Connection Formula for the Jackson Integral of Type A

Connection Formula for the Jackson Integral of Type A

and Elliptic Lagrange Interpolation

Masahiko ITO ^† and Masatoshi NOUMI ^‡

† Department of Mathematical Sciences, University of the Ryukyus, Okinawa 903-0213, Japan E-mail: mito@sci.u-ryukyu.ac.jp

‡ Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan E-mail: noumi@math.kobe-u.ac.jp

Received January 23, 2018, in final form July 07, 2018; Published online July 24, 2018 https://doi.org/10.3842/SIGMA.2018.077

Abstract. We investigate the connection problem for the Jackson integral of typeAn. 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 ex- plicit determinant formula for a fundamental solution matrix of the associated system of q-difference equations.

Key words: Jackson integral of type A_n; q-difference equations; Selberg integral; Slater’s transformation formulas; elliptic Lagrange interpolation

2010 Mathematics Subject Classification: 33D52; 39A13

1 Introduction

Throughout this paper we fix the baseq ∈Cwith 0<|q|<1, and use the notation ofq-shifted factorials

(u)∞=

∞

Y

l=0

1−uq^l

, (a1, . . . , ar)∞= (a1)∞· · ·(ar)∞, (u)ν = (u)∞/ uq^ν

∞, (a1, . . . , ar)ν = (a1)ν· · ·(ar)ν.

For generic complex parameters a1, . . . , ar and b1, . . . , br, the bilateral basic hypergeometric series _rψ_r [9, equation (5.1.1)] is defined by

rψr

a1, . . . , ar

b₁, . . . , b_r;q, x

=

∞

X

ν=−∞

(a1, . . . , ar)ν

(b₁, . . . , b_r)_ν x^ν, where |b₁· · ·br/a1· · ·ar|<|x|<1.

An important summation for ₁ψ₁ series is Ramanujan’s theorem [9, equation (5.2.1)],

1ψ₁ a

b;q, x

= (ax, q, b/a, q/ax)∞

(x, b, q/a, b/ax)∞

. (1.1)

There is also a summation formula for a very-well-poised ₆ψ₆ series due to Bailey [9, equa- tion (5.3.1)],

6ψ6

q√ a,−q√

a, b, c, d, e

√a, −√

a,^aq_b ,^aq_c,^aq_d,^aq_e ;q, a²q bcde

= aq,^aq_bc,^aq_bd,^aq_be,^aq_cd,^aq_ce,^aq_de, q,^q_a

∞ aq

b ,^aq_c,^aq_d,^aq_e,^q_b,^q_c,^q_d,^q_e,_bcde^a2q

∞

, (1.2)

This paper is a contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications.

The full collection is available athttps://www.emis.de/journals/SIGMA/EHF2017.html

where|a²q/bcde|<1. Furthermore there are a lot of transformation formulas for therψr series.

In the paper [30] Slater proved the transformation formula

rψr

a₁, . . . , a_r b1, . . . , br;q, z

=

c1

a1, . . . ,^c_a¹

r, c₂, . . . , c_r,_c^q

2, . . . ,_c^q

r,^b_c^1q

1 , . . . ,^b_c^rq

1 , Ac₁z,_Ac^q

1z

∞ q

a1, . . . ,_a^q

r,^c_c¹

2, . . . ,^c_c¹

r,^c_c^2q

1 , . . . ,^c_c^rq

1 , b₁, . . . , b_r, Azq,_Az¹

∞

× _rψr

"_a1q

c1 , . . . ,^a_c^rq

1

b1q

c1 , . . . ,^b_c^rq

1

;q, z

#

+ idem(c1;c2, . . . , cr), (1.3) whereA=a1· · ·ar/c1· · ·crand|b₁· · ·br/a1· · ·ar|<|z|<1, which is called theSlater’s general transformation for a _rψ_r series [9, equation (5.4.3)]. Here the symbol “idem(c₁;c₂, . . . , c_r)”

stands for the sum of the r −1 expressions obtained from the preceding one by interchan- ging c1 with eachck, k= 2, . . . , r. Slater also proved a transformation formula for a very-well- poised _2rψ_2r series, which is expressed as

2rψ_2r q√

a,−q√

a, b₃, . . . , b_2r

√a,−√ a,^aq_b

3, . . . ,_b^aq

2r

;q,a^r−1q^r−2 b₃. . . b_2r

= a4, . . . , ar,_a^q

4, . . . ,_a^q

r,^a_a⁴, . . . ,^a_a^r,^aq_a

4, . . . ,^aq_a

r,^a_b^3q

3 , . . . ,^a_b^3q

2r,_a^aq

3b3, . . . ,_a^aq

3b2r

∞ q

b3, . . . ,_b^q

2r,^aq_b

3, . . . ,_b^aq

2r,^a_a⁴

3, . . . ,^a_a^r

3,^a_a^3q

4 , . . . ,^a_a^3q

r ,^a3_a^a4, . . . ,^a3_a^ar,_a^aq

3a4, . . . ,_a^aq

3ar

∞

× aq,^q_a

∞ a²₃q

a ,^aq_a2 3

∞ 2rψ2r

"_qa

√3

a,−^qa√_a³,^a3_a^b3, . . . ,^a3_a^b2r

a3

√a,−^√a3

a,^a_b^3q

3 , . . . ,^a_b^3q

2r

;q,a^r−1q^r−2 b3. . . b2r

#

+ idem(a3;a4, . . . , ar), (1.4) where |a^r−1q^r−2/b3· · ·b2r| < 1, which is called Slater’s transformation for a very-well-poised balanced _2rψ_2r series[9, equation (5.5.2)].

The aim of this paper is to give an explanation and a generalization of Ramanujan’s 1ψ1

summation (1.1) and Slater’s rψr transformation (1.3) in relation to the Selberg integral [29]

(see also the recent sources [7,8]), i.e., 1

n!

Z 1 0

· · · Z 1

0 n

Y

i=1

z_i^α−1(1−zi)^β−1 Y

1≤j<k≤n

|z_j−zk|^2τdz1dz2· · ·dzn

=

n

Y

j=1

Γ(α+ (j−1)τ)Γ(β+ (j−1)τ)Γ(jτ)

Γ(α+β+ (n+j−2)τ)Γ(τ) , (1.5)

where Re(α) > 0, Re(β) > 0 and Re(γ) > −min{1/n,Re(α)/(n−1),Re(β)/(n−1)}, which is a multi-dimensional generalization of the evaluation of the Euler beta integral in terms of products of gamma functions. (This is not the first such paper using Selberg integrals to obtain multiple analogues of some of Slater’s transformation formulas. For example, the paper [14]

about a generalization for Bailey’s very-well-poised₆ψ₆ summation (1.2) and Slater’s very-well- poised_2rψ_2r transformation (1.4) withBC_nsymmetry, is very similar in spirit and methodology to the current paper.) For this purpose we define some terminology about a q-analog of the Selberg type integral. For z= (z1, . . . , zn)∈(C^∗)ⁿ, let Φ(z) be specified by

Φ(z) = Φs(z;α, a, b;q, t)

= (z₁z₂· · ·z_n)^α

n

Y

i=1 s

Y

m=1

qa⁻¹_m z_i

∞

(b_mz_i)∞

Y

1≤j<k≤n

z_j^2τ−1 qt⁻¹z_k/z_j

∞

(tz_k/z_j)∞

, (1.6)

withα∈C,a= (a₁, . . . , a_s),b= (b₁, . . . , b_s)∈(C^∗)^s andt∈C^∗, whereτ ∈Cis given byt=q^τ. For z= (z1, . . . , zn)∈(C^∗)ⁿ, let ∆(z) be the Vandermonde product

∆(z) = Y

1≤j<k≤n

(z_j−z_k). (1.7)

For a functionϕ=ϕ(z) of z= (z1, . . . , zn)∈(C^∗)ⁿ, we denote by hϕ, zi=

Z z∞

0

ϕ(w)Φ(w)∆(w)dqw1

w₁ ∧ · · · ∧ dqwn

w_n

= (1−q)ⁿ X

ν∈Zⁿ

ϕ(zq^ν)Φ(zq^ν)∆(zq^ν), (1.8)

the Jackson integral associated with the multiplicative lattice zq^ν = (z₁q^ν1, . . . , z_nq^νn)∈(C^∗)ⁿ, ν = (ν₁, . . . , ν_n) ∈ Zⁿ. Note that the sum hϕ, zi is invariant under the shifts z → zq^ν for all ν ∈Zⁿwhen it converges. This sum is called theJackson integral of A-typein the context of [1].

For the general setting of Jackson integrals see [3].

We remark that for s= 1, ϕ≡ 1 andz = a₁, a₁t, . . . , a₁tⁿ⁻¹

∈(C^∗)ⁿ, the Jackson integ- ral (1.8) is expressed as

1, a1, a1t, . . . , a1tⁿ⁻¹

= (1−q)ⁿ

n

Y

j=1

a₁t^j−1α+2(n−j)τ (q)∞(t)∞ q^αa₁b₁t^n+j−2

∞

(t^j)∞ q^αt^j−1

∞ a₁b₁t^j−1

∞

, (1.9)

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 whereτ is a positive integer, while (1.9) for general complexτ was given by Aomoto [1].

See [13] for further details. Other closely related and relevant works are [17,18,33].

We denote byS_nthe symmetric group of degreen; this group acts on the field of meromorphic functions on (C^∗)ⁿ through the permutations of variables z₁, . . . , z_n. Setting

hhϕ, zii= hϕ, zi

Θ(z), Θ(z) =

n

Y

i=1

z_i^α

s

Q

m=1

θ(bmzi) Y

1≤j<k≤n

z_j^2τθ(zj/zk)

θ(tz_j/z_k) , (1.10)

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 hϕ, zi of (1.8) converges as a meromorphic function on (C^∗)ⁿ. Then the functionf(z) =hhϕ, ziidefined by (1.10)is anS_n-invariant holomorphic function on(C^∗)ⁿ. Fur- thermore it satisfies the quasi-periodicityTq,zif(z) =f(z)/(−z_i)^sq^αtⁿ⁻¹b1· · ·bs for i= 1, . . . , n, where T_q,zi stands for the q-shift operator inz_i.

For the proof of this lemma, see Lemma3.3.

We call hhϕ, zii 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_ziϕ(z) ≤ s−1, i= 1, . . . , n, it satisfies the condition of Lemma 1.1under the assumption (3.2) below.

In order to state our first main theorem we define some terminology. We set Z =Z_s,n =

µ= (µ₁, µ₂, . . . , µ_s)∈N^s|µ₁+µ₂+· · ·+µ_s =n , (1.11) where N ={0,1,2, . . .}, so that |Z_s,n|= ^s+n−1_n

. We use the symbol for the lexicographic order onZ_s,n. Namely, forµ, ν ∈Z_s,n, we denoteµ≺ν if there existsk∈ {1,2, . . . , n}such that µ₁ =ν₁, µ₂ =ν₂, . . . , µ_k−1 =ν_k−1 and µ_k < ν_k. For an arbitrary x= (x₁, x₂, . . . , x_s) ∈ (C^∗)^s, we consider the pointsxµ (µ∈Zs,n) in (C^∗)ⁿ specified by

xµ= x1, x1t, . . . , x1t^µ1−1

| {z }

µ1

, x2, x2t, . . . , x2t^µ2−1

| {z }

µ2

, . . . , xs, xst, . . . , xst^µs−1

| {z }

µs

∈(C^∗)ⁿ. (1.12)

Theorem 1.2(connection formula). Suppose thatϕ(z)is anS_n-invariant holomorphic function satisfying the condition of Lemma 1.1. Then, for genericx∈(C^∗)^s we have

hhϕ, zii= X

λ∈Zs,n

c_λhhϕ, x_λii, (1.13)

where the connection coefficients cλ are explicitly written as

c_λ = X

K1t···tKs

={1,2,...,n}

s

Y

i=1

Y

k∈K_i

"

θ

q^αb1· · ·bstⁿ⁻¹z_k Q

1≤l≤s l6=i

x_lt^λ(k−1)l

θ

q^αb₁· · ·b_stⁿ⁻¹

s

Q

l=1

x_lt^λ(k−1)l

× Y

1≤j≤s j6=i

θ zkx⁻¹_j t^−λ

(k−1)

j

θ x_it^λ(k−1)i x⁻¹_j t^−λ

(k−1)

j

#

, (1.14)

where λ^(k)_i =|K_i∩ {1,2, . . . , k}|, and the summation is taken over all partitionsK1t · · · tKs= {1,2, . . . , n} such that |K_i|=λ_i, i= 1,2, . . . , s.

We call the formula (1.13) the generalized Slater transformation. In fact, the connection formula (1.13) forn= 1 coincides with the transformation formula (1.3) ifϕ(z)≡1, as explained below.

Example 1.3. The case n = 1. Setting _i = (0, . . . ,0,

i

^

1,0, . . . ,0) for i = 1,2, . . . , s, we have Zs,1={₁, 2, . . . , s} and xi =xi ∈C^∗, i.e., we obtain

hhϕ, zii=

s

X

i=1

cihhϕ, x_iii, where ci = θ(q^αb1· · ·bsx1· · ·xsz/xi) θ(q^αb1· · ·bsx1· · ·xs)

Y

1≤j≤s j6=i

θ(z/x_j)

θ(xi/xj).(1.15) The formula (1.15) for ϕ ≡ 1 (which was first stated by Mimachi [27, theorem in Section 4]) coincides with Slater’s transformation in (1.3). For z = a_i, i = 1, . . . , s, and x_j = b⁻¹_j , j = 1, . . . , s, it was given by Aomoto and Kato [4, equation (2.12) in Corollary 2.1]. For further details about the formula (1.15), see also [15, p. 255, Theorem 5.1 in the Appendix].

Example 1.4. The case s= 1. We have Z1,n ={(n)}, which has only one element, and have x_(n)= (x, xt, . . . , xtⁿ⁻¹) for x∈C^∗. Then

hhϕ, zii=c_(n)hhϕ, x_(n)ii, where c_(n) =

n

Y

i=1

θ q^αb1tⁿ⁻¹zi

θ q^αb1tⁿ⁻¹xtⁱ⁻¹. (1.16) The formula (1.16) for ϕ≡1 was given by Aomoto in [1]. See also [13, Lemma 3.6].

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). LetB_s,n be the set of partitions defined by

B =B_s,n=

λ= (λ₁, λ₂, . . . , λ_n)∈Zⁿ|s−1≥λ₁ ≥λ₂ ≥ · · · ≥λ_n≥0 , (1.17) so that|B_s,n|= ^s+n−1_n

. We also use the symbolfor the lexicographic order ofB_s,n. For each λ∈B_s,n, we denote by s_λ(z) theSchur function

sλ(z) = det z_i^λj+n−j

1≤i,j≤n

det z_i^n−j

1≤i,j≤n

= det z_i^λj+n−j

1≤i,j≤n

∆(z) ,

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 det hhs_λ, xµii

λ∈B µ∈Z =C

n

Y

k=1

θ q^αx1· · ·xsb1· · ·bst^n+k−2(^s+k−2k−1 )

×

n

Y

k=1





n−k

Y

r=0

Y

1≤i<j≤s

xjt^rθ xix⁻¹_j t^n−k−2r



 (^s+k−3k−1 )

, (1.18)

where the rows and the columns of the matrix are arranged by the lexicographic orders ofλ∈Bs,n

and µ∈Z_s,n, respectively. Here C is a constant independent of x, which is explicitly written as

C =

n

Y

k=1











(1−q)^s(q)^s_∞ qt^−(n−k+1)s

∞ s

Q

i,j=1

qa⁻¹_i b⁻¹_j t^−(n−k)

∞

qt⁻¹s

∞ q^αt^n−k

∞

q^1−αt^−(n+k−2)

s

Q

i=1

a⁻¹_i b⁻¹_i

∞









 (^s+k−2k−1 )

.

Remark. When s= 1, the determinant formula (1.18), combined with the connection formula hh1, zii=c_(n)hh1, x_(n)ii of (1.16), implies the summation formula

hh1, zii= (1−q)ⁿ

n

Y

j=1

(q)∞ qt^−j

∞ qa⁻¹₁ b⁻¹₁ t^−(j−1)

∞

qt⁻¹

∞ q^αt^j−1

∞ q^1−αa⁻¹₁ b⁻¹₁ t^−(n+j−2)

∞

θ q^αb₁tⁿ⁻¹z_j ,

for the Jackson integral hh1, zii 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 Ra- manujan’s ₁ψ₁ 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 typeAn. LetO((C^∗)ⁿ) be theC-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^∗)ⁿconsisting of allS_n-invariant holomorphic functionsf(z) such thatT_q,zif(z) = f(z)/(−z_i)^sζ fori= 1, . . . , n, whereTq,zi stands for theq-shift operator in zi:

H_s,n,ζ =

f(z)∈ O((C^∗)ⁿ)^Sn|T_q,zif(z) =f(z)/(−z_i)^sζ fori= 1,2, . . . , n . (1.19) The dimension ofH_s,n,ζ as aC-vector space will be shown to be ^n+s−1_n

in Section2. Moreover, Theorem 1.6. For generic x∈(C^∗)^s there exists a unique C-basis {E_µ(x;z)|µ∈Z_s,n} of the space H_s,n,ζ such that

Eµ(x;xν) =δµν for µ, ν ∈Zs,n, (1.20)

where xν ∈(C^∗)ⁿ, ν∈Zs,n, are the reference points specified by (1.12) andδ_λµ is the Kronecker delta.

This theorem will be proved in the end of Section2.2. We callEµ(x;z) the elliptic Lagrange interpolation functions of type A_nassociated 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 Eµ(x;z), µ∈Zs,n, with coefficientsf(xµ):

f(z) = X

µ∈Zs,n

f(x_µ)E_µ(x;z). (1.21)

Theorem 1.7. The functionsE_λ(x;z) are expressed as

Eλ(x;z) = X

K1t···tKs

={1,2,...,n}

s

Y

i=1

Y

k∈K_i











 θ

zkζ Q

1≤l≤s l6=i

xlt^λ(k−1)l

θ ζ

s

Q

l=1

x_lt^λ(k−1)l Y

1≤j≤s j6=i

θ z_kx⁻¹_j t^−λ

(k−1)

j

θ x_it^λ(k−1)i x⁻¹_j t^−λ

(k−1)

j













, (1.22)

where λ^(k)_i =|K_i∩ {1,2, . . . , k}|, and the summation is taken over all partitionsK1t · · · tKs= {1,2, . . . , n} such that |K_i|=λi, i= 1,2, . . . , s.

This theorem will be proved in Section2 as Theorem2.4.

Once Theorems 1.6 and 1.7 have been established, the connection formula (1.13) in Theo- rem 1.2 is immediately obtained. In the setting of Theorem 1.2, the regularization hhϕ, zii = hϕ, zi/Θ(z) belongs toH_s,n,ζ withζ =q^αtⁿ⁻¹b₁· · ·b_sby Lemma1.1. Hence, by (1.21) we obtain

hhϕ, zii= X

µ∈Zs,n

hhϕ, x_µiiE_µ(x;z). (1.23)

This means that the coefficients in (1.13) are given byc_µ=E_µ(x;z). The explicit formula (1.14) follows from Theorem 1.7.

For the evaluation of the determinant of Theorem1.5, we make use of the following determi- nant formula for the elliptic Lagrange interpolation functions E_µ(x;z) withζ =q^αtⁿ⁻¹b₁· · ·b_s. Theorem 1.8. Suppose that x, y∈(C^∗)^s are generic. Then,

det Eµ(x;yν)

µ,ν∈Zs,n =

n

Y

k=1

"

θ q^αy1· · ·ysb1· · ·bst^n+k−2 θ q^αx₁· · ·x_sb₁· · ·b_st^n+k−2

#(^s+k−2k−1 )

×

n

Y

k=1





n−k

Y

r=0

Y

1≤i<j≤s

y_it^rθ t^(n−k)−2ry_jy_i⁻¹ xit^rθ t^(n−k)−2rxjx⁻¹_i



 (^s+k−3_k−1 )

,

where y_ν are specified as in (1.12).

This theorem will be proved as Theorem2.8 in Section2.

Applying the connection formula (1.13) in Theorem 1.2, we see that Theorem 1.5is reduced to the special case where x=a= (a₁, . . . , a_s). Since

hhϕ, zii= X

µ∈Z_s,n

hhϕ, a_µiiE_µ(a;z)

by (1.23), setting ϕ(z) =s_λ(z) (λ∈B_s,n) andz=x_ν (ν ∈Z_s,n), we have hhs_λ, xνii= X

µ∈Zs,n

hhs_λ, aµiiE_µ(a;xν) for λ∈Bs,n, ν∈Zs,n, so that

det hhs_λ, xνii

λ∈B ν∈Z

= det hhs_λ, aµii

λ∈B µ∈Z

det Eµ(a;xν)

µ∈Z ν∈Z

.

Since we already know the explicit form of det E_µ(a;x_ν)

µ∈Z,ν∈Zby Theorem1.8, the evaluation of det hhs_λ, x_νii

λ∈B,ν∈Z reduces to that of det hhs_λ, a_µii

λ∈B,µ∈Z.

Lemma 1.9. We have

det hhs_λ, aµii

λ∈B

µ∈Z =

n

Y

k=1

"

(1−q)^s(q)^s_∞ qt^−(n−k+1)s

∞

qt⁻¹s

∞

×

q^αt^n+k−2

s

Q

i=1

aibi

∞ s

Q

i=1 s

Q

j=1

qa⁻¹_i b⁻¹_j t^−(n−k)

∞

q^αt^n−k

∞

#(^s+k−2k−1 )

×

n

Y

k=1





n−k

Y

r=0

Y

1≤i<j≤s

ajt^rθ aia⁻¹_j t^n−k−2r



 (^s+k−3k−1 )

. (1.24)

Lemma1.9will be proved in Section 6.

Remark. Tarasov and Varchenko [32] have already obtained a determinant formula for a mul- tiple 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 Section2we investigate the tran- sition coefficients between two sets of elliptic interpolation functions. In particular we provide a proof of Theorem1.8 for the determinant of the transition matrix. A proof of Lemma1.1for 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 func- tions. 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.

2 The elliptic Lagrange interpolation functions of type A

In this section we give proofs of Theorems1.6,1.7 and 1.8.

LetP+ be the set of partitions of length at mostn specified by P+=

λ= (λ1, λ2, . . . , λn)∈Zⁿ|λ1≥λ2 ≥ · · · ≥λn≥0 .

Forµ= (µ1, . . . , µn)∈Zⁿ, we denote by z^µthe monomialz^µ₁¹· · ·zn^µn. For the partitionsλ∈P+

let m_λ(z) be the monomial symmetric functions [23] defined by m_λ(z) = X

µ∈Snλ

z^µ,

where S_nλ = {wλ|w ∈ S_n} is the S_n-orbit of λ. For a function f(z) = f(z1, z2, . . . , zn) on (C^∗)ⁿ, we define the action of the symmetric group S_n on f(z) by

(σf)(z) =f σ⁻¹(z)

=f z_σ(1), z_σ(2), . . . , z_σ(n)

for σ ∈S_n.

We say that a function f(z) on (C^∗)ⁿ is symmetric or skew-symmetric if σf(z) = f(z) or σf(z) = (sgnσ)f(z) for all σ ∈ S_n, respectively. We denote by Af(z) the alternating sum overS_n defined by

(Af)(z) = X

σ∈S_n

(sgnσ)σf(z), (2.1)

which is skew-symmetric.

For an arbitrarily fixed ζ ∈ C^∗, we consider the C-linear subspace H_s,n,ζ ⊂ O((C^∗)ⁿ) con- sisting of all S_n-invariant holomorphic functions f(z) such that T_q,zif(z) = f(z)/(−z_i)^sζ, i= 1, . . . , n, as in (1.19). In this section we use the symbol

e(u, v) =uθ(v/u), u, v∈C^∗.

Since θ(u) =θ(q/u) and θ(qu) =−u⁻¹θ(u), this symbol satisfies e(u, v) =−e(v, u), e(qu, v) = (−v/u)e(u, v).

In particular we have e(u, v)→u−v in the limit q→0.

2.1 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 functionsE_λ⁽ⁿ⁾(x;z) recursively with respect to the numbernof variablesz1, . . . , zn, and show that those E_λ⁽ⁿ⁾(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 Theorem1.6;

we show thatE_λ⁽ⁿ⁾(x;z)∈ H_s,n,ζ and E_λ⁽ⁿ⁾(x;xµ) =δ_λµ using the dual Cauchy kernel as in [22].

First of all we show the following lemma.

Lemma 2.1. dim_CH_s,n,ζ ≤ ^n+s−1_n .

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) = P

λ∈Zⁿcλz^λ. Since f(z) is symmetric, all coefficients off(z) are determined fromc_λ corresponding toλ∈P+. On the other hand, sincef(z) satisfies T_q,zif(z) =f(z)/(−z_i)^sζ fori= 1, . . . , n, we have

X

λ∈Zⁿ

q^λic_λz^λ= X

λ∈Zⁿ

c_λz^λ/(−z_i)^sζ.

Equating coefficients of z^λ on both sides, all coefficients of f(z) are determined from c_λ corre- sponding to λ satisfying s−1 ≥ λi ≥ 0, i = 1, . . . , n. Therefore, f(z) is determined by the coefficients c_λ corresponding to λ∈ Bs,n defined by (1.17). Since |B_s,n|= ^n+s−1_n

, we obtain dim_CH_s,n,ζ ≤ ^n+s−1_n

.

Before introducingE_λ⁽ⁿ⁾(x;z) we prove Theorem 1.6forn= 1 by independent means. From the definition (1.11), we have Zs,1 ={₁, 2, . . . , s}, where i is specified in Example 1.3, and have x_i =x_i ∈C^∗.

Lemma 2.2. Forx= (x₁, . . . , x_s)∈(C^∗)^s and z∈C^∗ the functions

Ei(x;z) = e

zζ

s

Q

k=1

xk, xi

e

x_iζ

s

Q

k=1

x_k, x_i Y

1≤j≤s j6=i

e(z, xj)

e(x_i, x_j), i= 1, . . . , s, (2.2) satisfy Ei(x;xj) =δij. The set {E_i(x;z)|i= 1, . . . , s} is a basis of the C-linear space H_s,1,ζ. In particular dim_CH_s,1,ζ =s.

Proof . It is directly confirmed that Ei(x;xj) =δij from (2.2). It is obvious that Ei(x;z) ∈ H_s,1,ζ and the linearly independence of {E_i(x;z)|i = 1, . . . , s} follows since E_i(x;x_j) = δ_ij. Since we have dim_CH_s,1,ζ ≤ s by Lemma 2.1, we see that {E_i(x;z)|i = 1, . . . , s} is a basis

of H_s,1,ζ, and we obtain dim_CH_s,1,ζ =s.

Definition 2.3. For x = (x₁, . . . , x_s) ∈ (C^∗)^s and z = (z₁, . . . , z_n) ∈ (C^∗)ⁿ, let E_λ⁽ⁿ⁾(x;z), λ ∈ Zs,n, be functions defined inductively by E⁽¹⁾k (x;z1) = E_k(x;z1), k = 1, . . . , s, and for n≥2 by

E_λ⁽ⁿ⁾(x;z₁, . . . , z_n) = X

1≤k≤s λk>0

E⁽ⁿ⁻¹⁾_λ−

k (x;z₁, . . . , zn−1)E⁽¹⁾_k xt^λ−k;z_n

, (2.3)

where

xt^µ= x1t^µ1, x2t^µ2, . . . , xst^µs

∈(C^∗)^s. (2.4)

By the repeated use of (2.3) we have E_λ⁽ⁿ⁾(x;z) = X

(i1,...,in)

∈{1,...,s}ⁿ, i1+···+_in=λ

Ei1(x;z1)Ei2 xtⁱ¹;z2

Ei3 xtⁱ¹⁺ⁱ²;z3

· · ·E_in xt^{i1+···+in−1};zn

= X

(i1,...,in)∈{1,...,s}ⁿ i1+···+_in=λ

n

Y

k=1

E_ik xt^λ(k−1);z_k

, (2.5)

where λ^(k) = _i1 +· · ·+_ik. Rewriting this formula as in [14, p. 373, Theorem 3.4] we obtain the following.

Theorem 2.4. The functionsE⁽ⁿ⁾_λ (x;z) defined by (2.3) are expressed explicitly as E_λ⁽ⁿ⁾(x;z) = X

K1t···tKs

={1,2,...,n}

s

Y

i=1

Y

k∈K_i

Ei xt^λ(k−1);zk

,

where λ^(k)= λ^(k)₁ , . . . , λ^(k)s

∈Z_s,k, λ^(k)_i =|K_i∩ {1,2, . . . , k}| and the summation is taken over all index sets K_i, i= 1,2, . . . , s satisfying |K_i|=λ_i and K₁t · · · tK_s ={1,2, . . . , n}.

We remark that these functions forλ=ni ∈Zs,nhave simple factorized forms; this fact will be used in the next subsection.

Corollary 2.5. For n_i ∈Z_s,n, i= 1, . . . , s, one has

E_n⁽ⁿ⁾_i(x;z) =

n

Q

k=1

e

zkζ

s

Q

m=1

xm, xi

e

x_iζ

s

Q

m=1

x_m, x_i

n

Y

1≤j≤s j6=i

n

Q

k=1

e(zk, xj)

e(x_i, x_j)_n , (2.6)

where

e(u, v)r=e(u, v)e(ut, v)· · ·e ut^r−1, v

for r= 0,1,2, . . . . (2.7)

Proof . If we put λ=ni in the formula (2.5) then the right-hand side reduces to a single term with (i₁, i₂, . . . , i_n) = (i, i, . . . , i). Therefore, using (2.2) we obtain

E_n⁽ⁿ⁾_i(x;z) =E_i(x;z₁)E_i xtⁱ;z₂

E_i xt²ⁱ;z₃

· · ·E_i xt⁽ⁿ⁻¹⁾ⁱ;z_n ,

which coincides with (2.6).

We simply write E_λ(x;z) = E_λ⁽ⁿ⁾(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.

Forx= (x1, . . . , xs)∈(C^∗)^sandw= (w1, . . . , ws)∈(C^∗)^s, letFµ(x;w),µ∈N^s, be functions specified by

Fµ(x;w) =

s

Y

i=1 s

Y

j=1

e(xi, wj)µi, (2.8)

where e(u, v)r is given by (2.7). By definition the functionsFµ(x;w) satisfy

F_µ(x;w)F_ν(xt^µ;w) =F_µ+ν(x;w), (2.9)

where xt^µ is given by (2.4). For z = (z₁, . . . , z_n) ∈ (C^∗)ⁿ and w = (w₁, . . . , w_s) ∈ (C^∗)^s, let Ψ(z;w) be function specified by

Ψ(z;w) =

n

Y

i=1 s

Y

j=1

e(z_i, w_j),

which we call thedual Cauchy kernel. IfζQs

i=1wi= 1, then by definition Ψ(z;w) as a holomor- phic function of z ∈ (C^∗)ⁿ satisfies Ψ(z;w) ∈ H_s,n,ζ. Note that Fµ(x;w) of (2.8) is expressed as

F_µ(x;w) = Ψ(x_µ;w).

Lemma 2.6 (duality). Under the conditionζQs

i=1wi= 1, Ψ(z;w) expands as Ψ(z;w) = X

µ∈Zs,n

Eµ(x;z)Fµ(x;w). (2.10)

Proof . We proceed by induction onn, the cardinality of z. We consider the case n= 1 as the base case. Under the condition ζQs

i=1wi = 1, we have Ψ(z1;w) =Qs

j=1e(z1, wj)∈ H_s,1. Then, from Lemma2.2, Ψ(z₁;w) is expanded as Ψ(z₁;w) =Pn

i=1Ψ(x_i;w)E_i(x;z), whose coefficient Ψ(x_i;w) is written as Ψ(x_i;w) = Ψ(x_i;w) =Qs

j=1e(x_i, w_j) =F_i(x;w). This indicates (2.10) of the case n= 1.

Next we supposen≥2. We assume (2.10) holds for the number of variables forzless thann.

Denoting z_bn= (z1, . . . , zn−1)∈(C^∗)ⁿ⁻¹ forz= (z1, . . . , zn)∈(C^∗)ⁿ, (2.3) is rewritten as E_λ⁽ⁿ⁾(x;z) =

s

X

i=1

E_λ−⁽ⁿ⁻¹⁾

i (x;z

bn)E⁽¹⁾_i xt^λ−i;z_n

, (2.11)

where we regard E_λ−⁽ⁿ⁻¹⁾

i (x;z

bn) = 0 if λ−_i 6∈Zs,n−1. Then, we obtain Ψ(z;w) =

n

Y

i=1 s

Y

j=1

e(zi, wj) =

n−1

Y

i=1 s

Y

j=1

e(zi, wj)×

s

Y

j=1

e(zn, wj)

= X

µ∈Zs,n−1

E_µ⁽ⁿ⁻¹⁾(x;z

nb)F_µ(x;w)

X

ν∈Z_s,1

E_ν⁽¹⁾(xt^µ;z_n)F_ν(xt^µ;w)

(by the assumption of induction)

= X

µ∈Zs,n−1

ν∈Z_s,1

E_µ⁽ⁿ⁻¹⁾(x;z_nb)E_ν⁽¹⁾(xt^µ;zn)Fµ(x;w)Fν(xt^µ;w)

= X

µ∈Zs,n−1

ν∈Z_s,1

E_µ⁽ⁿ⁻¹⁾(x;z

nb)E_ν⁽¹⁾(xt^µ;z_n)F_µ+ν(x;w) (by (2.9))

= X

λ∈Zs,n

X

µ∈Zs,n−1, ν∈Zs,1

µ+ν=λ

E_µ⁽ⁿ⁻¹⁾(x;z_bn)E_ν⁽¹⁾(xt^µ;zn)

Fλ(x;w)

= X

λ∈Zs,n

E_µ⁽ⁿ⁾(x;z)F_λ(x;w), (by (2.11))

as desired.

When we consider the family of functions Fµ(x;w), µ ∈ Zs,n, on the hypersurface w ∈ (C^∗)^s|ζQs

i=1w_i = 1 , we regard F_µ(x;w) as a function of s−1 variables (w₁, . . . , ws−1) ∈ (C^∗)^s−1 with the relation w_s = ζQs−1

i=1w_i−1

. Here, for x = (x₁, . . . , x_s) ∈ (C^∗)^s and ν = (ν1, . . . , νs)∈Zs,n, we define a special pointην(x) on the hypersurface as

η_ν(x) =



x₁t^ν1, . . . , xs−1t^νs−1, ζ

s−1

Y

i=1

x_it^νi

!−1

.

Lemma 2.7(triangularity). For eachµ, ν ∈Zs,n,Fµ(x;ην(x)) = 0unlessµi≤νi fori= 1, . . . , s−1. In particular, F_µ(x;η_ν(x)) = 0 for µν with respect to the lexicographic order of Z_s,n. Moreover, if x∈(C^∗)^s is generic, then F_µ(x;η_µ(x))6= 0 for all µ∈Z_s,n.

This lemma implies that the matrix F = Fµ(x;ην(x))

µ,ν∈Z is upper triangular, and also invertible if x∈C^s is generic.

Proof . If there existsj ∈ {1,2, . . . , s−1}such that νj < µj, thenFµ(x;ην(x)) = 0. In fact, in the expression

F_µ(x;η_ν(x)) =

s

Y

i=1



e



x_i, ζ

s−1

Y

i=1

x_it^νi

!−1



µi

s−1

Y

j=1

e x_i, x_jt^νj

µi



, the function e(x_j, x_jt^νj)_µj has the factorθ(t^νj)θ t^νj−1

· · ·θ t^{νj−(µj−1)}

= 0 ifν_j < µ_j. Ifν ≺µ, then νi < µi for some i∈ {1,2, . . . , s−1} by definition, and hence we obtain Fµ(x;ην(x)) = 0 ifν ≺µ. Forν =µ,

F_µ(x;η_µ(x)) =

s

Y

i=1

"

x^µ_iⁱt(^µi₂)

µi−1

Y

j=0

θ



x⁻¹_i t^−j ζ

s−1

Y

k=1

x_kt^µk

!−1



×

s−1

Y

j=1

θ x⁻¹_i x_jt^µj

θ x⁻¹_i x_jt^µj−1

· · ·θ x⁻¹_i x_jt^µj−µi+1

#

does not vanish if we impose an appropriate genericity condition on x∈(C^∗)^s.