Connection Formula for the Jackson Integral of Type A n and Elliptic Lagrange Interpolation

. We investigate the connection problem for the Jackson integral of type A n . Our connection formula implies a Slater type expansion of a bilateral multiple basic hypergeo-metric series as a linear combination of several speciﬁc multiple series. Introducing certain elliptic Lagrange interpolation functions, we determine the explicit form of the connection coeﬃcients. We also use basic properties of the interpolation functions to establish an explicit determinant formula for a fundamental solution matrix of the associated system of q -diﬀerence equations.

For the general setting of Jackson integrals see [3].
We remark that for s = 1, ϕ ≡ 1 and z = a 1 , a 1 t, . . ., a 1 t n−1 ∈ (C * ) n , the Jackson integral (1.8) is expressed as 1, a 1 , a 1 t, . . ., a 1 t n−1 whose limiting case where q → 1 is equivalent to the Selberg integral (1.5).In this sense the Jackson integral (1.8) includes the Selberg integral as a special case.The formula (1.9) was discovered by Askey [5] and proved by Habsieger [11], Kadell [16], Evans [6] and Kaneko [19] in the case where τ is a positive integer, while (1.9) for general complex τ was given by Aomoto [1].See [13] for further details.Other closely related and relevant works are [17,18,33].We denote by S n the symmetric group of degree n; this group acts on the field of meromorphic functions on (C * ) n through the permutations of variables z 1 , . . ., z n .Setting where θ(u) = (u) ∞ qu −1 ∞ , we have the following.Lemma 1.1.Let ϕ(z) be an S n -invariant holomorphic function on (C * ) n , and suppose that the Jackson integral ϕ, z of (1.8) converges as a meromorphic function on (C * ) n .Then the function f (z) = ϕ, z defined by (1.10) is an S n -invariant holomorphic function on (C * ) n .Furthermore it satisfies the quasi-periodicity T q,z i f (z) = f (z)/(−z i ) s q α t n−1 b 1 • • • b s for i = 1, . . ., n, where T q,z i stands for the q-shift operator in z i .
For the proof of this lemma, see Lemma 3.3.We call ϕ, z the regularized Jackson integral of A-type, which is the main object of this paper.We remark that, when ϕ(z) is a symmetric polynomial such that deg z i ϕ(z) ≤ s − 1, i = 1, . . ., n, it satisfies the condition of Lemma 1.1 under the assumption (3.2) below.
In order to state our first main theorem we define some terminology.We set where N = {0, 1, 2, . ..}, so that |Z s,n | = s+n−1 n .We use the symbol for the lexicographic order on Z s,n .Namely, for µ, ν ∈ Z s,n , we denote µ ≺ ν if there exists k ∈ {1, 2, . . ., n} such that Theorem 1.2 (connection formula).Suppose that ϕ(z) is an S n -invariant holomorphic function satisfying the condition of Lemma 1.1.Then, for generic x ∈ (C * ) s we have where the connection coefficients c λ are explicitly written as where λ , . . ., k}|, and the summation is taken over all partitions We call the formula (1.13) the generalized Slater transformation.In fact, the connection formula (1.13) for n = 1 coincides with the transformation formula (1.3) if ϕ(z) ≡ 1, as explained below.
As we will see below the connection coefficients c µ in (1.14) are characterized, as functions of z ∈ (C * ) n , by an interpolation property.
We next apply the connection formula (1.13) in Theorem 1.2 to establish a determinant formula associated with the Jackson integral (1.8).Let B s,n be the set of partitions defined by . We also use the symbol for the lexicographic order of B s,n .For each λ ∈ B s,n , we denote by s λ (z) the Schur function which is a S n -invariant polynomial.Our second main theorem is the following.
Theorem 1.5 (determinant formula).Suppose that x ∈ (C * ) s is generic.Then we have where the rows and the columns of the matrix are arranged by the lexicographic orders of λ ∈ B s,n and µ ∈ Z s,n , respectively.Here C is a constant independent of x, which is explicitly written as .
Remark.When s = 1, the determinant formula (1.18), combined with the connection formula for the Jackson integral 1, z for an arbitrary z ∈ (C * ) n .This formula for n = 1 coincides with Ramanujan's formula (1.1).This is another multi-dimensional bilateral extension of Ramanujan's 1 ψ 1 summation theorem, which is different from the Milne-Gustafson summation theorem [10,25].Another class of extension relates to the theory of Macdonald polynomials; see for example [19,20,26,33] as cited in [34].
In this paper, we prove Theorems 1.2 and 1.5 from the viewpoint of the elliptic Lagrange interpolation functions of type A n .Let O((C * ) n ) be the C-vector space of holomorphic functions on (C * ) n .In view of Lemma 1.1, fixing a constant ζ ∈ C * we consider the C-linear subspace H s,n,ζ ⊂ O((C * ) n consisting of all S n -invariant holomorphic functions f (z) such that T q,z i f (z) = f (z)/(−z i ) s ζ for i = 1, . . ., n, where T q,z i stands for the q-shift operator in z i : ( The dimension of H s,n,ζ as a C-vector space will be shown to be n+s−1 n in Section 2.Moreover, Theorem 1.6.For generic x ∈ (C * ) s there exists a unique C-basis where x ν ∈ (C * ) n , ν ∈ Z s,n , are the reference points specified by (1.12) and δ λµ is the Kronecker delta.
This theorem will be proved in the end of Section 2.2.We call E µ (x; z) the elliptic Lagrange interpolation functions of type A n associated with the set of reference points x ν , ν ∈ Z s,n .Note that an arbitrary function f (z) ∈ H s,n,ζ can be written as a linear combination of E µ (x; z), µ ∈ Z s,n , with coefficients f (x µ ): (1.21) Theorem 1.7.The functions E λ (x; z) are expressed as where λ , . . ., k}|, and the summation is taken over all partitions This theorem will be proved in Section 2 as Theorem 2.4.Once Theorems 1.6 and 1.7 have been established, the connection formula (1.13) in Theorem 1.2 is immediately obtained.In the setting of Theorem 1.2, the regularization ϕ, z = ϕ, z /Θ(z) belongs to H s,n,ζ with ζ = q α t n−1 b 1 For the evaluation of the determinant of Theorem 1.5, we make use of the following determinant formula for the elliptic Lagrange interpolation functions , where y ν are specified as in (1.12).This theorem will be proved as Theorem 2.8 in Section 2.
Applying the connection formula (1.13) in Theorem 1.2, we see that Theorem 1.5 is reduced to the special case where x = a = (a 1 , . . ., a s ).Since Since we already know the explicit form of det E µ (a; x ν ) µ∈Z,ν∈Z by Theorem 1.8, the evaluation of det s λ , x ν λ∈B,ν∈Z reduces to that of det s λ , a µ λ∈B,µ∈Z .
Remark.Tarasov and Varchenko [32] have already obtained a determinant formula for a multiple contour integrals of hypergeometric type, which is similar to (1.24).See [32, Theorem 5.9] for the details.
In the succeeding sections, we give proofs for Theorems 1.6, 1.7, 1.8 and Lemmas 1.1, 1.9, which we use for proving our main theorems.
This paper is organized as follows.In the first part of Section 2 we prove Theorems 1.6 and 1.7 on the basis of an explicit construction of the elliptic Lagrange interpolation functions by means of a kernel function as in [22].In the second part of Section 2 we investigate the transition coefficients between two sets of elliptic interpolation functions.In particular we provide a proof of Theorem 1.8 for the determinant of the transition matrix.A proof of Lemma 1.1 for the regularized Jackson integrals will be given in Section 3. In Section 4 we introduce certain interpolation polynomials which are a limiting case of the elliptic Lagrange interpolation functions.These polynomials are used in Section 5 for the construction of the q-difference system satisfied by the Jackson integrals.In particular we establish two-term difference equations with respect to α → α + 1 for the determinant of the Jackson integrals.Section 6 is devoted to analyzing the boundary condition for difference equations through asymptotic analysis of the Jackson integrals as α → +∞, which completes the proof of Lemma 1.9.In Appendix A, we provide a detailed proof of Lemma 5.4 which is omitted in Section 5.In Appendix B we give proofs for some propositions in Section 4 by using the kernel function in the similar way as in Section 2.

The elliptic Lagrange interpolation functions of type A
In this section we give proofs of Theorems 1.6, 1.7 and 1.8.
Let P + be the set of partitions of length at most n specified by For µ = (µ 1 , . . ., µ n ) ∈ Z n , we denote by z µ the monomial z µ 1 1 • • • z µn n .For the partitions λ ∈ P + let m λ (z) be the monomial symmetric functions [23] defined by where S n λ = {wλ | w ∈ S n } is the S n -orbit of λ.For a function f (z) = f (z 1 , z 2 , . . ., z n ) on (C * ) n , we define the action of the symmetric group S n on f (z) by We say that a function f (z) on (C * ) n is symmetric or skew-symmetric if σf (z) = f (z) or σf (z) = (sgn σ)f (z) for all σ ∈ S n , respectively.We denote by Af (z) the alternating sum over S n defined by which is skew-symmetric.
For an arbitrarily fixed ζ ∈ C * , we consider the C-linear subspace H s,n,ζ ⊂ O((C * ) n ) consisting of all S n -invariant holomorphic functions f (z) such that T q,z i f (z) = f (z)/(−z i ) s ζ, i = 1, . . ., n, as in (1.19).In this section we use the symbol Since θ(u) = θ(q/u) and θ(qu In particular we have e(u, v) → u − v in the limit q → 0.

Construction of the interpolation functions
This subsection is devoted to providing a proof of Theorem 1.6.In the first half, we define a family of functions E (n) λ (x; z) recursively with respect to the number n of variables z 1 , . . ., z n , and show that those E (n) λ (x; z) are expressed explicitly as (1.22) in Theorem 1.7.In the second half, we prove that they are in fact the elliptic interpolation functions in the sense of Theorem 1.6; we show that x µ ) = δ λµ using the dual Cauchy kernel as in [22].First of all we show the following lemma.Proof .For an arbitrary f (z) ∈ H s,n,ζ , since f (z) is a holomorphic on (C * ) n , f (z) may be expanded as Laurent series as f (z) = λ∈Z n c λ z λ .Since f (z) is symmetric, all coefficients of f (z) are determined from c λ corresponding to λ ∈ P + .On the other hand, since f (z) satisfies T q,z i f (z) = f (z)/(−z i ) s ζ for i = 1, . . ., n, we have Equating coefficients of z λ on both sides, all coefficients of f (z) are determined from c λ corresponding to λ satisfying s − 1 ≥ λ i ≥ 0, i = 1, . . ., n.Therefore, f (z) is determined by the coefficients c λ corresponding to λ ∈ B s,n defined by (1.17 Before introducing E (n) λ (x; z) we prove Theorem 1.6 for n = 1 by independent means.From the definition (1.11), we have Z s,1 = { 1 , 2 , . . ., s }, where i is specified in Example 1.3, and have ) where By the repeated use of ( 2.3) we have where Rewriting this formula as in [14, p. 373, Theorem 3.4] we obtain the following.

k}| and the summation is taken over all index sets K
We remark that these functions for λ = n i ∈ Z s,n have simple factorized forms; this fact will be used in the next subsection.
We simply write E λ (x; z) = E (n) λ (x; z) when there is no fear of misunderstanding.In the remaining part of this subsection we show that E λ (x; z) ∈ H s,n,ζ and E λ (x; x µ ) = δ λµ .

Lemma 2.6 (duality). Under the condition ζ s
i=1 w i = 1, Ψ(z; w) expands as Proof .We proceed by induction on n, the cardinality of z.We consider the case n = 1 as the base case.Under the condition Next we suppose n ≥ 2. We assume (2.10) holds for the number of variables for z less than n.
where we regard E (n−1) as desired.
When we consider the family of functions , we define a special point η ν (x) on the hypersurface as In particular, F µ (x; η ν (x)) = 0 for µ ν with respect to the lexicographic order of Z s,n .Moreover, if x ∈ (C * ) s is generic, then F µ (x; η µ (x)) = 0 for all µ ∈ Z s,n .
This lemma implies that the matrix F = F µ (x; η ν (x)) µ,ν∈Z is upper triangular, and also invertible if x ∈ C s is generic.

Transition coefficients for the interpolation functions
In this subsection we investigate the transition coefficients between two sets of interpolation functions with different parameters.Theorem 1.8 in the Introduction follows from Theorem 2.8 below when For generic x, y ∈ (C * ) s , the interpolation functions where the coefficients C µν (x; y) are independent of z.From the property (1.20) of the interpolation functions, we immediately see that C µν (x; y) may be expressed in terms of special value of E µ (x; z) as For x, y ∈ (C * ) s , we denote the transition matrix from (E λ (x; z)) λ∈Zs,n to (E λ (y; z)) λ∈Zs,n by where the rows and the columns are arranged in the total order ≺ of Z s,n .By definition, for generic x, y, w ∈ (C * ) s we have Theorem 2.8.For generic x, y ∈ (C * ) s the determinant of the transition matrix E(x; y) is given explicitly as Proof .The proof of this theorem is reduced to a special case indicated as Lemma 2.9 below.The details are the same as in the case of the BC n interpolation functions of [14, p. 377, proof of Theorem 4.1].
Lemma 2.9.For x, y ∈ (C * ) s suppose that In particular, E(x, y) is a lower triangular matrix with the diagonal entries Moreover the determinant of E(x, y) for y = (x 1 , . . ., x s−1 , y s ) is expressed as 3 Jackson integral of A-type
By definition the function Φ(z) satisfies the quasi-symmetric property that where U σ (z), as given by (3.8), is invariant under the q-shift z i → qz i .If ϕ(z) is a symmetric holomorphic function on (C * ) n , then we have This implies that In fact Lemma 3.2.Suppose ϕ(z) is symmetric and holomorphic.Under the condition τ ∈ Z, if z i = z j for some i and j, 1 ≤ i < j ≤ n, then ϕ, z = 0.
For the regularized Jackson integral ϕ, z defined in (1.10), we have the following.Proof .By the definition (1.10), Θ(z) satisfies that σΘ(z) = (sgn σ)U σ (z)Θ(z).Combining this and (3.17), we obtain that ϕ, z = ϕ, z /Θ(z) is symmetric if ϕ(z) is symmetric.We now check ϕ, z is holomorphic.Taking account of the poles of Φ(z), we have the expression where f (z) is some holomorphic function on (C * ) n .Since ϕ, z is an invariant under the qshift z i → qz i , 1 ≤ i ≤ n, Lemma 3.2 implies that ϕ, z is divisible by 1≤i<j≤n z i θ(z j /z i ) if τ ∈ Z.This indicates that f (z) = g(z) 1≤i<j≤n z i θ(z j /z i ), where g(z) is a holomorphic function on (C * ) n .From (1.10) and (3.18), we therefore obtain that ϕ, z = g(z) is a holomorphic function on (C * ) n .

The Lagrange interpolation polynomials of type A
In this section we introduce a class of interpolation polynomials of type A which correspond to the elliptic Lagrange interpolation functions discussed in Section 2. These polynomials play essential roles in constructing the q-difference equations satisfied by the Jackson integrals in the succeeding sections.Here we enumerate some properties of the interpolation polynomials without proof (see Appendix B for their detailed proofs).Let H z s,n be the C-linear subspace of C[z 1 , . . ., z n ] consisting of all S n -invariant polynomials f (z) such that deg z i f (z) ≤ s for i = 1, . . ., n: Note that dim C H z s,n = n+s n .For µ = (µ 0 , µ 1 , . . ., µ s ) ∈ Z s+1,n and x = (x 1 , . . ., x s ) ∈ (C * ) s we denote by x µ 0 µ 1 ,...,µs the point in (C * ) n defined as x µ 0 µ 1 ,...,µs = z 1 , z 2 , . . ., z µ 0 µ 0 , x 1 , x 1 t, . . ., x 1 t µ 1 −1 µ 1 , . . ., x s , x s t, . . ., x s t µs−1 µs leaving the first µ 0 variables unspecialized.
Proof .See Theorem B.1 in Appendix B.

Remark.
The polynomial E λ 0 λ 1 ,...,λs (x; z) may be expanded as where the leading term m ((s−1) n−λ 0 s λ 0 ) (z) is the monomial symmetric polynomial of type ((s − 1) n−λ 0 s λ 0 ) = (s, . . ., s ) ∈ P + and the remaining terms are of lower order with respect to the lexicographic order of P + .The coefficient C λ 0 λ 1 ,...,λs of the leading term is given by The above result will be proved in Lemma B.12 in Appendix B.
Corollary 4.5.We have where L = (l µν ) is a lower triangular matrix of size s+n−1 n .

Difference equations with respect to α
The aim of this section is to give a proof of the following proposition.
Then, the difference system (5.1) is transformed into This implies that by ( 5.3).
In the remaining part of this section we complete the proof of Proposition 5.1.We first explain a fundamental idea for deriving difference equations.Let Φ(z) be the function defined by (1.6) as before.For each i = 1, . . ., n we introduce the operator ∇ i by setting for a function ϕ(z), where T q,z i stands for the q-shift operator z i → qz i , i.e., T q,z i f (. . ., z i , . ..) = f (. . ., qz i , . ..).We remark that the ratio T q,z i Φ(z)/Φ(z) is expressed as where converge for all σ ∈ S n , then ξ∞ 0 Φ(z)A∇ i ϕ(z) q = 0 for i = 1, . . ., n, (5.9) where A indicates the skew-symmetrization defined in (2.1).
The statement (5.8) is equivalent to the q-shift invariance of the Jackson integral and (5.9) follows from the quasi-symmetry (3.15) of Φ(z).
We denote by C the right-hand side of (1.24).Then we have the following: Lemma 6.3.We have .
Proof .From (6.5) and the explicit form of C, we can immediately confirm the result.
We now prove Lemma 1.9.
On the other hand it is easily confirmed that C satisfies the same equation as above, i.e., .
This means that the ratio J /C is invariant under the shift T α with respect to α → α + 1.Thus J /C is also invariant under the shift T N α , N = 1, 2, . ... From Lemmas 6.1 and 6.3 we therefore obtain which is the claim of Lemma 1.9.
This also completes the proof of our determinant formula of Theorem 1.5.
In the same way as (4.1) we define x i t k−1 − x j t ν j .Thus if there exists i ∈ {1, . . ., s} such that ν i < µ i , then F µ (x; xt ν ) = 0.If ν ≺ µ, then ν i < µ i for some i ∈ {1, 2, . . ., s} by definition, and hence we obtain F µ (x; xt ν ) = 0 if ν ≺ µ.Moreover, from (B.3), we obtain  ; w , we have the expansion In particular, the leading term of E λ (x; z) with |λ| = r as a symmetric polynomial in z is equal to m ((s−1) r s n−r ) (z) up to a constant.See also [13,Appendix] for an application of these polynomials to the q-Selberg integral.
Proof of Lemma B.8. Since we have s j=1 (z 1 − w j ) = s i=0 E i (x; z 1 )F i (x; w) from (B.8), the function Ψ(z; w) may be expanded as which is equivalent to the identity [24, p. 46, Lemma 1.51] with (x i , y i ) → t λ i , x i t λ i .
Remark.The identity (B.21) is very well known and has played a very important role in the theory of multiple basic hypergeometric series.In the paper [28], Rosengren extended this identity to its elliptic form and applied it to his multiple elliptic hypergeometric series.It is really remarkable that, according to his paper, this identity in elliptic form already appeared in Tannery and Molk's book [31, p. 34] (which was published in 1898) and also in Whittaker and Watson [35, p. 451].See his very detailed discussion about this identity in [28].
Here, from the definition (B.3) of F µ (x; w), we have

Lemma 3 . 3 .
Suppose that τ ∈ Z.If ϕ(z) is a symmetric holomorphic function on (C * ) n , then ϕ, z is also a symmetric holomorphic function on (C * ) n .

λ
(x) the coefficient C λ (x) in (B.18).When r = 1, we immediately findC (1) i (x) = 1≤j≤s j =i 1 x i − x j for i = 1, . . .,s from the explicit expression (B.2) of E i (x; z).From the recursion formula (B.17) we have the relation C 19) holds for r = 1, it suffices to show that the right-hand side of (B.19) satisfies the same recurrence formula as (B.20).One can directly verify that the corresponding formula for the right-hand side of (B.19) reduces to the identity 1 = s k=1