b M )( b l , q ) ∞ M Y i = l +1 θ ( t i b i )( b i , q ) ∞ F L ; k,lN,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19)) (3.55)if | t i | < ≤ i ≤ L ), (cid:12)(cid:12)(cid:12) c ··· c N qa ··· a N b i t i (cid:12)(cid:12)(cid:12) < L + 1 ≤ i ≤ M ) and (cid:12)(cid:12)(cid:12) qt i b j t j (cid:12)(cid:12)(cid:12) < ≤ i ≤ M − , L + 1 ≤ j ≤ M, i < j ). Remark 3.1.
When M = 1, we have N +1 ϕ N (cid:18) a , . . . , a N +1 c , . . . , c N ; t (cid:19) = N +1 X k =1 N Y j =1 ( c j /a k ) ∞ ( c j ) ∞ Y j = k ≤ j ≤ N +1 ( a j ) ∞ ( a j /a k ) ∞ θ ( ta k ) θ ( t ) N +1 ϕ N (cid:18) { qa k /c j } ≤ j ≤ N , a j { qa k /a j } ≤ j ≤ N +1 , j = k ; t (cid:19) . (3.56)This formula was obtained in [4], pp. 285 (7), pp. 289 (11a).12 emark 3.2. P. O. M. Olsson [5] obtained connection formulas for the Appell hypergeomet-ric function F . He also indicated that connection formulas of the Lauricella hypergeometricfunction F (3) D are obtained in the same way as F . Here, the function F ( M ) D is the Lauricellahypergeometric function F D of M variables. S. I. Bezrodnykh [6] obtained connection formulasof F ( M ) D for any M . When N = 1, the formula (3.49) is a q -analog of Olsson’s formulas [5], pp.423 (17), pp. 425 (22) and Bezrodnykh’s formula [6], pp. 132 (13). q -difference system E N,M
In this subsection, we show fundamental solutions of the q -difference system E N,M which con-verge locally.
Definition 3.3.
We define a series F L ; k,lN,M as F L ; k,lN,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19) = X m i ≥ ( N Y j =1 ( qa j /c k ) m L ( qc j /c k ) m L l − Y i =1 ( b i ) m i ( q ) m i L Y i = l +1 ( b i ) m i − ( q ) m i − M Y i = L +1 ( b i ) m i ( q ) m i × ( c k /qb l +1 · · · b M ) m ( l − ( c k /b l · · · b M ) m ( l − l − Y i =1 (cid:18) qt i b l t l (cid:19) m i L Y i = l +1 (cid:18) b l t l b i t i (cid:19) m i − M Y i = L +1 (cid:18) b l t l b i t i (cid:19) m i (cid:18) b l t l q (cid:19) m L ) , (3.57)where 0 ≤ L ≤ M , 1 ≤ k ≤ N and 1 ≤ l ≤ L .The series (3.57) converges in n | t l | < , (cid:12)(cid:12)(cid:12) qt i b l t l (cid:12)(cid:12)(cid:12) < ≤ i ≤ l − , (cid:12)(cid:12)(cid:12) qt l b i t i (cid:12)(cid:12)(cid:12) < l + 1 ≤ i ≤ M ) o . Proposition 3.3.
For 0 ≤ L ≤ M , the functions M Y i = L +1 θ ( t i b i ) θ ( t i ) F LN,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19) , (3.58) θ ( t l c k /qb l +1 · · · b M ) θ ( t l ) M Y i = l +1 θ ( t i b i ) θ ( t i ) F L ; k,lN,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19) (1 ≤ k ≤ N, ≤ l ≤ L ) , (3.59) θ ( t l a k /b l +1 · · · b M ) θ ( t l ) M Y i = l +1 θ ( t i b i ) θ ( t i ) F L ; k,lN,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19) (1 ≤ k ≤ N, L + 1 ≤ l ≤ M ) , (3.60)satisfy the q -difference system E N,M . Proof.
We can check them easily. Here, we check that the function (3.58) satisfies the q -differenceequations (2.14). First, we have θ ( aqx ) θ ( qx ) = 1 a θ ( ax ) θ ( x ) . (3.61)Thus, we obtain t s N Y j =1 (1 − a j T )(1 − b s T s ) − N Y j =1 (1 − c j q − T )(1 − T s ) M Y i = L +1 θ ( t i b i ) θ ( t i )13 M Y i = L +1 θ ( t i b i ) θ ( t i ) t s N Y j =1 (cid:18) − a j b L +1 · · · b M T (cid:19) (1 − b s T s ) − N Y j =1 (cid:18) − c j q − b L +1 · · · b M T (cid:19) (1 − T s ) (1 ≤ s ≤ L ) , M Y i = L +1 θ ( t i b i ) θ ( t i ) t s N Y j =1 (cid:18) − a j b L +1 · · · b M T (cid:19) (1 − T s ) − N Y j =1 (cid:18) − c j q − b L +1 · · · b M T (cid:19) (cid:18) − b s T s (cid:19) ( L + 1 ≤ s ≤ M ) , (3.62)as an operator. For 1 ≤ s ≤ L , we have t s N Y j =1 (cid:18) − a j b L +1 · · · b M T (cid:19) (1 − b s T s ) F LN,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19) = t s N Y j =1 (cid:18) − a j b L +1 · · · b M T (cid:19) (1 − b s T s ) X m i ≥ N Y j =1 ( a j /b L +1 · · · b M ) m ( L ) ( c j /b L +1 · · · b M ) m ( L ) M Y i =1 ( b i ) m i ( q ) m i L Y i =1 t m i i M Y i = L +1 (cid:18) qb i t i (cid:19) m i = X m i ≥ t s N Y j =1 − a j q m ( L ) b L +1 · · · b M ! (1 − b s q m s ) N Y j =1 ( a j /b L +1 · · · b M ) m ( L ) ( c j /b L +1 · · · b M ) m ( L ) M Y i =1 ( b i ) m i ( q ) m i L Y i =1 t m i i M Y i = L +1 (cid:18) qb i t i (cid:19) m i = X m i ≥ N Y j =1 ( a j /b L +1 · · · b M ) m ( L )+1 ( c j /b L +1 · · · b M ) m ( L ) Y ≤ i ≤ Mi = s ( b i ) m i ( q ) m i ( b s ) m s +1 ( q ) m s Y ≤ i ≤ Li = s t m i i t m s +1 s M Y i = L +1 (cid:18) qb i t i (cid:19) m i = X m i ≥ i = s X m s ≥− ( N Y j =1 ( a j /b L +1 · · · b M ) m ( L )+1 ( c j /b L +1 · · · b M ) m ( L )+1 N Y j =1 − c j q − q m ( L )+1 b L +1 · · · b M ! Y ≤ i ≤ Mi = s ( b i ) m i ( q ) m i × ( b s ) m s +1 ( q ) m s +1 (1 − q m s +1 ) Y ≤ i ≤ Li = s t m i i t m s +1 s M Y i = L +1 (cid:18) qb i t i (cid:19) m i ) = X m i ≥ N Y j =1 − c j q − q m ( L ) b L +1 · · · b M ! (1 − q m s ) N Y j =1 ( a j /b L +1 · · · b M ) m ( L ) ( c j /b L +1 · · · b M ) m ( L ) M Y i =1 ( b i ) m i ( q ) m i L Y i =1 t m i i M Y i = L +1 (cid:18) qb i t i (cid:19) m i = N Y j =1 (cid:18) − c j q − Tb L +1 · · · b M (cid:19) (1 − T s ) F LN,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19) , (3.63)and for L + 1 ≤ s ≤ M , we have t s N Y j =1 (cid:18) − a j b L +1 · · · b M T (cid:19) (1 − T s ) F LN,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19) = X m i ≥ t s N Y j =1 − a j q m ( L ) b L +1 · · · b M ! (1 − q − m s ) N Y j =1 ( a j /b L +1 · · · b M ) m ( L ) ( c j /b L +1 · · · b M ) m ( L ) M Y i =1 ( b i ) m i ( q ) m i L Y i =1 t m i i M Y i = L +1 (cid:18) qb i t i (cid:19) m i = X m i ≥ i = s X m s ≥ ( N Y j =1 ( a j /b L +1 · · · b M ) m ( L )+1 ( c j /b L +1 · · · b M ) m ( L )+1 N Y j =1 − c j q − q m ( L )+1 b L +1 · · · b M ! Y ≤ i ≤ Mi = s ( b i ) m i ( q ) m i ( b s ) m s − ( q ) m s − (1 − b s q m s − )( − q − m s ) L Y i =1 t m i i Y L +1 ≤ i ≤ Mi = s (cid:18) qb i t i (cid:19) m i qb s (cid:18) qb s t s (cid:19) m s − ) = X m i ≥ N Y j =1 − c j q − q m ( L ) b L +1 · · · b M ! (cid:18) − q − m s b s (cid:19) N Y j =1 ( a j /b L +1 · · · b M ) m ( L ) ( c j /b L +1 · · · b M ) m ( L ) M Y i =1 ( b i ) m i ( q ) m i L Y i =1 t m i i M Y i = L +1 (cid:18) qb i t i (cid:19) m i = N Y j =1 (cid:18) − c j q − b L +1 · · · b M T (cid:19) (cid:18) − b s T s (cid:19) F LN,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19) . (3.64)Similar to these calculations, we can check Proposition 3.3 directly. Remark 3.3.
From Proposition 3.2, the functions (3.58), (3.60) can be taken as parts of fun-damental solutions of the q -difference system E N,M . In particular, the function M Y i =1 θ ( t i b i ) θ ( t i ) F N,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19) = M Y i =1 θ ( t i b i ) θ ( t i ) F N,M { qb · · · b M /c j } , { b i }{ qb · · · b M /a j } ; N Y j =1 c j a j qb i t i (3.65)can be taken as that, too. By applying Proposition 3.2 to F N,M (cid:18) { qb · · · b M /c j } , { b i }{ qb · · · b M /a j } ; nQ Nj =1 c j a j qb i t i o(cid:19) ,we find that the functions (3.59) can be taken as parts of fundamental solutions this way as well.For 0 ≤ L ≤ M , the functions (3.58), (3.59), (3.60) converge in the region | t i | < ≤ i ≤ L ) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N Y j =1 c j a j qb i t i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < L + 1 ≤ i ≤ M ) , (cid:12)(cid:12)(cid:12)(cid:12) qt i b j t j (cid:12)(cid:12)(cid:12)(cid:12) < ≤ i < j ≤ M ) . (3.66)Also, if a function f (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19) satisfies the q -difference system E N,M , f (cid:18) { a j } , { b σ ( i ) }{ c j } ; { t σ ( i ) } (cid:19) satisfies the same system for σ ∈ S M . Therefore we have the following proposition. Proposition 3.4.
For 0 ≤ L ≤ M and σ ∈ S M , let u L,σ = M Y i = L +1 θ ( t σ ( i ) b σ ( i ) ) θ ( t σ ( i ) ) F LN,M (cid:18) { a k } , { b σ ( i ) }{ c k } ; { t σ ( i ) } (cid:19) , (3.67) u L,σk,l = θ ( t σ ( l ) c k /qb σ ( l +1) · · · b σ ( M ) ) θ ( t σ ( l ) ) M Y i = l +1 θ ( t σ ( i ) b σ ( i ) ) θ ( t σ ( i ) ) F L ; k,lN,M { a j } , { b σ ( i ) }{ c j } ; { t σ ( i ) } ! (1 ≤ k ≤ N, ≤ l ≤ L ) ,θ ( t σ ( l ) a k /b σ ( l +1) · · · b σ ( M ) ) θ ( t σ ( l ) ) M Y i = l +1 θ ( t σ ( i ) b σ ( i ) ) θ ( t σ ( i ) ) F L ; k,lN,M { a j } , { b σ ( i ) }{ c j } ; { t σ ( i ) } ! (1 ≤ k ≤ N, L + 1 ≤ l ≤ M ) , (3.68)and let u L,σ = t ( u L,σ , u L,σ , , . . . , u L,σ ,M , u L,σ , , . . . , u L,σN,M ) . (3.69)15hen u L,σ is a fundamental solution of the q -difference system E N,M in the region D L,σ if theparameters { a j } , { b i } , { c j } are generic, where D L,σ = | t σ ( i ) | < ≤ i ≤ L ) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N Y j =1 c j a j qb σ ( i ) t σ ( i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < L + 1 ≤ i ≤ M ) , (cid:12)(cid:12)(cid:12)(cid:12) qt σ ( i ) b σ ( j ) t σ ( j ) (cid:12)(cid:12)(cid:12)(cid:12) < ≤ i < j ≤ M ) . (3.70) In this subsection, we consider the connection problem between u L ,σ and u L ,σ , where 0 ≤ L , L ≤ M and σ , σ ∈ S M . We can solve this problem in principle by calculating the followingmatrices: • the matrix which connects u L, id with u L +1 , id (0 ≤ L ≤ M − • the matrix which connects u L, id with u L − , id (1 ≤ L ≤ M ), • the matrix which connects u M, id with u M,s r (0 ≤ r ≤ M − s r = ( r, r + 1) ∈ S M . First, we consider the matrix which connects u L, id with u L +1 , id . If l = L + 1, then we have u L, id k,l = u L +1 , id k,l (3.71)easily. Hence, we should calculate the analytic continuations of u L, id0 , u L, id k,L +1 . By rewriting thedefinition (3.23), we have F LN,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19) = X m i ≥ N Y j =1 ( a j /b L +1 · · · b M ) m ( L ) ( c j /b L +1 · · · b M ) m ( L ) M Y i =1 ( b i ) m i ( q ) m i L Y i =1 t m i i M Y i = L +1 (cid:18) qb i t i (cid:19) m i = X m i ≥ i = L +1 ( N Y j =1 ( a j /b L +1 · · · b M ) m ( L ) ′ ( c j /b L +1 · · · b M ) m ( L ) ′ Y i = L +1 ( b i ) m i ( q ) m i L Y i =1 t m i i M Y i = L +2 (cid:18) qb i t i (cid:19) m i × N +1 ϕ N { qb L +1 · · · b M /a j q m ( L ) ′ } ≤ j ≤ N , b L +1 { qb L +1 · · · b M /c j q m ( L ) ′ } ≤ j ≤ N ; N Y j =1 c j a j qb L +1 t L +1 ) . (3.72)Here and in the following, we use the notation m ( l ) ′ = l X i =1 m i − M X i = l +2 m i , (3.73)for m = ( m , . . . , m M ) and 0 ≤ l ≤ M . By applying the formula (3.56) to N +1 ϕ N , we obtain F LN,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19) = N Y j =1 ( qb L +2 · · · b M /a j , qb L +1 · · · b M /c j ) ∞ ( qb L +1 · · · b M /a j , qb L +2 · · · b M /c j ) ∞ θ ( t L +1 a · · · a N /c · · · c N ) θ ( t L +1 b L +1 a · · · a N /c · · · c N ) F L +1 N,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19) + N X d =1 ( N Y j =1 ( c d /a j ) ∞ ( qb L +1 · · · b M /a j ) ∞ Y j = d ( qb L +1 · · · b M /c j ) ∞ ( c d /c j ) ∞ ( b L +1 ) ∞ ( c d /qb L +2 · · · b M ) ∞ θ ( t L +1 a · · · a N c d /qb L +2 · · · b M c · · · c N ) θ ( t L +1 b L +1 a · · · a N /c · · · c N ) F L +1; d,L +1 N,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19)) . (3.74)Similarly, we have F L ; k,L +1 N,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19) = Y j = k ( qb L +2 · · · b M /a j ) ∞ ( qa k /a j ) ∞ ( q/b L +1 ) ∞ ( qa k /b L +1 · · · b M ) ∞ N Y j =1 ( qa k /c j ) ∞ ( qb L +2 · · · b M /c j ) ∞ × θ ( t L +1 b L +1 · · · b M a · · · a N /a k c · · · c N ) θ ( t L +1 b L +1 a · · · a N /c · · · c N ) F L +1 N,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19) + N X d =1 (Y j = k ( c d /a j ) ∞ ( qa k /a j ) ∞ ( c d /b L +1 · · · b M ) ∞ ( qa k /b L +1 · · · b M ) ∞ Y j = d ( qa k /c j ) ∞ ( c d /c j ) ∞ ( a k /b L +2 · · · b M ) ∞ ( c d /qb L +2 · · · b M ) ∞ × θ ( t L +1 b L +1 a · · · a N c d /qc · · · c N a k ) θ ( t L +1 b L +1 a · · · a N /c · · · c N ) F L +1; d,L +1 N,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19)) . (3.75)Therefore, let A L, id = A L, id (cid:18) { a j } , { b i }{ c j } ; t L +1 (cid:19) = A , A , · · · A ,N A , A , · · · A ,N ... ... ... A N, A N, · · · A N,N , (3.76) A , = N Y j =1 ( qb L +2 · · · b M /a j , qb L +1 · · · b M /c j ) ∞ ( qb L +1 · · · b M /a j , qb L +2 · · · b M /c j ) ∞ θ ( t L +1 a · · · a N /c · · · c N ) θ ( t L +1 b L +1 ) θ ( t L +1 b L +1 a · · · a N /c · · · c N ) θ ( t L +1 ) , (3.77) A ,d = ( A , (1 ,d ) , A , (2 ,d ) , . . . , A , ( M,d ) ) = (0 , . . . , , A , ( L +1 ,d ) , , . . . , , (3.78) A , ( L +1 ,d ) = N Y j =1 ( c d /a j ) ∞ ( qb L +1 · · · b M /a j ) ∞ Y j = d ( qb L +1 · · · b M /c j ) ∞ ( c d /c j ) ∞ ( b L +1 ) ∞ ( c d /qb L +2 · · · b M ) ∞ × θ ( t L +1 a · · · a N c d /qb L +2 · · · b M c · · · c N ) θ ( t L +1 b L +1 ) θ ( t L +1 b L +1 a · · · a N /c · · · c N ) θ ( t L +1 c d /qb L +2 · · · b M ) , (3.79) A k, = t ( A (1 ,k ) , , A (2 ,k ) , , . . . , A ( N,k ) , ) = t (0 , . . . , , A ( L +1 ,k ) , , , . . . , , (3.80) A ( L +1 ,k ) , = Y j = k ( qb L +2 · · · b M /a j ) ∞ ( qa k /a j ) ∞ ( q/b L +1 ) ∞ ( qa k /b L +1 · · · b M ) ∞ N Y j =1 ( qa k /c j ) ∞ ( qb L +2 · · · b M /c j ) ∞ × θ ( t L +1 b L +1 · · · b M a · · · a N /a k c · · · c N ) θ ( t L +1 a k /b L +2 · · · b M ) θ ( t L +1 b L +1 a · · · a N /c · · · c N ) θ ( t L +1 ) , (3.81) A k,d = I L O OO A ( L +1 ,k ) , ( L +1 ,d ) OO O I M − L − , (3.82) A ( L +1 ,k ) , ( L +1 ,d ) = Y j = k ( c d /a j ) ∞ ( qa k /a j ) ∞ ( c d /b L +1 · · · b M ) ∞ ( qa k /b L +1 · · · b M ) ∞ Y j = d ( qa k /c j ) ∞ ( c d /c j ) ∞ ( a k /b L +2 · · · b M ) ∞ ( c d /qb L +2 · · · b M ) ∞ × θ ( t L +1 b L +1 a · · · a N c d /qc · · · c N a k ) θ ( t L +1 a k /b L +2 · · · b M ) θ ( t L +1 b L +1 a · · · a N /c · · · c N ) θ ( t L +1 c d /qb L +2 · · · b M ) , (3.83)where 1 ≤ k, d ≤ N , I n is the unit matrix of degree n and O is the null matrix. Then we have u L, id = A L, id u L +1 , id . (3.84)17econdly, we consider the matrix which connects u L, id with u L − , id . This can be calculated bya similar method as given above. Let B L, id = B L, id (cid:18) { a j } , { b i }{ c j } ; t L (cid:19) = B , B , · · · B ,N B , B , · · · B ,N ... ... ... B N, B N, · · · B N,N , (3.85) B , = N Y j =1 ( a j /b L +1 · · · b M , c j /b L · · · b M ) ∞ ( a j /b L · · · b M , c j /b L +1 · · · b M ) ∞ , (3.86) B ,d = ( B , (1 ,d ) , B , (2 ,d ) , . . . , B , ( M,d ) ) = (0 , . . . , , B , ( L,d ) , , . . . , , (3.87) B , ( L,d ) = N Y j =1 ( c j /a d ) ∞ ( c j /b L +1 · · · b M ) ∞ Y j = d ( a j /b L +1 · · · b M ) ∞ ( a j /a d ) ∞ ( b L ) ∞ ( b L · · · b M /a d ) ∞ , (3.88) B k, = t ( B (1 ,k ) , , B (2 ,k ) , , . . . , B ( M,k ) , ) = t (0 , . . . , , B ( L,k ) , , , . . . , , (3.89) B ( L,k ) , = Y j = k ( c j /b L · · · b M ) ∞ ( qc j /c k ) ∞ ( q/b L ) ∞ ( q b L +1 · · · b M /c k ) ∞ N Y j =1 ( qa j /c k ) ∞ ( a j /b L · · · b M ) ∞ × θ ( t L c k /qb L +1 · · · b M ) θ ( t L qb L · · · b M /c k ) θ ( t L ) θ ( t L b L ) , (3.90) B k,d = I L − O OO B ( L,k ) , ( L,d ) OO O I M − L , (3.91) B ( L,k ) , ( L,d ) = Y j = k ( c j /a d ) ∞ ( qc j /c k ) ∞ ( qb L +1 · · · b M /a d ) ∞ ( q b L +1 · · · b M /c k ) ∞ Y j = d ( qa j /c k ) ∞ ( a j /a d ) ∞ ( qb L · · · b M ) ∞ ( b L · · · b M ) ∞ × θ ( t L c k /qb L +1 · · · b M ) θ ( t L qb L · · · b M /c k ) θ ( t L ) θ ( t L b L ) , (3.92)where 1 ≤ k, d ≤ N , then we have u L, id = B L, id u L − , id . (3.93)Finally, we consider the matrix which connects u M, id with u M,s r . We have u M,s r = u M, id0 , (3.94) u M,s r k,l = u M, id k,l (3.95)easily if l = r, r + 1. Thus we should calculate the analytic continuations of u M,s r k,r , u M,s r k,r +1 . Wehave F M ; k,rN,M (cid:18) { a j } , { b s r ( i ) }{ c j } ; { t s r ( i ) } (cid:19) = X m i ≥ ( N Y k =1 ( qa k /c j ) m M ( qc k /c j ) m M r − Y i =1 ( b i ) m i ( q ) m i M Y i = r +2 ( b i ) m i − ( q ) m i − ( b r ) m r ( q ) m r × ( c k /qb r b r +2 · · · b M ) m ( r − ( c k /b r · · · b M ) m ( r − r − Y i =1 (cid:18) qt i b r +1 t r +1 (cid:19) m i M Y i = r +2 (cid:18) b r +1 t r +1 b i t i (cid:19) m i − (cid:18) b r +1 t r +1 b r t r (cid:19) m r (cid:18) b r +1 t r +1 q (cid:19) m M ) = X m i ≥ i = r ( N Y k =1 ( qa k /c j ) m M ( qc k /c j ) m M r − Y i =1 ( b i ) m i ( q ) m i M Y i = r +2 ( b i ) m i − ( q ) m i − ( c k /qb r b r +2 · · · b M ) m ( r − ′ ( c k /b r · · · b M ) m ( r − ′ r − Y i =1 (cid:18) qt i b r +1 t r +1 (cid:19) m i M Y i = r +2 (cid:18) b r +1 t r +1 b i t i (cid:19) m i − (cid:18) b r +1 t r +1 q (cid:19) m M ϕ (cid:18) b r , qb r · · · b M /c k q m ( r − ′ q b r b r +2 · · · b M /c k q m ( r − ′ ; qt r +1 b r t r (cid:19)) , (3.96)and by applying the formula (3.56) to ϕ , we obtain F M ; k,rN,M (cid:18) { a j } , { b s r ( i ) }{ c j } ; { t s r ( i ) } (cid:19) = ( q/b r +1 , b r ) ∞ ( q b r b r +2 · · · b M /c k , c k /qb r +1 · · · b M ) ∞ θ ( t r c k /t r +1 qb r +1 · · · b M ) θ ( t r b r /t r +1 ) F M ; k,rN,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19) + ( q b r +2 · · · b M /c k , qb r · · · b M /c k ) ∞ ( q b r b r +2 · · · b M /c k , q r +1 · · · b M /c k ) ∞ θ ( t r /t r +1 ) θ ( t r b r /t r +1 ) F M ; j,r +1 N,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19) . (3.97)Similarly, we have F M ; k,r +1 N,M (cid:18) { a j } , { b s r ( i ) }{ c j } ; { t s r ( i ) } (cid:19) = ( c k /b r · · · b M , c k /qb r +2 · · · b M ) ∞ ( c k /b r b r +2 · · · b M , c k /qb r +1 · · · b M ) ∞ θ ( t r b r /t r +1 b r +1 ) θ ( t r b r /t r +1 ) F M ; k,rN,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19) + ( q/b r , b r +1 ) ∞ ( c k /b r b r +2 · · · b M , qb r +1 · · · b M /c k ) ∞ θ ( t r qb r b r +2 · · · b M /t r +1 c k ) θ ( t r b r /t r +1 ) F M ; k,r +1 N,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19) . (3.98)Therefore, let S M, id s r = S M, id s r (cid:18) { b i }{ c j } ; t r , t r +1 (cid:19) = O · · · · · · OO S r O · · · O ... O S r ...... ... . . . OO O · · ·
O S Nr , (3.99) S kr = I r − O O OO S kr,r S kr,r +1 OO S kr +1 ,r S kr +1 ,r +1 OO O O I M − r − , (3.100) S kr,r = ( q/b r +1 , b r ) ∞ ( q b r b r +2 · · · b M /c k , c k /qb r +1 · · · b M ) ∞ θ ( t r c k /t r +1 qb r +1 · · · b M ) θ ( t r b r ) θ ( t r +1 c k /qb r b r +2 · · · b M ) θ ( t r b r /t r +1 ) θ ( t r c k /qb r +1 · · · b M ) θ ( t r +1 b r +1 ) , (3.101) S kr,r +1 = ( q b r +2 · · · b M /c k , qb r · · · b M /c k ) ∞ ( q b r b r +2 · · · b M /c k , qb r +1 · · · b M /c k ) ∞ θ ( t r /t r +1 ) θ ( t r b r ) θ ( t r +1 c k /qb r b r +2 · · · b M ) θ ( t r b r /t r +1 ) θ ( t r ) θ ( t r +1 c j /qb r +2 · · · b M ) , (3.102) S kr +1 ,r = ( c k /b r · · · b M , c k /qb r +2 · · · b M ) ∞ ( c k /b r b r +2 · · · b M , c k /qb r +1 · · · b M ) ∞ θ ( t r b r /t r +1 b r +1 ) θ ( t r c k /qb r +2 · · · b M ) θ ( t r +1 ) θ ( t r b r /t r +1 ) θ ( t r c k /qb r +1 · · · b M ) θ ( t r +1 b r +1 ) , (3.103) S kr +1 ,r +1 = ( q/b r , b r +1 ) ∞ ( c k /b r b r +2 · · · b M , qb r +1 · · · b M /c k ) ∞ θ ( t r qb r b r +2 · · · b M /t r +1 c k ) θ ( t r c k /qb r +2 · · · b M ) θ ( t r +1 ) θ ( t r b r /t r +1 ) θ ( t r ) θ ( t r +1 c k /qb r +2 · · · b M ) , (3.104)where 1 ≤ k ≤ N , and then we have u M,s r = S M, id s r u M, id . (3.105)19oreover, for σ ∈ S M , let A L,σ = A L, id (cid:18) { a j } , { b σ ( i ) }{ c j } ; t σ ( L +1) (cid:19) , (3.106) B L,σ = B L, id (cid:18) { a j } , { b σ ( i ) }{ c j } ; t σ ( L ) (cid:19) , (3.107) S M,σs r = S M, id s r (cid:18) { b σ ( i ) }{ c j } ; t σ ( r ) , t σ ( r +1) (cid:19) . (3.108)Then we have u L,σ = A L,σ u L +1 ,σ , (3.109) u L,σ = B L,σ u L − ,σ , (3.110) u M,s r σ = S M,σs r u M,σ . (3.111)Therefore, we obtain the following theorem: Theorem 3.1.
For 0 ≤ L , L ≤ M and σ , σ ∈ S M , we have u L ,σ = A L ,σ A L +1 ,σ · · · A M − ,σ S M,s r ··· s rI σ s r S M,s r ··· s rI σ s r · · · S M,σ s rI B M,σ B M − ,σ · · · B L +1 ,σ u L ,σ (3.112)if σ = s r · · · s r I σ , where s r = ( r, r + 1) ∈ S M . In this section, we assume b i = q β i (1 ≤ i ≤ M ) and c j = q γ j (1 ≤ j ≤ N ). We obtain an ellipticsolution of the Yang-Baxter equation as an application of Theorem 3.1. For α ∈ C , the function C ( x ) = x α θ ( q α x ) θ ( x ) (4.1)is a pseudo constant: C ( qx ) = C ( x ) . (4.2)Namely, the functions v σ = F N,M (cid:18) { a j } , { b i }{ c j } ; { t i } (cid:19) , (4.3) v σk,l = t β σ ( l ) + ··· + β σ ( M ) − γ k σ ( l ) M Y i = l +1 t − β σ ( i ) σ ( i ) F M ; k,lN,M (cid:18) { a j } , { b σ ( i ) }{ c j } ; { t σ ( i ) } (cid:19) , (4.4)where 1 ≤ k ≤ N , 1 ≤ l ≤ M and σ ∈ S M , are solutions of the q -difference system E N,M in theregion D M,σ = (cid:26) | t i | < ≤ i ≤ M ) , (cid:12)(cid:12)(cid:12)(cid:12) qt σ ( i ) b σ ( j ) t σ ( j ) (cid:12)(cid:12)(cid:12)(cid:12) < ≤ i < j ≤ M ) (cid:27) . (4.5)Similar to the calculation of the matrix (3.99), let v σ = t ( v σ , v σ , , . . . , v σ ,M , v σ , , . . . , v σN,M ) , (4.6)20nd let˜ S id s r = ˜ S id s r (cid:18) { b i }{ c j } ; t r t r +1 (cid:19) = O · · · · · · OO ˜ S r O · · · O ... O ˜ S r ...... ... . . . OO O · · · O ˜ S Nr , (4.7)˜ S kr = ˜ S r (cid:18) { b i } c k ; t r t r +1 (cid:19) = I r − O O OO ˜ S kr,r ˜ S kr,r +1 OO ˜ S kr +1 ,r ˜ S kr +1 ,r +1 OO O O I M − r − , (4.8)˜ S kr,r = ( q/b r +1 , b r ) ∞ ( q b r b r +2 · · · b M /c k , c k /qb r +1 · · · b M ) ∞ θ ( t r c k /t r +1 qb r +1 · · · b M ) θ ( t r b r /t r +1 ) (cid:18) t r t r +1 (cid:19) − − β r −···− β M + γ k , (4.9)˜ S kr,r +1 = ( q b r +2 · · · b M /c k , qb r · · · b M /c k ) ∞ ( q b r b r +2 · · · b M /c k , qb r +1 · · · b M /c k ) ∞ θ ( t r /t r +1 ) θ ( t r b r /t r +1 ) (cid:18) t r t r +1 (cid:19) − β r , (4.10)˜ S kr +1 ,r = ( c k /b r · · · b M , c k /qb r +2 · · · b M ) ∞ ( c k /b r b r +2 · · · b M , c k /qb r +1 · · · b M ) ∞ θ ( t r b r /t r +1 b r +1 ) θ ( t r b r /t r +1 ) (cid:18) t r t r +1 (cid:19) − β r +1 , (4.11)˜ S kr +1 ,r +1 = ( q/b r , b r +1 ) ∞ ( c k /b r b r +2 · · · b M , qb r +1 · · · b M /c k ) ∞ θ ( t r qb r b r +2 · · · b M /t r +1 c k ) θ ( t r b r /t r +1 ) (cid:18) t r t r +1 (cid:19) β r +2 + ··· + β M − γ k , (4.12)where 1 ≤ k ≤ N and s r = ( r, r + 1) ∈ S M , then we have v s r = ˜ S id s r v id . (4.13)In addition, let ˜ S σs r = ˜ S id s r (cid:18) { b σ ( i ) }{ c j } ; t σ ( r ) t σ ( r +1) (cid:19) , (4.14)for σ ∈ S M , and then we have v s r σ = ˜ S σs r v σ . (4.15) Remark 4.1.
The matrices ˜ S id s r (1 ≤ r ≤ M −
1) depend only on t r /t r +1 and the parameters { b i } , { c j } .By the braid relation ( r, r + 2) = s r s r +1 s r = s r +1 s r s r +1 , we have v ( r,r +2) = ˜ S s r +1 s r s r ˜ S s r s r +1 ˜ S id s r v id = ˜ S s r s r +1 s r +1 ˜ S s r +1 s r ˜ S id s r +1 v id . (4.16)In particular, we find that the matrices ˜ S r satisfy the Yang-Baxter equation˜ S r (cid:18) { b s r +1 s r ( i ) } c k ; u (cid:19) ˜ S r +1 (cid:18) { b s r ( i ) } c k ; uv (cid:19) ˜ S r (cid:18) { b i } c k ; v (cid:19) = ˜ S r +1 (cid:18) { b s r s r +1 ( i ) } c k ; v (cid:19) ˜ S r (cid:18) { b s r +1 ( i ) } c k ; uv (cid:19) ˜ S r +1 (cid:18) { b i } c k ; u (cid:19) , (4.17)where u = t r +1 /t r +2 , v = t r /t r +1 . 21 emark 4.2. For the details of the Yang-Baxter equation, see Jimbo’s text [9].
Remark 4.3.
K. Aomoto, Y. Kato and K. Mimachi [10] obtained an elliptic solution of theYang-Baxter equation by considering the connection matrices of a holonomic q -difference systemwhich was studied in [11]. They obtained that the matrices P i ( u ) = I i − O OO W ( α ′ + ( i − β ′ , β ′ ; u ) OO O I n − i − , (4.18) W ( α, β ; u ) = u α +3 β +1 θ ( q − β ) θ ( uq α +2 β +1 ) θ ( q − α − β ) θ ( uq − β ) q β +1 u β θ ( u ) θ ( q − α − β +1 ) θ ( q α +3 β +2 ) θ ( q − α − β ) θ ( uq − β ) u β θ ( u ) θ ( uq − β ) u − α − β θ ( q − β ) θ ( uq − α − β ) θ ( q − α − β ) θ ( uq − β ) , , (4.19)where 1 ≤ i ≤ n −
1, satisfy the Yang-Baxter equation P i ( u ) P i +1 ( uv ) P i ( v ) = P i +1 ( v ) P i ( uv ) P i +1 ( u ) . (4.20)They also found that the matrix W is identified as the matrix W ′ with entires of Boltzmannweight κµ (cid:3) σν of the A (1)1 face model discussed by M. Jimbo, T. Miwa and M. Okado [12]. Thematrix W ′ is expressed as W ′ = [ a − u ][ a ] [ u ][ a + 1][ a − a ] [ u ][1] [ a + u ][ a ] , (4.21)where [ u ] = θ ( πu/L, q ), L = 0 is an arbitrary complex parameter and θ ( u, q ) = 2 q / sin u ∞ Y k =1 (1 − q k cos 2 u + q k )(1 − q k ) = 2 q / sin u ( e √− u q, e − √− u q, q ) ∞ . (4.22)The matrix W is equivalent to W ′ as follows: x g c (cid:18) x − g a +1 x − g a − (cid:19) W ′ (cid:18) x − g a +1 x − g a − (cid:19) = θ ( xq − β ) θ ( q − β ) W ( α, β ; x ) , (4.23)with e π √− u/L = x , e π √− /L = q β +1 , e π √− a/L = q − α − β , g a − = α + β , g a +1 = − α − β ,2 g c = . On the other hand, by specializing parameters b i as b = · · · = b M = q β , the matrices˜ S r ( u ) = ˜ S r (cid:18) q β c k ; u (cid:19) satisfy the Yang-Baxter equation˜ S r ( u ) ˜ S r +1 ( uv ) ˜ S r ( v ) = ˜ S r +1 ( v ) ˜ S r ( uv ) ˜ S r +1 ( u ) . (4.24)By definition (4.8), we have˜ S r ( u ) = I r − O OO ˜ W ( γ k − − ( M − r − β, − β ; u ) OO O I M − r − , (4.25)22 W ( α, β ; u ) = u α +3 β +1 θ ( q − β ) θ ( uq α +2 β +1 ) θ ( q − α − β ) θ ( uq − β ) u β θ ( u )( q − α − β , q − α − β − ) ∞ θ ( uq − β )( q − α − β , q − α − β − ) ∞ u β θ ( u )( q α +3 β +2 , q α + β +1 ) ∞ θ ( uq − β )( q α +2 β +2 , q α +2 β +1 ) ∞ u − α − β − θ ( uq − α − β − ) θ ( q − β ) θ ( uq − β ) θ ( q − α − β − ) , (4.26)and by means of easy calculations, we find that the matrices W and ˜ W are conjugate as follows: W ( α, β ; u ) = A ( α, β ) − ˜ W ( α, β ; u ) A ( α, β ) = B ( α, β ) ˜ W ( α, β ; u ) B ( α, β ) − , (4.27)where A ( α, β ) = (cid:18) f ( α, β ) (cid:19) , (4.28) B ( α, β ) = (cid:18) f ( α, β ) 00 1 (cid:19) , (4.29) f ( α, β ) = ( q α +3 β +2 , q α + β +1 ) ∞ ( q α +2 β +2 , q α +2 β +1 ) ∞ . (4.30)In conclusion, our matrix ˜ W is identified as Jimbo, Miwa and Okado’s matrix W ′ and with n = M , α ′ = γ k − − ( M − β and β ′ = − β , our matrices ˜ S r ( u ) and Aomoto, Kato andMimachi’s matrices P r ( u ) are conjugate. In this paper, we obtained one main result, and as steps for deriving it, we obtained threesecondary results which are also important. A summary of these results is as follows.The main result of this paper is Theorem 3.1, which gives the connection formulas forfundamental solutions of the q -difference system E N,M (2.14) and (2.15). To solve the connectionproblem, we obtained the following three results (i), (ii) and (iii) step-by-step.(i) We obtained a Barnes-type integral representation ϕ N,M (3.13) of the q -hypergeometricfunction F N,M . Whereas the function F N,M converges if | t i | < ≤ i ≤ M , theintegral is defined for ( t , . . . , t M ) ∈ C M .(ii) By calculating the Barnes-type integral, we obtained connection formulas (3.49) of F N,M .The function F N,M is an extension of N +1 ϕ N and ϕ D , thus these formulas contain con-nection formulas of N +1 ϕ N and also contain connection formulas of ϕ D . Our connectionformulas contain Watson’s formula [4] and a q -analog of Olsson’s formula [5] and Bezrod-nykh’s formula [6].(iii) We obtained fundamental solutions u L,σ (3.69) of the q -difference system E N,M . Here, for0 ≤ L ≤ M and σ ∈ S M , u L,σ converges in the region {| t σ (1) | ≪ · · · ≪ | t σ ( L ) | ≪ ≪| t σ ( L +1) | ≪ · · · ≪ | t σ ( M ) |} .In addition, as an application of Theorem 3.1, we obtained a solution of the Yang-Baxter equationby considering the connection matrix between u M, ( r,r +2) and u M, id . Also we showed that ourmatrix (4.26) is identified as Jimbo, Miwa, Okado’s matrix (4.21), and our solution and Aomoto,Kato, Mimachi’s solution [10] are conjugate.There are many problems related to our results. We mention three of them here.23i) By taking the limit q → a j = q α j , b i = q β i , c j = q γ j , we obtain fundamentalsolutions of Tsuda’s hypergeometric equations [1] t s ( β s + δ s ) N Y j =1 ( α j + D ) − δ s N Y j =1 ( γ j − D ) y = 0 (1 ≤ s ≤ M ) , { t r ( β r + δ r ) δ s − t s ( β s + δ s ) δ r } y = 0 (1 ≤ r < s ≤ M ) , where δ s = t s ∂∂t s and D = P Ms =1 δ s . Similar to the method of Theorem 3.1, it is expectedthat the connection problem of Tsuda’s equations will be solved. In this case, it mustbe noted that solutions are multivalued functions. The analytic continuation of Tsuda’shypergeometric function F N +1 ,M , which is a solution of Tsuda’s equations, depends on thepath on X = { ( t , . . . , t M ) ∈ C M ; t i = t j ( i = j ) , t i = 0 , } . In [13], a path of analyticcontinuation for solutions of GG system [14] was discussed by S-J. Matsubara-Heo. Thusthe connection problem of Tsuda’s equations will also be solved by Matsubara’s method.(ii) Our fundamental solutions u L,σ are given by series. On the other hand, a solution F N,M has the Euler-type Jackson integral representation (2.12). It is also expected that othersolutions of the q -difference system E N,M have the Euler-type Jackson integral represen-tation with suitable domain of integration.(iii) In recent years, the theory of elliptic difference equations has progressed. For example,considering discrete isomonodromic deformations of a linear difference system, an ellipticGarnier system which is a generalization of elliptic Painlev´e equation defined by H. Sakai[15] was obtained by C. H. Ormerod, E. M. Rains [16] and Y. Yamada [17], and using rep-resentation theory of the elliptic quantum group U q,p ( b sl N ), an explicit formula for elliptichypergeometric integral solutions of the face type elliptic q -KZ equation was obtained byH. Konno [18]. We hope that an elliptic analog of the hypergeometric function F N,M andtheir related isomonodromic system will be obtained and our result will be extended tothe elliptic hypergeometric function.
Acknowledgements
The author would like to thank Professor Yasuhiko Yamada for useful discussions, valuable sug-gestions and his encouragement. He also thanks Assistant Professor Saiei-Jaeyeong Matsubara-Heo for useful comments. He is also grateful to Professor Wayne Rossman for careful readingand corrections in the manuscript.
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