A certain generalization of q -hypergeometric functions and their related monodromy preserving deformation II
AA certain generalization of q -hypergeometric functions and their related monodromy preservingdeformation II Kanam Park
Departmant of Mathematics, Graduate School of Science, Kobe University , Japan
Email: [email protected]
Abstract
We define a nonlinear q -difference system P N, ( M − ,M + ) as monodromy pre-serving deformations of a certain linear equation. We study its relation to aseries F N,M defined as a certain generalization of q -hypergeometric functions. Keywords : generalized q -hypergeometric function; q -Garnier system; q -differences;a linear Pfaffian systems. In the previous work [10], we defined a series F N,M F N,M (cid:16) { a j } , { b i }{ c j } ; { y i } (cid:17) = (cid:88) m i ≥ N (cid:89) j =1 ( a j ) | m | ( c j ) | m | M (cid:89) i =1 ( b i ) m i ( q ) m i M (cid:89) i =1 y m i i , (1.1)as a certain generalization of q -hypergeometric functions, where, ( a ) n = ( a ) ∞ ( q n a ) ∞ ,( a ) ∞ = (cid:81) ∞ i =0 (1 − q i a ) and 0 < | q | <
1. The aim of this paper is to study the systemof q -difference non-linear equations which admits a particular solution in terms ofthe function F N,M . This problem was solved in cases of (
N, M ) = (1 , M ) in [10],and we will achieve our goal for all (
N, M ) in this paper. We remark that theseresults can be considered as a natural q -analog of Tsuda’s results [14].The contents of this paper is as follows. In the next section, we formulatea system of nonlinear q -difference equations P N, ( M − ,M + ) which is the monodromypreserving deformation we aim at. In section 3, we review some facts on the series F N,M from the previous work [10]. In section 4, we derive a Pfaffian system whichthe series F N,M from an integral representation of it. In section 5, we construct thePfaffian system which F N,M satisfies and derive solution of the system P N +1 , ( M,M ) .1 a r X i v : . [ n li n . S I] M a y A monodromy preserving deformation P N, ( M − ,M + )In this section, we define a monodromy preserving deformation P N, ( M − ,M + ) . Itsrelation to a generalization of q -hypergeometric functions F N,M will be given in thenext section.
Definition 2.1.
We define nonlinear q -difference system P N, ( ε ,ε , ··· ,ε M ) , ε i = ± qz ) = Ψ( z ) A ( z ) , A ( z ) = DX ε ( z ) X ε ( z ) · · · X ε M M ( z ) , (2.1)where the matrices D and X ε i i ( z ) stand for D = diag[ d , d , · · · , d N ] ,X i ( z ) = diag[ u ,i , u ,i , · · · u N,i ] + Λ , Λ = z , (2.2)and { u j,i } are dependent variables and { d j , c i } are parameters such that N (cid:89) j =1 u j,i = c i . (2.3)As is shown by the following Proposition, except for the dependence on de-formation direction, the equation ( 2.1) system P N, ( ε ,ε , ··· ,ε M ) essentialy dependsonly on M − , M + , where M ± = { ε i | ε i = ± } . Namely, A ( z ) for different (cid:126)ε are equivalent if M − , M + are the same. Therefore we sometimes use notation P N, ( ε ,ε , ··· ,ε M ) = P N, ( M − ,M + ) . Proposition 2.1.
For any permutation σ ∈ S M , there exists a unique set of variables { u (cid:48) j,i } such that the following equation holds X ε ( z ) X ε ( z ) · · · X ε M M ( z ) = X ε σ (1) σ (1) (cid:48) X ε σ (2) σ (2) (cid:48) · · · X ε σ ( M ) σ ( M ) (cid:48) , det X k = det X (cid:48) k = z − c k , (2.4)where the matrix X (cid:48) i stands for a matrix X i whose variables u j,i are replaced by u (cid:48) j,i . Proof . We show the existence first. Define a transformation s ε k ,ε l k,l : s ε k ,ε l k,l ( u j,k , u j,l ) = ( u j,k Q ε k ,ε l j +1 ,k Q ε k ,ε l j,k , u j,l Q ε k ,ε l j,k Q ε k ,ε l j +1 ,k ) , (2.5)where Q j,i is polynomial in u j,i , u j,i +1 given by Q + , + j,i = N (cid:88) a =1 ( a − (cid:89) k =1 u j + k,i N (cid:89) k = a +1 u j + k,i +1 ) ,Q − , − j,i = 1 Q j,i (cid:12)(cid:12)(cid:12)(cid:12) i ↔ i +1 ,Q + , − j,i = u j,i − u j,i +1 ,Q − , + j,i = ( u j − ,i − u j − ,i +1 ) − . (2.6)2uch defined a birational mapping s ε k ,ε l k,l satisfies s ε k ,ε l k,l ( X ε k k X ε l l ) = X ε l l (cid:48) X ε k k (cid:48) . For anypermutation σ ∈ S M , by composing the equations in ( 2.5), we obtain a birationalmapping which satisfies ( 2.4).To show the uniqueness, we note that the kernel of the left hand side of the firstequation ( 2.4) at z = c σ ( M ) should be equal to that of the right hand side. Thiscondition determine the matrix X ε σ ( M ) σ ( M ) uniquely. Remark 2.1.
It is easily shown that the system P , (0 , M ) and P N +2 , (0 , are re-spectively equivalent to the 2( M −
1) dimensional q -Garnier system [12] and the q - P ( N +1 ,N +1) [13] in the sense that the matrix A in ( 2.1) is equivalent to the matrix A given in [8]. Remark 2.2.
The equation ( 2.1) for the system P N, ( M − ,M + ) is an extended versionof the system in [4] [9] obtained as the N -reduced q -KP hierarchy through a similaritycondition. F N,M as an extension of q -hypergeometricfunctions In this subsection, we will recall some results in [10] which are needed in the followingsubsections.
Definition 3.1. ([10], Definition 2.1) We define a series F N,M as F N,M (cid:16) { a j } , { b i }{ c j } ; { y i } (cid:17) = (cid:88) m i ≥ N (cid:89) j =1 ( a j ) | m | ( c j ) | m | M (cid:89) i =1 ( b i ) m i ( q ) m i M (cid:89) i =1 y m i i , (3.1)where ( a ) n = ( a ) ∞ ( q n a ) ∞ and 0 < | q | <
1. Here and in what follows the symbol ( a ) ∞ means ( a ) ∞ = (cid:81) ∞ i =0 (1 − q i a ). The series ( 3.1) converges in the region | y i | < | y i | ≥ N = 1 or M = 1, the series ( 3.1) is equal to the q -Appell-Lauricellafunction ϕ D or the generalized q -hypergeometric function N +1 ϕ N , respectively [2]: F ,M (cid:18) a, { b i } c ; { y i } (cid:19) = (cid:88) m i ≥ ( a ) | m | ( c ) | m | M (cid:89) i =1 ( b i ) m i ( q ) m i y m i i = ϕ D (cid:18) a, { b i } c ; { y i } (cid:19) , (3.2) F N, (cid:18) { a j } , b { c j } ; y (cid:19) = (cid:88) n ≥ N (cid:89) j =1 ( a j ) m ( b ) m ( c ) m ( q ) m y m = N +1 ϕ N (cid:18) { a j } , b { c j } ; y (cid:19) . (3.3)There is a duality relation between the series F N,M and F M,N as follows:
Proposition 3.1. ([10], Proposition 2.1) The series F N,M satisfies the relation F N,M (cid:18) { y j } , { a i }{ b j y j } ; { x i } (cid:19) = N (cid:89) j =1 ( y j ) ∞ ( b j y j ) ∞ M (cid:89) i =1 ( a i x i ) ∞ ( x i ) ∞ F M,N (cid:18) { x i } , { b j }{ a i x i } ; { y j } (cid:19) . (3.4)3 emark 3.1. When N = 1 or M = 1, the relation ( 3.4) is known (see [1] [3] forexample).We can interpret the equation ( 3.4) as an integral representation of F N,M asfollows.
Corollary 3.1. ([10], Corollary 2.1.) With y j = q γ j , the relation ( 3.4) can berewritten as F N,M (cid:18) { q γ j } , { a i }{ b j q γ j } ; { x i } (cid:19) = N (cid:89) j =1 ( q γ j , b j ) ∞ ( b j q γ j , q ) ∞ N (cid:89) j =1 (cid:90) d q t j M (cid:89) i =1 ( a i x i (cid:81) Nj =1 t j ) ∞ ( x i (cid:81) Nj =1 t j ) ∞ N (cid:89) j =1 ( qt j ) ∞ ( b j t j ) ∞ t γ j − j − q , (3.5)where the Jackson integral is defined as (cid:90) c d q tf ( t ) = c (1 − q ) (cid:88) n ≥ f ( cq n ) q n . (3.6) Proposition 3.2. ([10], Proposition 2.2) The series F = F N,M (cid:16) { a j } , { b i }{ c j } ; { y i } (cid:17) sat-isfies the q -difference equations (cid:110)(cid:81) Nj =1 (1 − c j q − T ) · (1 − T y s ) − y s (cid:81) Nj =1 (1 − a j T ) · (1 − b s T y s ) (cid:111) F = 0 (1 ≤ s ≤ M ) , { y r (1 − b r T y r )(1 − T y s ) − y s (1 − b s T y s )(1 − T y r ) }F = 0 (1 ≤ r < s ≤ M ) , (3.7)where T y s is the q -shift operator for the variable y s and T = T y · · · T y M .We consider a representation by a Pfaffian system of the equation ( 3.7) in thenext subsection. F N,M
In this section, we derive a Pfaffian system of size (
M N + 1) × ( M N + 1) from anintegral representation of F N,M . From Corollary 3.1, the integral representation of F N,M is given in ( 3.5). We compute the Pfaffian system for the integral ( 3.5). Todo this, we follow the method given in [5], [6] in the same way as [10]. We denotethe integrand of ( 3.5) as Φ( { u j } Nj =1 )Φ( { u j } Nj =1 ) = N (cid:89) j =1 u ν j j ( qu j /u j − ) ∞ ( b j u j /u j − ) ∞ M (cid:89) i =1 ( a i x i u N ) ∞ ( x i u N ) ∞ , (4.1)where u j stands for (cid:81) jk =1 t k and we put the parameter ν j := γ j − γ j +1 ( ν N +1 = 0).We define functions Ψ , Ψ j,i (1 ≤ j ≤ N, ≤ i ≤ M ) asΨ = (cid:104) Φ p (cid:105) , Ψ j,i = (cid:104) Φ p j,i (cid:105) (4.2)where p = 1, p j,i = u j − − b j u j − a i x i u N i − (cid:89) k =1 − x k u N − a k x k u N and (cid:104) (cid:105) means a kind of Jacksonintegral for u , u , · · · , u N . Namely, (cid:104) f ( { u j } ) (cid:105) = (cid:88) n j ∈ Z f ( { q n j } ). We will see that { Ψ , Ψ , , · · · , Ψ N,M } is a basis of the solutions of the Pfaffian system.4 emark 4.1. The basis { p , p j,i } defined above has been used in many literatures(See for example [5], [7], [11]). Such a kind of basis is convenient for a computationbecause it is directly related to a shift of function Φ as Φ p j,i = u j − T a i T b j T x i − T x i − · · · T x Φ.Therefore the following equations holdΨ = ( { b j q γ j } , { q } ) ∞ ( { q γ j } , { b j } ) ∞ F , Ψ j,i = (1 − b j )(1 − b j q γ j ) j − (cid:89) l =1 (1 − q γ l )(1 − b l q γ l ) T a i T x T x · · · T x i − T b j F , (4.3) F = F N,M (cid:18) { q γ j } , { a i }{ b j q γ j } ; { x i } (cid:19) ( 1 ≤ j ≤ N ≤ i ≤ M ) (4.4)We define an exchange operator σ i (1 ≤ i ≤ M ) acting on a function f of { x i , a i } as σ i ( f ) = f | x i ↔ x i +1 ,a i ↔ a i +1 . (4.5)We note that σ i (Φ( { u j } )) = Φ( { u j } ). When the operator σ i acts on functions p , p j,i (1 ≤ j ≤ N, ≤ i ≤ M ), we have the following relations. Proposition 4.1.
For 1 ≤ i ≤ M , we have σ i (Ψ j,k ) = (1 − a i ) x i x i − a i +1 x i +1 Ψ j,i + a i x i − a i +1 x i +1 x i − a i +1 x i +1 Ψ j,i +1 , ( k = i ) x i − x i +1 x i − a i +1 x i +1 Ψ j,i + (1 − a i +1 ) x i +1 x i − a i +1 x i +1 Ψ j,i +1 , ( k = i + 1)Ψ j,k . ( k (cid:54) = i, i + 1) (4.6) Proof.
The last equation of ( 4.6) is obvious. The other equations of ( 4.6) is provedin the same way as Proposition 3.1. in the previous work [10]. Namely we considerthe equations σ i ( p j,i ) = s p j,i + s p j,i +1 ,σ i ( p j,i +1 ) = s p j,i + s p j,i +1 . (4.7)There exist unique coefficients s , s , s , s independent of u j satisfying these rela-tions.For the action of the q -shift operator T x M on Ψ , · · · , Ψ N,M , we obtain thefollowing equations.
Proposition 4.2.
We have T x M (Ψ ) = x M − βa M x M − β ρ (Ψ ) + ( a M − x M a M x M − β N (cid:88) j =1 ( j − (cid:89) k =1 b k ) ρ (Ψ j, ) ,T x M (Ψ j,i ) = ρ (Ψ j,i +1 ) ( i (cid:54) = M ) ,T x M (Ψ j,M ) = q − (cid:80) Nk = j γ k (1 − b j ) a M x M − β ( N (cid:89) l = j +1 b l ) (cid:20) ρ (Ψ ) − j − (cid:88) k =1 ( k (cid:89) t =1 b t ) ρ (Ψ k, )+ a M x M − β/b j (1 − b j ) (cid:81) Nk = j b k ρ (Ψ j, ) − a M x M N (cid:88) k = j +1 ( N (cid:89) t = k b − t ) ρ (Ψ k, ) (cid:21) , (4.8)where β = (cid:81) Nj =1 b j , ρ = σ M − · · · σ . 5 roof . First, we make a shift by T x M on Φ p , Φ p j,i (1 ≤ j ≤ N, ≤ i ≤ M ). Weeasily obtain the following equations T x M (Φ p j,i ) = Φ ρ ( p j,i +1 ) ( i (cid:54) = M ) ,T x M (Φ p ) = Φ 1 − x M u N − a M x M u N ,T x M T − u j T − u j +1 · · · T − u N (Φ p j,M ) = q − γ j Φ u j − − u j − a M x M u N . (4.9)The right-hand side of the second and third equations in ( 4.9), can be rewritten asa linear combination of ρ (Φ p ) = Φ p and ρ (Φ p j, ) = Φ u j − − b j u j − a M x M u N respectively,that is, T x M (Φ p ) = x M − βa M x M − β ρ (Φ p ) + ( a M − x M a M x M − β N (cid:88) j =1 ( j − (cid:89) k =1 b k ) ρ (Φ p j, ) ,T x M (Φ p j,M ) = q − (cid:80) Nk = j γ k (1 − b j ) a M x M − β ( N (cid:89) l = j +1 b l ) (cid:20) ρ (Φ p ) − j − (cid:88) k =1 ( k (cid:89) t =1 b t ) ρ (Φ p k, )+ a M x M − β (1 − b j ) (cid:81) Nk = j b k ρ (Φ p j, ) − a M x M N (cid:88) k = j +1 ( N (cid:89) t = k b − t ) ρ (Φ p k, ) (cid:21) , (4.10)where β = (cid:81) Nj =1 b j , ρ = σ M − · · · σ . Integrating the first equation of ( 4.9) andequations in ( 4.10) with respect to u , u , · · · , u N , we obtain the equations in ( 4.8).Combining Proposition 4.1 and Proposition 4.2, we obtain the following Theo-rem. Theorem 4.1.
The vector −→ Ψ = [Ψ A ] A ∈ I , I = { } ∪ { ( j, i ) | ≤ j ≤ N, ≤ i ≤ M } satisfies the Pfaffian system of rank M N + 1 T x M −→ Ψ = ρµ −→ Ψ = σ M − σ M − · · · σ µ −→ Ψ , (4.11)where the operators µ and σ i (1 ≤ i ≤ M ) are µ ( x ) = qx , µ (Ψ ) = x − βa x − β Ψ + ( a − x a x − β N (cid:88) j =1 ( j − (cid:89) k =1 b k )Ψ j, ,µ (Ψ k,l ) = Ψ k,l +1 ( l (cid:54) = M ) ,q − (cid:80) Nk = j γ k (1 − b j ) a x − β ( N (cid:89) l = j +1 b l ) (cid:20) Ψ − j − (cid:88) k =1 ( k (cid:89) t =1 b t )Ψ k, + a x − β/b j (1 − b j ) (cid:81) Nk = j b k Ψ j, − a x N (cid:88) k = j +1 ( N (cid:89) t = k b − t )Ψ k, (cid:21) ( l = M ) , (4.12)6 i ( a i ) = a i +1 , σ i ( a i +1 ) = a i , σ i ( x i ) = x i +1 , σ i ( x i +1 ) = x i ,σ i (Ψ ) = Ψ , σ i (Ψ k,l ) = (1 − a i ) x i x i − a i +1 x i +1 Ψ k,i + a i x i − a i +1 x i +1 x i − a i +1 x i +1 Ψ k,i +1 , ( l = i ) x i − x i +1 x i − a i +1 x i +1 Ψ k,i + (1 − a i +1 ) x i +1 x i − a i +1 x i +1 Ψ k,i +1 , ( l = i + 1)Ψ k,l , ( l (cid:54) = i, i + 1)(4.13) Proof.
The results follows by direct computation using ( 4.6) ( 4.8).
Remark 4.2.
The equations for the shift of the other variables T x i −→ Ψ = −→ Ψ A i , (4.14)for i = 1 , · · · , M − { σ i } in Proposition4.1. For example, the equation for the shift of the variable x M − is obtained as follows T x M − −→ Ψ = σ M − T x M σ M − −→ Ψ = −→ Ψ R M − · σ M − ( A M ) · σ M − ( R M − ( qz )) . (4.15)By construction, the coefficient matrices in ( 4.11) and ( 4.14) satisfy a compatibilitycondition A i ( T x i A j ) = A j ( T x j A i ) . (4.16) ( N + 1) × ( N + 1) form In this section we reduce the equation ( 4.14) into ( N + 1) × ( N + 1) form. To dothis, we specialize the parameter a M to be 1. Then the integrand ( 4.1), and henceΨ j,i (1 ≤ j ≤ N, ≤ i ≤ M −
1) and Ψ , become independent of x M . Therefore wecan consider Ψ j,i (1 ≤ j ≤ N, ≤ i ≤ M −
1) as Ψ times r j,i , where r j,i is a rationalfunction in x , · · · , x M − . The explicit form of r j,i is as follows (see ( 4.3)) r j,i = 1 − b j − q γ j j (cid:89) l =1 − q γ l − b l q γ l N (cid:89) k =1 ( q γ k , b k ) ∞ ( b k q γ k , q ) ∞ T a i T x T x · · · T x i − T b j FF (5.1) F = F N,M − (cid:18) { q γ j } , { a i }{ b j q γ j } ; { x i } (cid:19) . (5.2) Theorem 5.1.
Specializing a M = 1 and setting z = x M , t = x M − and Ψ j,i = r j,i Ψ (1 ≤ j ≤ N, ≤ i ≤ M − T z −→ Ψ red = −→ Ψ red (cid:18) M − (cid:89) i =1 z − a M − i x M − i z − x M − i X M − i ) − X M − i ) − (cid:19) X M − − X M D ,T t −→ Ψ red = −→ Ψ red z − qtz − a M − qt X M − ( z/q ) X M − − ( z/q ) D , (5.3)7here −→ Ψ red is a vector with N +1 components: −→ Ψ red = [Ψ red0 , Ψ red1 ,M , Ψ red2 ,M , · · · , Ψ red N,M ],Ψ red j,M = Ψ j,M | a M =1 , D = diag[1 , q − (cid:80) Nj =1 γ j , q − (cid:80) Nj =2 γ j , · · · , q − γ N ] , D = diag[ d , , · · · , ,X k = u ,k u ,k . . .. . . 1 z u N +1 ,k , (cid:81) N +1 j =1 u j, i − = ( − N x M − i , (cid:81) N +1 j =1 u j, i = ( − N a M − i x M − i . (5.4)Here { u j,i } are independent of z and they are given as a rational functions in { r j,i } .Explicit forms of { u j,i } are given in the proof. Proof . When a M = 1, obviously we have T z (Ψ ) = Ψ . We will compute T z (Ψ j,M )(1 ≤ j ≤ N ). In the equation ( 4.11), we have T z −→ Ψ red = ρµ −→ Ψ red , = ρ ( −→ Ψ (1) Q ) , = σ M − σ M − · · · σ ( −→ Ψ (2) R Q ) , = σ M − σ M − · · · σ ( −→ Ψ (3) R R Q )...= −→ Ψ ( M ) R M − R M − M − R M − ,M − M − · · · R M − ,M − , ··· , ρ ( Q ) , (5.5)where we put a vector −→ Ψ ( i ) := (cid:2) Ψ Ψ ,i Ψ ,i · · · Ψ N,i (cid:3) (1 ≤ i ≤ M ) and( ∗ ) k ,k , ··· ,k l stands for σ k l σ k l − · · · σ k ( ∗ ). The matrices Q and R i come from theequations ( 4.12) and ( 4.13). The matrix Q is a representation matrix of the trans-formation µ for the vector −→ Ψ red and −→ Ψ (1) . By the equation ( 4.12), we have µ −→ Ψ red = −→ Ψ (1) Q = −→ Ψ (1) − b ... − b ...... ... − b N − a x b N − − − x × diag[1 , q − (cid:80) Nj =1 γ j , q − (cid:80) Nj =2 γ j , · · · , q − γ N ] . (5.6)The matrix R i is a representation matrix of the transformation σ i for the vector −→ Ψ ( i ) and −→ Ψ ( i +1) . In the first equation of the equation ( 4.13), we put r j,i := Ψ j,i / Ψ σ i −→ Ψ ( i ) = −→ Ψ ( i +1) R i = −→ Ψ ( i +1) a i x i − a i +1 x i +1 x i − a i +1 x i +1 × − x i r N,i − r ,i ... − r ,i r ,i ...... 1 a i +1 x i +1 − r N − ,i r N,i − a i x i r N,i − r ,i ... − r ,i r ,i ...... ... 1 a i +1 x i +1 − r N − ,i r N,i − (5.7)The second equation for T t −→ Ψ is obtained similarly by considering T t = σ M − T z σ M − .In fact, by the equations ( 4.10) and ( 4.6), we have T t −→ Ψ = −→ Ψ z − qtz − qa M − t v z − qa M − tz − qt v z − qt · · · v N +1 z − qt = −→ Ψ z − qtz − qa M − t v , v , . . .. . . 1 z/q v N +1 , v , v , . . .. . . 1 z/q v N +1 , − d (cid:48) = −→ Ψ z − qtz − qa M − t X (cid:48) M − ( z/q ) X (cid:48)− M − ( z/q ) D (cid:48) , (5.8)where (cid:81) N +1 j =1 v j, = ta M − , (cid:81) N +1 j =1 v j, = t . And v j,i are rational functions which areindependent of z . Remark 5.1.
The equations ( 5.5), ( 5.8) satisfy a compatibility condition by con-struction. Therefore, the rational function r j,i satisfies certain difference equation.For instance, in case of ( M, N ) = (2 , r , = − q − γ ( a tr , + b r , − b r , − r , − b + 1) − a b tr , − a tr , + b tr , + tr , + b b − t ,r , = q − γ − γ ( − a b tr , − a tr , + a tr , + b r , + b b − b ) − a b tr , − a tr , + b tr , + tr , + b b − t . (5.9)These equations are interpreted as multivariable Riccatti equations for the specialsolution of the corresponding system P , (2 , . The solution is given in ( 5.1).In order to relate the above results to the monodromy preserving deformations P N +1 , ( M , M +) , we specify the deformation equation as follows. First we choose (cid:126)ε as (cid:126)ε = (+ , − , + , − , · · · , + , − , − , +). Namely we consider a linear equation ( 2.1)Ψ( qz ) = Ψ( z ) A ( z ) , A ( z ) = DX ( z ) X − ( z ) · · · X M − ( z ) X − M − ( z ) X M − ( z ) − X M ( z ) . (5.10)9s a deformation of this, we consider the following equationΨ = Ψ B ( z ) , B ( z ) = X M ( z/q ) X − M − ( z/q ) , (5.11)where the shift of parameters are given by a k = a k ( k = 1 , , · · · , M ) ,b j = b j ( j = 1 , , · · · , N ) ,c l = c l ( l = 1 , , · · · , M − , c i = qc i ( i = 2 M − , M ) ,d j = d j ( j = 1 , , · · · , N ) . (5.12)We denote by P N +1 , ( M,M ) the nonlinear equation arising as the comatibility condition A ( z ) B ( qz ) = B ( z ) A ( z ) of ( 5.10) ( 5.11). Then we have Theorem 5.2.
Under the specialization d = 1, the nonlinear equation P N +1 , ( M,M ) ( 2.1) admits a particular solution given in terms of a generalized hypergeometricfunctions F N,M − as follows. u , M − i ) − = − r N,i x i d d N +1 ,u , M − i ) = − r N,i a i x i d d N +1 ,u , M − = d d N +1 ,u , M = d d N +1 , (1 ≤ i ≤ M ) u j, M − i ) − = − r j − ,i d j r j − ,i d j − ,u j, M − i ) = − r j − ,i d j r j − ,i d j − ,u j, M − = − b j − d j d j − ,u j, M = − d j d j − , (cid:16) ≤ i ≤ M ≤ j ≤ N (cid:17) u N +1 , M − i ) − = − r N +1 − ,i d N +1 r N,i d N ,u N +1 , M − i ) = − r N − ,i d N +1 r N,i d N ,u N +1 , M − = − b N d N +1 d N ,u N +1 , M = d d N (1 ≤ i ≤ M ) , (5.13)and c i − = ( − N x M − i , c i = ( − N a M − i x M − i ,d = 1 , d k = q − (cid:80) Nj = k − γ j (1 ≤ i ≤ M, ≤ k ≤ N + 1) , (5.14)where r j,i is a ratio of the hypergeometric function as given ( 5.1) r j,i = 1 − b j − q γ j j (cid:89) l =1 − q γ l − b l q γ l N (cid:89) k =1 ( q γ k , b k ) ∞ ( b k q γ k , q ) ∞ T a i T x T x · · · T x i − T b j FF (5.15)10nd the function F stands for F = F N,M − (cid:16) { q γj } , { a i }{ b j q γj } ; { x i } (cid:17) (1 ≤ j ≤ N, ≤ i ≤ M − Proof.
Then, the system of equations ( 5.3) is a specialization of the equation ( 5.10)via a gauge transformation (cid:98)
Ψ = M − (cid:89) i =1 ( a i x i q/z ) ∞ ( x i q/z ) ∞ Ψ . (5.16)In more detail, we define a transformation δ ( ε i ,ε i +1 ) k,l of variables { u j,k } and { d j } suchthat δ ( ε k ,ε l ) k,l ( DX ε k k ( z ) X ε l l ( z )) = X ε k k ( z ) X ε l l ( z ) D. (5.17)This translation is uniquely given as follows δ ± , ∓ k,l ( u j,k , u j,l , d j ) = ( d j − d j u j,k , d j − d j u j,l , d j ) . (5.18)We apply the transformation ( 5.18) on the equation ( 5.10) repeatedly. Then, thefollowing equation holds δ − , +2 M − , M δ + , − M − , M − · · · δ + , − , δ + , − , ( DX X − · · · X M − X − M − X − M − X M )= Y Y − · · · Y M − Y − M − Y − M − Y M D, (5.19)where a matrix Y i stands for a matrix X i whose variable u j,i replaced y j,i = d j − d j u j,i .Comparing the coefficients of the right-hand side of the equation ( 5.19) and of thefirst equation of the equation ( 5.3), we obtain the following result ( 5.13) where d = 1 , d k = q − (cid:80) Nj = k − γ j (2 ≤ k ≤ N + 1), r ,i = 1. We recall that the rationalfunction r j,i is given as in ( 5.1) Remark 5.2.
Theorem 5.1 gives a q -analogue of an argument to reduce a rank ofa linear Pfaffian system in section 5 of [15]. Remark 5.3.
The system ( 2.1) for P , ( M − ,M + ) ( M − + M + = 2 M ) is equivalent tothe Lax equation for 2( M −
1) dimensional q -Garnier system [8] [12] via the followinggauge transformation (cid:98) Ψ = M − (cid:89) i =1 (cid:18) ( qc i /z ) ∞ ( − z ) ∞ ( − q/z ) ∞ (cid:19) Ψ . (5.20) Acknowledgement
The author would like to express her gratitude to Professor Yasuhiko Yamada forvaluable suggestions and encouragement. She also thanks supports from JSPS KAK-ENHI Grant Numbers 17H06127 and 26287018 for the travel expenses in accomplish-ing this study. 11 ibliography [1] Andrews, George, E. (1972) Summations and transformations for basic Appellseries,
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