q-Difference Systems for the Jackson Integral of Symmetric Selberg Type
We provide an explicit expression for the first-order -difference system for the Jackson integral of symmetric Selberg type. The q-difference system gives a generalization of the -analog of contiguous relations for the Gauss hypergeometric function. As a basis of the system, we use a set of symmet...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2020 |
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Інститут математики НАН України
2020
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| Цитувати: | q-Difference Systems for the Jackson Integral of Symmetric Selberg Type. Masahiko Ito. SIGMA 16 (2020), 113, 31 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859802899163381760 |
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| author | Ito, Masahiko |
| author_facet | Ito, Masahiko |
| citation_txt | q-Difference Systems for the Jackson Integral of Symmetric Selberg Type. Masahiko Ito. SIGMA 16 (2020), 113, 31 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We provide an explicit expression for the first-order -difference system for the Jackson integral of symmetric Selberg type. The q-difference system gives a generalization of the -analog of contiguous relations for the Gauss hypergeometric function. As a basis of the system, we use a set of symmetric polynomials introduced by Matsuo in his study of the -KZ equation. Our main result is an explicit expression for the coefficient matrix of the -difference system in terms of its Gauss matrix decomposition. We introduce a class of symmetric polynomials called interpolation polynomials, which includes Matsuo's polynomials. By repeated use of three-term relations among the interpolation polynomials, we compute the coefficient matrix.
|
| first_indexed | 2026-03-16T07:30:58Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 113, 31 pages
q-Difference Systems for the Jackson Integral
of Symmetric Selberg Type
Masahiko ITO
Department of Mathematical Sciences, University of the Ryukyus, Okinawa 903-0213, Japan
E-mail: mito@sci.u-ryukyu.ac.jp
Received April 29, 2020, in final form October 29, 2020; Published online November 08, 2020
https://doi.org/10.3842/SIGMA.2020.113
Abstract. We provide an explicit expression for the first order q-difference system for the
Jackson integral of symmetric Selberg type. The q-difference system gives a generalization
of q-analog of contiguous relations for the Gauss hypergeometric function. As a basis of
the system we use a set of the symmetric polynomials introduced by Matsuo in his study
of the q-KZ equation. Our main result is an explicit expression for the coefficient matrix of
the q-difference system in terms of its Gauss matrix decomposition. We introduce a class
of symmetric polynomials called interpolation polynomials, which includes Matsuo’s poly-
nomials. By repeated use of three-term relations among the interpolation polynomials we
compute the coefficient matrix.
Key words: q-difference equations; Selberg type integral; contiguous relations; Gauss de-
composition
2020 Mathematics Subject Classification: 33D60; 39A13
1 Introduction
The Gauss hypergeometric function
2F1
(
a, b
c
;x
)
=
Γ(c)
Γ(a)Γ(c− a)
∫ 1
0
za−1(1− z)c−a−1(1− xz)−b dz,
where Re c > Re a > 0 and |x| < 1, satisfies the contiguous relations
2F1
(
a, b
c
;x
)
= 2F1
(
a, b+ 1
c+ 1
;x
)
− xa(c− b)
c(c+ 1)
2F1
(
a+ 1, b+ 1
c+ 2
;x
)
(1.1)
and
2F1
(
a, b
c
;x
)
= 2F1
(
a+ 1, b
c+ 1
;x
)
− xb(c− a)
c(c+ 1)
2F1
(
a+ 1, b+ 1
c+ 2
;x
)
. (1.2)
These contiguous relations for the Gauss hypergeometric function are extended to a difference
system for a function defined by multivariable integral with respect to the Selberg type kernel [34]
Ψ(z) :=
n∏
i=1
zα−1i (1− zi)β−1(x− zi)γ−1
∏
1≤j<k≤n
|zj − zk|2τ .
For the integral
〈ei〉 :=
∫
C
ei(z)Ψ(z) dz1 · · · dzn, i = 0, 1, . . . , n,
This paper is a contribution to the Special Issue on Elliptic Integrable Systems, Special Functions and Quan-
tum Field Theory. The full collection is available at https://www.emis.de/journals/SIGMA/elliptic-integrable-
systems.html
mailto:mito@sci.u-ryukyu.ac.jp
https://doi.org/10.3842/SIGMA.2020.113
https://www.emis.de/journals/SIGMA/elliptic-integrable-systems.html
https://www.emis.de/journals/SIGMA/elliptic-integrable-systems.html
2 M. Ito
where ei(z) is the function specified by
ei(z) :=
n−i∏
j=1
(x− zj)
n∏
k=n−i+1
(1− zk)
and C is some suitable region, the (n + 1)-tuple (〈e0〉, 〈e1〉, . . . , 〈en〉) satisfies the following
difference system. Let δij be the symbol of Kronecker’s delta.
Proposition 1.1 ([14, Theorem 2.2]). Let Tα be the shift operator with respect to α → α + 1,
i.e., Tαf(α) = f(α+ 1) for an arbitrary function f : C→ C. Then
Tα(〈e0〉, 〈e1〉, . . . , 〈en〉) = (〈e0〉, 〈e1〉, . . . , 〈en〉)M, (1.3)
where the (n+ 1)× (n+ 1) matrix M is written in terms of its Gauss matrix decomposition as
M = LDU = U ′D′L′. (1.4)
Here L = (lij)0≤i,j≤n, D = (djδij)0≤i,j≤n, U = (uij)0≤i,j≤n are the lower triangular, diagonal,
upper triangular matrices, respectively, given by
lij = (−x)i−j
(
n− j
n− i
)
(γ + jτ ; τ)i−j
(α+ γ + 2jτ ; τ)i−j
,
dj =
xj(α; τ)j(α+ γ + 2jτ ; τ)n−j
(α+ γ + (j − 1)τ ; τ)j(α+ β + γ + (n+ j − 1)τ ; τ)n−j
,
uij = (−1)j−i
(
j
i
)
(β + (n− j)τ ; τ)j−i
(α+ γ + 2iτ ; τ)j−i
,
and U ′ = (u′ij)0≤i,j≤n, D′ = (d′jδij)0≤i,j≤n, L′ = (l′ij)0≤i,j≤n are the upper triangular, diagonal,
lower triangular matrices, respectively, given by
u′ij = (−x−1)j−i
(
j
i
)
(β + (n− j)τ ; τ)j−i
(α+ β + 2(n− j)τ ; τ)j−i
,
d′j =
xj(α+ β + 2(n− j)τ ; τ)j(α; τ)n−j
(α+ β + γ + (2n− j − 1)τ ; τ)j(α+ β + (n− j − 1)τ ; τ)n−j
,
l′ij = (−1)i−j
(
n− j
n− i
)
(γ + jτ ; τ)i−j
(α+ β + 2(n− i)τ ; τ)i−j
,
where (x; τ)0 := 1 and (x; τ)i := x(x+ τ)(x+ 2τ) · · · (x+ (i− 1)τ) for i = 1, 2, . . ..
In particular, when n = 1 the system (1.3) is given by
Tα(〈e0〉, 〈e1〉) = (〈e0〉, 〈e1〉)
(
α+ γ 0
−xγ xα
)(
α+ β + γ β
0 α+ γ
)−1
, (1.5a)
Tα(〈e0〉, 〈e1〉) = (〈e0〉, 〈e1〉)
(
α −β
0 x(α+ β)
)(
α+ β 0
γ α+ β + γ
)−1
. (1.5b)
The system (1.5b) can be rewritten as the following system of three-term equations
(α+ β + γ)Tα〈e1〉 = −β〈e0〉+ x(α+ β)〈e1〉, (1.6)
and
γTα〈e1〉+ (α+ β)Tα〈e0〉 = α〈e0〉. (1.7)
q-Difference Systems for the Jackson Integral of Symmetric Selberg Type 3
Since, for n = 1
〈e0〉 =
∫ 1
0
zα−1(1− z)β−1(x− z)γ dz and 〈e1〉 =
∫ 1
0
zα−1(1− z)β(x− z)γ−1 dz,
under the conditions Reα > 0, Reβ > 0 and |x| > 1 we see that the equations (1.6) and (1.7)
exactly coincide with the contiguous relations (1.1) and (1.2), respectively, after the substitutions
α→ a, β → c−a, γ → −b and x→ 1/x. Therefore the difference system (1.3) expressed in terms
of Gauss matrix decomposition (1.4) can be regarded as a natural extension of the contiguous
relations (1.1) and (1.2). For further applications of the difference system (1.3) in random matrix
theory, see [14]. In the discussion of the result being a generalization of contiguous relations of
the Gauss hypergeometric function, the analog of the equation (1.1) for the Selberg integral can
be found in [15, 21].
Next we would like to discuss a q-analogue of the difference system (1.3) in Proposition 1.1.
This is one of the aims of this paper. For an arbitrary c ∈ C∗ we use the c-shifted factorial for
x ∈ C
(x; c)i =
(1− x)(1− cx) · · ·
(
1− ci−1x
)
if i = 1, 2, . . . ,
1 if i = 0,
1(
1− c−1x
)(
1− c−2x
)
· · ·
(
1− cix
) if i = −1,−2, . . . ,
and the c-binomial coefficient[
i
j
]
c
=
(c; c)i
(c; c)i−j(c; c)j
.
We also use the symbol (x; c)∞ :=
∏∞
i=0
(
1 − cix
)
for |c| < 1. Throughout this paper we fix
q ∈ C∗ with |q| < 1. For a point ξ = (ξ1, . . . , ξn) ∈ (C∗)n and a function f(z) = f(z1, . . . , zn)
on (C∗)n we define the following sum over the lattice Zn by∫ ξ∞
0
f(z)
dqz1
z1
∧ · · · ∧ dqzn
zn
:= (1− q)n
∑
(ν1,...,νn)∈Zn
f
(
ξ1q
ν1 , . . . , ξnq
νn
)
, (1.8)
if it converges. We call it the Jackson integral of f(z). By definition the Jackson integral (1.8)
is invariant under the q-shift ξi → qξi (i = 1, . . . , n). Let Φn,m(z) and ∆(z) be the functions
on (C∗)n specified by
Φn,m(z) :=
n∏
i=1
{
zαi
m∏
r=1
(
qa−1r zi; q
)
∞
(brzi; q)∞
} ∏
1≤j<k≤n
z2τ−1j
(
qt−1zk/zj ; q
)
∞
(tzk/zj ; q)∞
, (1.9)
∆(z) :=
∏
1≤i<j≤n
(zi − zj),
where t = qτ . For a point ξ = (ξ1, . . . , ξn) ∈ (C∗)n and an arbitrary symmetric function
φ(z) = φ(z1, . . . , zn) on (C∗)n we set
〈φ, ξ〉 :=
∫ ξ∞
0
φ(z)Φn,m(z)∆(z)
dqz1
z1
∧ · · · ∧ dqzn
zn
,
which we call the Jackson integral of symmetric Selberg type. In the study of q-difference de
Rham cohomology associated with Jackson integrals [4, 7], Aomoto and Kato [8] showed that
the Jackson integral of symmetric Selberg type satisfies q-difference systems of rank
(
n+m−1
m−1
)
4 M. Ito
when the parameters are generic. When m = 1 the Jackson integral of symmetric Selberg type
is equivalent to the q-Selberg integral defined by Askey [11] and proved by others, see [6, 13,
16, 20] for instance. See also recent references [15, Section 2.3] and [18]. q-Selberg integral is
a very active area of research with important connections to special functions, combinatorics,
mathematical physics and orthogonal polynomials (see [1, 23, 24, 25, 32, 35] and [17, Section 5]).
Using the Jackson integral of symmetric Selberg type for m = 2, Matsuo [28, 29] constructed
a set of solutions of the q-KZ equation. Varchenko [33] extended Matsuo’s construction to
more general setting of the q-KZ equation using the Jackson integral of symmetric Selberg type
for general m. Writing ar = xr, br = qβrx−1r in (1.9), the q-KZ equation they studied can
be regarded as the q-difference system with respect to the q-shift xr → qxr (r = 1, . . . ,m).
In another context, writing ar = qx−1r , br = qµrxr in (1.9), Kaneko [22] showed an explicit
expression for the q-difference system with respect to the q-shift xr → qxr (r = 2, . . . ,m)
satisfied by the Jackson integral of symmetric Selberg type for general m with special constraints
µ2 = · · · = µm = 1 or µ2 = · · · = µm = −τ . With these constraints the q-difference system
degenerates to be very simple and it can also be regarded as a generalization of the second order
q-difference equation satisfied by Heine’s 2φ1 q-hypergeometric function.
In this paper, we fix m = 2 for (1.9), and study two types of q-difference systems for the
Jackson integral of symmetric Selberg type for Φ(z) = Φn,2(z). One is the q-difference system
with respect to the shift α → α + 1, and the other is the system with respect to the q-shifts
ai → qai and bi → q−1bi simultaneously. For these purposes, we define the set of symmetric
polynomials {ei(a, b; z) | i = 0, 1, . . . , n}, where
ei(a, b; z) :=
1
∆(z)
×A
n−i∏
j=1
(1− bzj)
n∏
j=n−i+1
(
1− a−1zj
) ∏
1≤k<l≤n
(
zk − t−1zl
) , (1.10)
which we call Matsuo’s polynomials. The symbol A means the skew-symmetrization (see the
definition (2.1) of A in Section 2). With these symmetric polynomials, we denote
〈ei(a, b), ξ〉 :=
∫ ξ∞
0
ei(a, b; z)Φ(z)∆(z)
dqz1
z1
∧ · · · ∧ dqzn
zn
. (1.11)
We assume that∣∣qa−11 a−12 b−11 b−12
∣∣ < ∣∣qα∣∣ < 1 and
∣∣qa−11 a−12 b−11 b−12
∣∣ < ∣∣qαt2n−2∣∣ < 1
for convergence of the Jackson integrals (1.11). (See [19, Lemma 3.1] for details of convergence.)
For the polynomials (1.10) let R be the (n+ 1)× (n+ 1) matrix defined by(
en(a2, b1; z), en−1(a2, b1; z), . . . , e0(a2, b1; z)
)
=
(
e0(a1, b2; z), e1(a1, b2; z), . . . , en(a1, b2; z)
)
R. (1.12)
The transition matrix R is called the R-matrix in the context of [29]. Matsuo [29] gave the
q-difference system with respect to the q-shifts ai → qai and bi → q−1bi simultaneously, using
Matsuo’s polynomials as follows.
Proposition 1.2 (Matsuo). Let Tq,u be the q-shift operator with respect to u→ qu, and T−1q,bi
Tq,ai
(i = 1, 2) denote the q-shift operator with respect to ai → qai and bi → q−1bi simultaneously.
Then, the Jackson integrals of symmetric Selberg type satisfy the q-difference system with respect
to T−1q,bi
Tq,ai (i = 1, 2) given by
T−1q,bi
Tq,ai
(
〈en(a2, b1), ξ〉, 〈en−1(a2, b1), ξ〉, . . . , 〈e0(a2, b1), ξ〉
)
q-Difference Systems for the Jackson Integral of Symmetric Selberg Type 5
=
(
〈en(a2, b1), ξ〉, 〈en−1(a2, b1), ξ〉, . . . , 〈e0(a2, b1), ξ〉
)
Ki, (1.13)
whose coefficient matrices Ki are expressed as K1 = R−1D1 and K2 = D2
(
T−1q,b2
Tq,a2R
)
, where R
is the (n+ 1)× (n+ 1) matrix given by (1.12), and D1, D2 are the diagonal matrices given by
D1 =
((
qαtn−1
)n−i
δij
)
0≤i,j≤n, D2 =
((
qαtn−1
)i
δij
)
0≤i,j≤n.
Remark 1.3. If we replace ai and bi as ai = xi and bi = qβix−1i , respectively, then (1.13)
simplifies to the case considered by Matsuo, and then T−1q,bi
Tq,ai in (1.13) becomes the single q-
shift operator Tq,xi , and the system (1.13) coincides with the q-KZ equation (see [28, 29, 33]). For
the problem of finding the explicit form of the coefficient matrix Ki for the system (1.13) Aomoto
and Kato used the information of a connection matrix [9] between two kinds of fundamental
solutions of (1.13) specified by their asymptotic behaviors. Based on Birkhoff’s classical theory
they introduced a way to derive the explicit form of the coefficient matrix for a linear ordinary
q-difference system from its connection matrix. They call their method the Riemann–Hilbert
approach for q-difference equations from connection matrices [5], and they presentedKi explicitly
when n = 1 and 2 as an example of their method (see [10, p. 272, examples]). The problem
of finding the explicit form of the coefficient matrix for the equivalent q-difference system with
respect to Tq,xi was also studied by Mimachi [30, 31]. As a basis of the system Mimachi [31]
introduced a family of Schur polynomials different from Matsuo’s polynomials, and he calculated
the entries of the coefficient matrix explicitly when n = 1, 2 and 3.
From Proposition 1.2, if we want to know the coefficient matrices Ki of the above q-difference
systems, it suffices to give the explicit expression for the transition matrix R or its inverse R−1.
Theorem 1.4 ([12, 26]). The matrix R is written in terms of its Gauss matrix decomposition
as
R = LRDR UR = U ′RD
′
R L
′
R, (1.14)
where LR =
(
lRij
)
0≤i,j≤n, DR =
(
dR
j δij
)
0≤i,j≤n, UR =
(
uR
ij
)
0≤i,j≤n are the lower triangular,
diagonal, upper triangular matrices, respectively, given by
lRij =
[
n− j
n− i
]
t−1
(−1)i−jt−(i−j
2 )(a2b2tj ; t)i−j(
a−11 a2t−(n−2j−1); t
)
i−j
, (1.15a)
dR
j =
(
a1a
−1
2 t−j ; t
)
n−j(a2b1; t)j
(a1b2; t)n−j
(
a−11 a2t−(n−j); t
)
j
, (1.15b)
uR
ij =
[
j
i
]
t−1
(
a1b1t
n−j ; t
)
j−i(
a1a
−1
2 tn−i−j ; t
)
j−i
, (1.15c)
and U ′R =
(
uR ′
ij
)
0≤i,j≤n, D′R =
(
dR ′
j δij
)
0≤i,j≤n, L′R =
(
lR ′ij
)
0≤i,j≤n are the upper triangular,
diagonal, lower triangular matrices, respectively, given by
uR ′
ij =
[
j
i
]
t
(−1)j−it(
j−i
2 )(a−11 b−11 t−(n−i−1); t
)
j−i(
b−11 b2ti+j−n; t
)
j−i
, (1.16a)
dR ′
j =
(
b1b
−1
2 tn−2j+1; t
)
j
(
a−12 b−11 t−(n−j−1); t
)
n−j(
a−11 b−12 t−(j−1); t
)
j
(
b−11 b2t−(n−2j−1); t
)
n−j
, (1.16b)
lR ′ij =
[
n− j
n− i
]
t
(
a−12 b−12 t−(i−1); t
)
i−j(
b1b
−1
2 tn−2i+1; t
)
i−j
. (1.16c)
6 M. Ito
One of the main aims of this paper is to give a proof of the above result, which we will do
in Section 6. The explicit expression for R−1 in terms of its Gauss matrix decomposition is also
presented as Corollary 6.1 in Section 6.
Remark 1.5. After completing of earlier version of this paper, the author was informed that
Theorem 1.4 previously appeared implicitly in [12, Section 5] and [26]. Our proof of the Theorem
is, however, very different to that of [12] and [26].
From Theorem 1.4 we immediately obtain a closed-form expression for the determinant of R
(or Ki).
Corollary 1.6. The determinant of the transition matrix R evaluates as
detR = dR
0 d
R
1 · · · dR
n =
(
−a1a−12
)(n+1
2 )
n∏
i=1
(a2b1; t)i
(a1b2; t)i
.
The determinants of the coefficient matrices K1 and K2 given in (1.13) evaluate as
detK1 = det
(
R−1D1
)
=
(
−a2a−11 qαtn−1
)(n+1
2 )
n∏
i=1
(a1b2; t)i
(a2b1; t)i
and
detK2 = det
(
D2
(
T−1q,b2
Tq,a2R
))
=
(
−a1a−12 qα−1tn−1
)(n+1
2 )
n∏
i=1
(qa2b1; t)i(
q−1a1b2; t
)
i
.
Next, we focus on the q-difference system with respect to the shift α→ α+ 1 for the Jackson
integral of symmetric Selberg type. Using Matsuo’s polynomials {ei(a1, b2; z) | i = 0, 1, . . . , n},
this q-difference system is given explicitly in terms of its Gauss matrix decomposition.
Theorem 1.7. Let Tα be the shift operator with respect to α→ α+ 1, i.e., Tαf(α) = f(α+ 1)
for an arbitrary function f(α) of α ∈ C. Then
Tα
(
〈e0(a1, b2), ξ〉, 〈e1(a1, b2), ξ〉, . . . , 〈en(a1, b2), ξ〉
)
=
(
〈e0(a1, b2), ξ〉, 〈e1(a1, b2), ξ〉, . . . , 〈en(a1, b2), ξ〉
)
A, (1.17)
where the coefficient matrix A is written in terms of its Gauss matrix decomposition as
A = LADA UA = U ′AD
′
A L
′
A.
Here LA =
(
lAij
)
0≤i,j≤n, DA =
(
dA
j δij
)
0≤i,j≤n, UA =
(
uA
ij
)
0≤i,j≤n are the lower triangular, diag-
onal, upper triangular matrices, respectively, given by
lAij = (−1)i−jt(
n−i
2 )−(n−j
2 )
[
n− j
n− i
]
t
(
a2b2t
j ; t
)
i−j(
qαa2b2t2j ; t
)
i−j
, (1.18a)
dA
j = an−j1 aj2t
(j2)+(n−j
2 )
(
qα; t
)
j
(
qαa2b2t
2j ; t
)
n−j(
qαa2b2tj−1; t
)
j
(
qαa1a2b1b2tn+j−1; t
)
n−j
, (1.18b)
uA
ij =
(
−qαa−11 a2
)j−i
t(
j
2)−(i
2)
[
j
i
]
t
(
a1b1t
n−j ; t
)
j−i(
qαa2b2t2i; t
)
j−i
, (1.18c)
q-Difference Systems for the Jackson Integral of Symmetric Selberg Type 7
and U ′A =
(
uA ′
ij
)
0≤i,j≤n, D′A =
(
dA ′
j δij
)
0≤i,j≤n, L′A =
(
lA ′ij
)
0≤i,j≤n are the upper triangular,
diagonal, lower triangular matrices, respectively, given by
uA ′
ij =
(
−qα
)j−i
t(
n−i
2 )−(n−j
2 )
[
j
i
]
t
(
a1b1t
n−j ; t
)
j−i(
qαa1b1t2(n−j); t
)
j−i
, (1.19a)
dA ′
j = an−j1 aj2t
(j2)+(n−j
2 )
(
qαa1b1t
2(n−j); t
)
j
(
qα; t
)
n−j(
qαa1a2b1b2t2n−j−1; t
)
j
(
qαa1b1tn−j−1; t
)
n−j
, (1.19b)
lA ′ij =
(
−a1a−12
)i−j
t(
j
2)−(i
2)
[
n− j
n− i
]
t
(
a2b2t
j ; t
)
i−j(
qαa1b1t2(n−i); t
)
i−j
. (1.19c)
The first part of Theorem 1.7 will be proved in Section 5, while the latter part of Theorem 1.7
will be explained in the Appendix. Note that, from this theorem we immediately have the
following.
Corollary 1.8. The determinant of the coefficient matrix A evaluates as
detA = dA
0 d
A
1 · · · dA
n = (a1a2)
(n+1
2 )t2(
n+1
3 )
n∏
i=1
(
qα; t
)
i(
qαa1a2b1b2t2n−i−1; t
)
i
.
Remark 1.9. The expression (1.17) for the q-difference equation is equivalent to
Tα
(
〈e0(a1, b2), ξ〉, 〈e1(a1, b2), ξ〉, . . . , 〈en(a1, b2), ξ〉
)
U−1A
=
(
〈e0(a1, b2), ξ〉, 〈e1(a1, b2), ξ〉, . . . , 〈en(a1, b2), ξ〉
)
LADA
or
Tα
(
〈e0(a1, b2), ξ〉, 〈e1(a1, b2), ξ〉, . . . , 〈en(a1, b2), ξ〉
)
L′−1A
=
(
〈e0(a1, b2), ξ〉, 〈e1(a1, b2), ξ〉, . . . , 〈en(a1, b2), ξ〉
)
U ′AD
′
A.
Here the entries of U−1A and L′−1A are also factorized into binomials like UA and L′A, respectively.
We will see the explicit expression for U−1A in Section 4 as Proposition 4.4. For the explicit
expression for L′A, see Proposition A.4.
Remark 1.10. If we consider the q → 1 limit after replacing ai and bi as a1 = 1, a2 = x,
b1 = qβ and b2 = qγx−1 on Theorem 1.7, then ei
(
1, qγx−1; z
)
Φ(z)∆(z)
dqz1
z1
· · · dqznzn
tends to
e′i(z)Ψ
′(z)dz1 · · · dzn, where
Ψ′(z) =
n∏
i=1
zα−1i (1− zi)β−1(1− zi/x)γ−1
∏
1≤j<k≤n
(zj − zk)2τ ,
e′i(z) = A
n−i∏
j=1
(1− zj/x)
n∏
j=n−i+1
(1− zj)
.
This confirms that Theorem 1.7 in the q → 1 limit is consistent with the result presented in
Proposition 1.1.
The paper is organized as follows. After defining some basic terminology in Section 2, we
characterize in Section 3 Matsuo’s polynomials by their vanishing property (Proposition 3.1),
and define a family of symmetric polynomials of higher degree, which includes Matsuo’s polyno-
mials. We call such polynomials the interpolation polynomials, which are inspired from Aomoto’s
method [2, Section 8], [3], which is a technique to obtain difference equations for the Selberg
8 M. Ito
integrals (see also [18] for a q-analogue of Aomoto’s method). We state several vanishing prop-
erties for the interpolation polynomials, which are used in subsequent sections. In Section 4 we
present three-term relations (Lemma 4.1) among the interpolation polynomials. These are key
equations for obtaining the coefficient matrix of the q-difference system with respect to the shift
α → α + 1. By repeated use of these three-term relations we obtain a proof of Theorem 1.7.
Section 5 is devoted to the proof of Lemma 4.1. In Section 6 we explain the Gauss decom-
position of the transition matrix R. For this purpose, we introduce another set of symmetric
polynomials called the Lagrange interpolation polynomials of type A in [19], which are different
from Matsuo’s polynomials. Both upper and lower triangular matrices in the decomposition can
be understood as a transition matrix between Matsuo’s polynomials and the other polynomials.
In the Appendix we explain the proof of the latter part of Theorem 1.7.
Finally we would like to make some remarks about the original motivation for the current
paper. Although the author already knew the results of this paper before publishing [14], many
years have passed since then. The author recently learned of an interesting application of the q-
difference systems of this paper in collaboration with Yasuhiko Yamada. They intend to publish
the detail in a forthcoming paper.
2 Notation
Let Sn be the symmetric group on {1, 2, . . . , n}. For a function f : (C∗)n → C we define an
action of the symmetric group Sn on f by
(σf)(z) := f
(
σ−1(z)
)
= f(zσ(1), zσ(2), . . . , zσ(n)) for σ ∈ Sn.
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 ∈ Sn, respectively. We denote by Af(z) the alternating sum
over Sn defined by
(Af)(z) :=
∑
σ∈Sn
(sgnσ)(σf)(z), (2.1)
which is skew-symmetric. Let P be the set of partitions defined by
P :=
{
(λ1, λ2, . . . , λn) ∈ Zn |λ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0
}
.
We define the lexicographic ordering < on P as follows. For λ, µ ∈ P , we denote λ < µ if
there exists a positive integer k such that λi = µi for all i < k and λk < µk. For λ =
(λ1, λ2, . . . , λn) ∈ Zn, we denote by zλ the monomial zλ11 zλ22 · · · zλnn . For λ ∈ P the monomial
symmetric polynomials mλ(z) are defined by
mλ(z) :=
∑
µ∈Snλ
zµ,
where Snλ := {σλ |σ ∈ Sn} is the Sn-orbit of λ. For λ ∈ P , we denote by mi the multiplicity
of i in λ, i.e., mi = #{j |λj = i}, see [27] for instance. It is convenient to use the notation
λ =
(
1m12m2 · · · rmr · · ·
)
: for example,
(
1322
)
= (2, 2, 1, 1, 1, 0) and z(1
322) = z21z
2
2z3z4z5.
3 Interpolation polynomials
In this section we define a family of symmetric functions which extends Matsuo’s polynomials.
For a, b ∈ C∗ and z = (z1, . . . , zn) ∈ (C∗)n let Ek,i(a, b; z) (k, i = 0, 1, . . . , n) be functions
q-Difference Systems for the Jackson Integral of Symmetric Selberg Type 9
specified by
Ek,i(a, b; z) := z1z2 · · · zk∆(t; z)
n−i∏
j=1
(1− bzj)
n∏
j=n−i+1
(
1− a−1zj
)
, (3.1)
where
∆(t; z) :=
∏
1≤i<j≤n
(
zi − t−1zj
)
= t−(n2)
∏
1≤i<j≤n
(tzi − zj),
and let Ẽk,i(a, b; z) (k, i = 0, 1, . . . , n) be the symmetric functions of z ∈ (C∗)n specified by
Ẽk,i(a, b; z) :=
AEk,i(a, b; z)
∆(z)
, (3.2)
which, in particular, satisfy
Ẽ0,i(a, b; z) = ei(a, b; z) and Ẽn,i(a, b; z) = z1z2 · · · znei(a, b; z),
as special cases. We sometimes abbreviate Ẽk,i(a, b; z) to Ẽk,i(z). The leading term of the
symmetric polynomial Ẽk,i(z) is m(1n−k2k)(z), i.e.,
Ẽk,i(z) = Ckim(1n−k2k)(z) + lower order terms,
where the coefficient Cki of the monomial m(1n−k2k)(z) is expressed as
Cki = (−1)n
(
t−1; t−1
)
k
(
t−1; t−1
)
n−k
aib−(n−i)
(
1− t−1
)n .
For arbitrary x, y ∈ C∗, we set
ζj(x, y) :=
(
yt−(j−1), yt−(j−2), . . . , yt−1, y︸ ︷︷ ︸
j
, x, xt, xt2, . . . , xtn−j−1︸ ︷︷ ︸
n−j
)
∈ (C∗)n. (3.3)
The following gives another characterization of Matsuo’s polynomials ei(a, b; z) = Ẽ0,i(z).
Proposition 3.1. The leading term of the function Ẽ0,i(z) is m(1n)(z) up to a multiplicative
constant. The functions Ẽ0,i(z), i = 0, 1, . . . , n, satisfy
Ẽ0,i
(
ζj
(
a, b−1
))
= ciδij , (3.4)
where the constant ci is given by
ci =
(
abti; t
)
n−i
(
a−1b−1t−(i−1); t
)
i
(t; t)i(t; t)n−i
t(
n
2)(1− t)n
(3.5a)
= (ab; t)n−i
(
a−1b−1t−(n−1); t
)
i
(
t−1; t−1
)
i
(
t−1; t−1
)
n−i(
1− t−1
)n . (3.5b)
Remark 3.2. The set of symmetric functions
{
Ẽ0,i(z) | i = 0, 1, . . . , n
}
forms a basis of the linear
space spanned by {mλ(z) |λ ≤ (1n)}. Conversely such basis satisfying the condition (3.4) is
uniquely determined. Thus we can take Proposition 3.1 as a definition of Matsuo’s polynomials,
instead of (1.10).
10 M. Ito
Proof. By definition Ẽ0,i(ζj(a, b
−1)) = 0 if i 6= j. Ẽ0,i(ζi(a, b
−1)) evaluates as
Ẽ0,i(ζi(a, b
−1)) =
E0,i
(
atn−i−1, . . . , at2, at, a, b−1, b−1t−1, . . . , b−1t−(i−1)
)
∆
(
atn−i−1, . . . , at2, at, a, b−1, b−1t−1, . . . , b−1t−(i−1)
) ,
which coincides with (3.5a) and (3.5b). �
Lemma 3.3 (triangularity). Suppose
ξj :=
(
b−1t−(j−1), b−1t−(j−2), . . . , b−1t−1, b−1︸ ︷︷ ︸
j
, z1, z2, . . . , zn−j
)
∈ (C∗)n.
Then
Ẽk,i(ξj) = 0 if 0 ≤ i < j ≤ n. (3.6)
Moreover, Ẽ0,i(ξi) evaluates as
Ẽ0,i(ξi) =
t−i(n−i)
(
t−1; t−1
)
i
(
t−1; t−1
)
n−i(
1− t−1
)n (
a−1b−1t−(i−1); t
)
i
n−i∏
j=1
(
1− zjbti
)
=
(t; t)i(t; t)n−i
t(
n
2)(1− t)n
(
a−1b−1t−(i−1); t
)
i
n−i∏
j=1
(
1− zjbti
)
. (3.7)
On the other hand, if ηj :=
(
z1, z2, . . . , zj , a, at, at
2, . . . , atn−j−1︸ ︷︷ ︸
n−j
)
∈ (C∗)n, then
Ẽk,i(ηj) = 0 if 0 ≤ j < i ≤ n. (3.8)
Moreover, Ẽ0,i(ηi) evaluates as
Ẽ0,i(ηi) =
(
t−1; t−1
)
i
(
t−1; t−1
)
n−i(
1− t−1
)n (ab; t)n−i
i∏
j=1
(
1− zj/atn−i
)
. (3.9)
Proof. By the definition (3.1) of Ẽk,i(z) it immediately follows that Ẽk,i(ξj) = 0 if i < j, and
Ẽk,i(ηj) = 0 if j < i. If we put zj = b−1t−i (j = 1, 2, . . . , n− i) in the polynomial Ẽk,i(ξi), then
we have Ẽk,i(ξi) = 0 because Ẽk,i(ξi) satisfies the condition of (3.6). This implies that Ẽk,i(ξi) is
divisible by
∏n−i
j=1(1− zjbti) up to a constant. Thus we write Ẽk,i(ξi) = c
∏n−i
j=1(1− zjbti), where
c is some constant independent of z1, . . . , zn−i. Next we determine the explicit form of c. If we
put z1 = a, z2 = at, . . . , zn−i = atn−i−1 in Ẽk,i(ξi), then Ẽk,i(ξi) = Ẽ0,i
(
ζi
(
a, b−1
))
. From (3.4),
we have ci = c(abti; t)n−i, where ci is given by (3.5b). Therefore the constant c is evaluated
as c = ci/(abt
i; t)n−i, i.e., we obtain the expression (3.7) for Ẽk,i(ξi). The evaluation (3.9) is
carried out in the same way as above. �
Lemma 3.4. For 0 ≤ j ≤ n, let ζj
(
x, b−1
)
∈ (C∗)n be the point specified by (3.3) with y = b−1.
Then
Ẽ0,i(ζj(x, b
−1)) =
(
xbti; t
)
n−i
(
xa−1; t
)
i−j
(
a−1b−1t−(j−1); t
)
j
(t; t)n
t(
n
2)(1− t)n
[
i
j
]
t[
n
j
]
t
. (3.10)
Moreover, if i+ k ≤ n (i.e., k ≤ n− i ≤ n− j), then
Ẽk,i
(
ζj
(
x, b−1
))
= xkt(n−j)k−(k+1
2 )Ẽ0,i
(
ζj
(
x, b−1
))
. (3.11)
q-Difference Systems for the Jackson Integral of Symmetric Selberg Type 11
Proof. If i ≤ j, Ẽk,i
(
ζj
(
x, b−1
))
= 0 is a special case of (3.6). Suppose j ≤ i. If we put x =
b−1t−i−k (k = 0, 1, . . . , n−i−1), then the polynomial Ẽ0,i
(
ζj
(
x, b−1
))
satisfies the condition (3.6)
of Lemma 3.3, so that it is equal to zero, which implies Ẽ0,i
(
ζj
(
x, b−1
))
is divisible by (xbti; t)n−i.
If we put x = at−k (k = 0, 1, . . . , i − j − 1), then the polynomial Ẽ0,i
(
ζj
(
x, b−1
))
satisfies the
condition (3.8) of Lemma 3.3, so that it is also equal to zero, which implies Ẽ0,i
(
ζj
(
x, b−1
))
is
divisible by
(
xa−1; t
)
i−j . Therefore we have
Ẽ0,i
(
ζj
(
x, b−1
))
= c
(
xbti; t
)
n−i
(
xa−1; t
)
i−j , (3.12)
where c is some constant independent of x. Next we determine the explicit form of c. Put
x = b−1t−(i−1) in (3.12). Then, using (3.7), the left-hand side of (3.12) is written as
Ẽ0,i
(
ζj
(
x, b−1
))∣∣∣
x=b−1t−(i−1)
= Ẽ0,i
(
ζi
(
x, b−1
))∣∣∣
x=b−1t−(j−1)
=
(t; t)i(t; t)n−i(t; t)n−j
t(
n
2)(1− t)n(t; t)i−j
(
a−1b−1t−(i−1); t
)
i
, (3.13)
while the right-hand side of (3.12) is
c
(
xbti; t
)
n−i
(
xa−1; t
)
i−j
∣∣∣
x=b−1t−(i−1)
= c(t; t)n−i
(
a−1b−1t−(i−1); t
)
i−j . (3.14)
Comparing with (3.13) and (3.14), we have
c =
(t; t)i(t; t)n−j
t(
n
2)(1− t)n(t; t)i−j
(
a−1b−1t−(j−1); t
)
j
.
Therefore we obtain (3.10). Moreover, if i+ k ≤ n, by definition we have
Ẽk,i
(
ζj
(
x, b−1
))
=
(
xt(n−j−1)
)(
xt(n−j−2)
)
· · ·
(
xt(n−j−k)
)
Ẽ0,i
(
ζj
(
x, b−1
))
,
which coincides with (3.11). �
As a counterpart of Lemma 3.4, we have the following.
Lemma 3.5. For 0 ≤ j ≤ n, let ζj(a, y) ∈ (C∗)n be the point specified by (3.3) with x = a.
Then
Ẽ0,i(ζj(a, y)) =
(
ybt−(j−i−1); t
)
j−i
(
y/atn−1; t
)
i
(ab; t)n−j
(
t−1; t−1
)
n(
1− t−1
)n
[
n− i
n− j
]
t−1[
n
j
]
t−1
. (3.15)
Moreover, if n ≤ i+ k (i.e., n− j ≤ n− i ≤ k), then
Ẽk,i(ζj(a, y)) = yk+j−nan−jt(
n−j
2 )−(k+j−n
2 )Ẽ0,i(ζj(a, y)).
Proof. This lemma can be proved in the same way as Lemma 3.4 using Lemma 3.3. �
12 M. Ito
4 Three-term relations
In this section we fix Ẽk,i(z) = Ẽk,i(a1, b2; z). In particular we have ei(a1, b2; z) = Ẽ0,i(a1, b2; z).
The following lemma is a technical key for computing the coefficient matrix A of (1.17) in
Theorem 1.7. We abbreviate
〈
Ẽk,i, x
〉
to
〈
Ẽk,i
〉
throughout this section.
Lemma 4.1 (three-term relations). Suppose i+ k ≤ n. Then,
1− qαa1a2b1b2t2n−k−1
a1tn−i−k
〈
Ẽk,i
〉
=
(
1− qαa2b2tn+i−k
)〈
Ẽk−1,i
〉
−
(
1− a2b2ti
)〈
Ẽk−1,i+1
〉
.(4.1)
On the other hand, if i+ k ≥ n, then,
tn−k
(
1− qαtn−k
)〈
Ẽk−1,i+1
〉
= a−11 qαtn+i−k
(
1− a1b1tn−i−1
)〈
Ẽk,i
〉
+ a−12
(
1− qαa2b2tn+i−k
)〈
Ẽk,i+1
〉
. (4.2)
The proof of this lemma will be given in Section 5. The rest of this section is devoted to
computing the Gauss matrix decomposition of A in Theorem 1.7 using Lemma 4.1. By repeated
use of Lemma 4.1, we have the following.
Corollary 4.2. Suppose i+ k ≤ n. For 0 ≤ l ≤ k,
〈
Ẽk,i
〉
is expressed as
〈
Ẽk,i
〉
=
l∑
j=0
Lk,ik−l,i+j
〈
Ẽk−l,i+j
〉
, (4.3)
where the coefficients Lk,ik−l,i+j is expressed as
Lk,ik−l,i+j =
[
l
j
]
t
(−1)j
(
a1t
n−i−k)lt(l−j
2 )(a2b2ti; t)j(qαa2b2tn+i+j−k; t)l−j(
qαa1a2b1b2t2n−k−1; t
)
l
.
On the other hand, if i+ k ≥ n, then,
〈
Ẽk,i
〉
=
l∑
j=0
Uk,ik−l+j,i−j
〈
Ẽk−l+j,i−j
〉
, (4.4)
where the coefficient Uk,ik−l+j,i−j is expressed as
Uk,ik−l+j,i−j =
[
l
j
]
t
(
−qαa−11 ti−l
)j(
a2t
n−k)lt(l
2)
(
a1b1t
n−i; t
)
j
(
qαtn−k; t
)
l−j(
qαa2b2tn+i−k−j−1; t
)
l−j
(
qαa2b2tn+i−k−2j+l; t
)
j
.
Proof. (4.3) and (4.4) follow by double induction on l and j using (4.1) and (4.2), respective-
ly. �
In particular, we immediately have the following as a special case of Corollary 4.2.
Corollary 4.3. For 0 ≤ j ≤ n, 〈En−j,j〉 is expressed as
〈
Ẽn−j,j
〉
=
n∑
i=j
l̃ij
〈
Ẽ0,i
〉
, (4.5)
q-Difference Systems for the Jackson Integral of Symmetric Selberg Type 13
where the coefficients l̃ij is expressed as
l̃ij = Ln−j,j0,i =
[
n− j
n− i
]
t
(−1)i−jan−j1 t(
n−i
2 )(a2b2tj ; t)i−j(qαa2b2ti+j ; t)n−i(
qαa1a2b1b2tn+j−1; t
)
n−j
, (4.6)
while, for 0 ≤ j ≤ n, 〈En,j〉 is expressed as
〈
Ẽn,j
〉
=
j∑
i=0
ũij
〈
Ẽn−i,i
〉
, (4.7)
where the coefficients ũij is expressed as
ũij = Un,jn−i,i =
[
j
i
]
t
(
−qαa−11
)j−i
aj2t
(j2)
(
qα; t
)
i
(
a1b1t
n−j ; t
)
j−i(
qαa2b2ti−1; t
)
i
(
qαa2b2t2i; t
)
j−i
. (4.8)
We now give the explicit expression for A in terms of Gauss decomposition.
Proof of Theorem 1.7. From (4.5), we have(〈
Ẽn,0
〉
,
〈
Ẽn−1,1
〉
, . . . ,
〈
Ẽ1,n−1
〉
,
〈
Ẽ0,n
〉)
=
(〈
Ẽ0,0
〉
,
〈
Ẽ0,1
〉
, . . . ,
〈
Ẽ0,n−1
〉
,
〈
Ẽ0,n
〉)
L̃,
where the matrix L̃ =
(
l̃ij
)
0≤i,j≤n is defined by (4.6). Moreover, from (4.7) we have(〈
Ẽn,0
〉
,
〈
Ẽn,1
〉
, . . . ,
〈
Ẽn,n−1
〉
,
〈
Ẽn,n
〉)
=
(〈
Ẽn,0
〉
,
〈
Ẽn−1,1
〉
, . . . ,
〈
Ẽ1,n−1
〉
,
〈
Ẽ0,n
〉)
Ũ (4.9a)
=
(〈
Ẽ0,0
〉
,
〈
Ẽ0,1
〉
, . . . ,
〈
Ẽ0,n−1
〉
,
〈
Ẽ0,n
〉)
L̃Ũ , (4.9b)
where the matrix Ũ =
(
ũij
)
0≤i,j≤n is defined by (4.8). Since TαΦ(z) = z1z2 · · · znΦ(z) and
z1z2 · · · znẼ0,i(z) = Ẽn,i(z), we have Tα
〈
Ẽ0,i
〉
=
〈
Ẽn,i
〉
, i.e.,
Tα
(〈
Ẽ0,0
〉
,
〈
Ẽ0,1
〉
, . . . ,
〈
Ẽ0,n−1
〉
,
〈
Ẽ0,n
〉)
=
(〈
Ẽn,0
〉
,
〈
Ẽn,1
〉
, . . . ,
〈
Ẽn,n−1
〉
,
〈
Ẽn,n
〉)
. (4.10)
From (4.9b) and (4.10), we obtain the difference system
Tα
(〈
Ẽ0,0
〉
,
〈
Ẽ0,1
〉
, . . . ,
〈
Ẽ0,n−1
〉
,
〈
Ẽ0,n
〉)
=
(〈
Ẽ0,0
〉
,
〈
Ẽ0,1
〉
, . . . ,
〈
Ẽ0,n−1
〉
,
〈
Ẽ0,n
〉)
L̃Ũ .
Comparing this with (1.17), we therefore obtain A = L̃Ũ = LADAUA, i.e.,
lAij =
l̃ij
l̃jj
, dA
j = l̃jj ũjj , uA
ij =
ũij
ũii
.
Corollary 4.2 implies that lAij , d
A
j and uA
ij coincide with (1.18a), (1.18b) and (1.18c), respectively.
The Gauss decomposition of A in the opposite direction, i.e., A = U ′AD
′
AL
′
A can also be given
in the same way as above. This will be explained in the Appendix. �
Finally we state the explicit forms for U−1A .
Proposition 4.4. The inverse matrix U−1A =
(
uA∗
ij
)
0≤i,j≤n is upper triangular, and is written
as
uA∗
ij =
(
qαa−11 a2t
j−1)j−i [j
i
]
t
(
a1b1t
n−j ; t
)
j−i(
qαa2b2tj+i−1; t
)
j−i
. (4.11)
14 M. Ito
Proof. Since A = L̃Ũ = LADAUA, we have Ũ = DŨUA, where DŨ is the diagonal matrix
defined by the diagonal elements of Ũ =
(
ũij
)
0≤i,j≤n, i.e., DŨ =
(
ũiiδij
)
0≤i,j≤n, where ũij is
given by (4.8). We first compute the explicit expression for Ũ−1. From (4.9a), Ũ−1 is regarded
as the transition matrix
(〈En,0〉, 〈En−1,1〉, . . . , 〈E1,n−1〉, 〈E0,n〉) = (〈En,0〉, 〈En,1〉, . . . , 〈En,n−1〉, 〈En,n〉)Ũ−1, (4.12)
namely, if we write Ũ−1 =
(
ṽij
)
0≤i,j≤n, then (4.12) is equivalent to
〈
Ẽn−j,j
〉
=
∑j
i=0 ṽij
〈
Ẽn,i
〉
.
Similar to Corollary 4.2, by repeated use of the three-term relation (4.4) inductively, 〈Ẽk,i〉 is
generally expressed as
〈
Ẽk,i
〉
=
l∑
j=0
V k,i
k+l,i−j
〈
Ẽk+l,i−j
〉
,
where the coefficients V k,i
k+l,i−j is given by
V k,i
k+l,i−j =
[
l
j
]
t
(
qαa−11 ti−1
)j
t(
l−j
2 )−(j2)
(
a1b1t
n−i; t
)
j
(
qαa2b2t
n+i−k−l−1; t
)
l−j(
a2tn−k−1
)l−j(
qαtn−k−l; t
)
l
.
In particular, the entries ṽij of Ũ−1 are explicitly expressed as
ṽij = V n−j,j
n,i =
[
j
i
]
t
(
qαa−11 tj−1
)j−i
t(
i
2)−(j−i
2 )(a1b1tn−j ; t)j−i(qαa2b2tj−1; t)i(
a2tj−1
)i(
qα; t
)
j
. (4.13)
Next we compute U−1A =
(
uA∗
ij
)
0≤i,j≤n. Since U−1A is expressed as U−1A = Ũ−1DŨ , using (4.8)
and (4.13), we obtain
uA∗
ij = ṽij ũjj
=
[
j
i
]
t
(
qαa−11 tj−1
)j−i
t(
i
2)−(j−i
2 )(a1b1tn−j ; t)j−i(qαa2b2tj−1; t)i(
a2tj−1
)i(
qα; t
)
j
aj2t
(j2)
(
qα; t
)
j(
qαa2b2tj−1; t
)
j
,
which coincides with (4.11). �
5 Proof of Lemma 4.1
The aim of this section is to give a proof of Lemma 4.1. Throughout this section we fix Ẽk,i(z) =
Ẽk,i(a1, b2; z). For Φ(z) = Φn,2, let ∇ be the operator specified by
(∇ϕ)(z) := ϕ(z)− Tq,z1Φ(z)
Φ(z)
Tq,z1ϕ(z),
where Tq,z1 is the q-shift operator with respect to z1 → qz1, i.e., Tq,z1f(z1, z2, . . . , zn) =
f(qz1, z2, . . . , zn) for an arbitrary function f(z1, z2, . . . , zn). Here the ratio Tq,z1Φ(z)/Φ(z) is
expressed explicitly as
Tq,z1Φ(z)
Φ(z)
=
qαt2(n−1)(1− b1z1)(1− b2z1)(
1− qa−11 z1
)(
1− qa−12 z1
) n∏
j=2
z1 − t−1zj
qz1 − tzj
=
G1(z)
Tz1F1(z)
,
q-Difference Systems for the Jackson Integral of Symmetric Selberg Type 15
where
F1(z) =
(
1− a−11 z1
)(
1− a−12 z1
) n∏
j=2
(z1 − tzj),
G1(z) = qαt2(n−1)(1− b1z1)(1− b2z1)
n∏
j=2
(
z1 − t−1zj
)
.
Lemma 5.1. Suppose that
∫ x∞
0 Φ(z)ϕ(z)$q converges for a meromorphic function ϕ(z), then∫ x∞
0
Φ(z)∇ϕ(z)$q = 0.
Moreover,∫ x∞
0
Φ(z)A∇ϕ(z)$q = 0.
Proof. See Lemma 5.3 in [19]. �
The rest of this section is devoted to the proof of Lemma 4.1. We show a further lemma
before proving Lemma 4.1. For this purpose we abbreviate Ẽk,i(a1, b2; z) to Ẽk,i(z). When we
need to specify the number of variables z1, . . . , zn, we use the notation Ẽ
(n)
k,i (z) = Ẽk,i(z) and
∆(n)(z) = ∆(z). We set ϕk,i(z) := F1(z)E
(n−1)
k−1,i (z2, . . . , zn). Then
∇ϕk,i(z) = (F1(z)−G1(z))E
(n−1)
k−1,i (z2, . . . , zn).
Let ϕ̃k,i(z) be the skew-symmetrization of ∇ϕk,i(z), i.e.,
ϕ̃k,i(z) := A∇ϕk,i(z) =
n∑
j=1
(−1)j−1(Fj(z)−Gj(z))Ẽ(n−1)
k−1,i (ẑj)∆
(n−1)(ẑj), (5.1)
where ẑj := (z1, . . . , zj−1, zj+1, . . . , zn) ∈ (C∗)n−1 for j = 1, . . . , n, and
Fi(z) =
(
1− a−11 zi
)(
1− a−12 zi
) ∏
1≤k≤n
k 6=i
(zi − tzk),
Gi(z) = qαt2(n−1)(1− b1zi)(1− b2zi)
∏
1≤k≤n
k 6=i
(
zi − t−1zk
)
,
which satisfy the following vanishing property at the point z = ζj
(
x, b−12
)
or z = ζj(a1, y).
Lemma 5.2. If i 6= 0 and i 6= j, then Fi+1
(
ζj
(
x, b−12
))
= 0. If i 6= n, then Gi
(
ζj
(
x, b−12
))
= 0.
Otherwise,
F1
(
ζj
(
x, b−12
))
=
(
1− a−11 b−12 t−(j−1)
)(
1− a−12 b−12 t−(j−1)
)(
b2tj−1
)n−1
(1− t)
(t; t)j
(
xb2t
j ; t
)
n−j , (5.2a)
Fj+1
(
ζj
(
x, b−12
))
= (−1)j
(
1− a−11 x
)(
1− a−12 x
)
xn−j−1
bj2t
(j−1
2 )−1(1− t)
(t, t)n−j
(
xb2t
−1; t
)
j
, (5.2b)
Gn
(
ζj
(
x, b−12
))
= qαt2(n−1)
(
1− b1xtn−j−1
)(
1− b2xtn−j−1
)
16 M. Ito
×
(
−b−12
)j
t−(j+1
2 )(xbtn−j ; t)
j
(
xtn−j−1
)n−j−1 (t−1; t−1)n−j
1− t−1
, (5.2c)
while, if i 6= 1, then Fi(ζj(a1, y)) = 0. If i 6= n and i 6= j, then Gi(ζj(a1, y)) = 0. Otherwise,
F1(ζj(a1, y)) =
(
1− ya−12 t−(j−1)
)
(−yt)j−1(−a1t)n−jt(
n−j
2 )−(j−1
2 )
×
(
ya−11 t−(n−1); t
)
n−j+1
(
t−1; t−1
)
j
1− t−1
, (5.3a)
Gj(ζj(a1, y)) = qαt2(n−1)(1− yb1)(1− yb2)
× yj−1(−a1t−1)n−jt(
n−j
2 )(ya−11 t−(n−j−2); t
)
n−j
(
t−1; t−1
)
j
1− t−1
, (5.3b)
Gn(ζj(a1, y)) = qαt2(n−1)
(
1− a1b1tn−j−1
)(
1− a1b2tn−j−1
)
×
(
a1t
n−j−1)n−1(ya−11 t−(n−1); t
)
j
(
t−1; t−1
)
n−j
1− t−1
. (5.3c)
Proof. The proof follows by direct computation and we omit the details. �
Since the leading term of the symmetric polynomial ϕ̃k,i(z)/∆
(n)(z) is equal to m(1n−k2k)(z)
up to a multiplicative constant, ϕ̃k,i(z)/∆
(n)(z) is expressed as the linear combination of the
symmetric polynomials Ẽ
(n)
l,j (z) in the following two ways:
ϕ̃k,i(z)
∆(n)(z)
=
k∑
l=0
n−l∑
j=0
cljẼ
(n)
l,j (z) =
k∑
l=0
n∑
j=n−l
dljẼ
(n)
l,j (z), (5.4)
where clj and dlj are some coefficients.
Lemma 5.3. Suppose i+ k ≤ n. Then, (5.4) is written as
ϕ̃k,i(z)
∆(n)(z)
= ck,iẼ
(n)
k,i (z) + ck−1,iẼ
(n)
k−1,i(z) + ck−1,i+1Ẽ
(n)
k−1,i+1(z), (5.5)
where
ck,i = −a−11 a−12 b−12 tk−1
(
1− qαa1a2b1b2t2n−k−1
)
= qαb1t
2n−2 − a−11 a−12 b−12 tk−1, (5.6a)
ck−1,i = a−12 b−12 tn−i−1
(
1− qαa2b2tn+i−k
)
= a−12 b−12 tn−i−1 − qαt2n−k−1, (5.6b)
ck−1,i+1 = −a−12 b−12 tn−i−1
(
1− a2b2ti
)
= tn−1
(
1− a−12 b−12 t−i
)
. (5.6c)
Suppose i+ k ≥ n. Then, (5.4) is written as
ϕ̃k,i(z)
∆(n)(z)
= dk,iẼ
(n)
k,i (z) + dk,i+1Ẽ
(n)
k,i+1(z) + dk−1,i+1Ẽ
(n)
k−1,i+1(z), (5.7)
where
dk,i = −a−11 qαtn+i−1
(
1− a1b1tn−i−1
)
= qαb1t
2n−2 − qαa−11 tn+i−1, (5.8a)
dk,i+1 = −a−12 tk−1
(
1− qαa2b2tn+i−k
)
= qαb2t
n+i−1 − a−12 tk−1, (5.8b)
dk−1,i+1 = tn−1
(
1− qαtn−k
)
= tn−1 − qαt2n−k−1. (5.8c)
Remark 5.4. Given Lemma 5.3, Lemma 4.1 immediately follows by Lemma 5.1. Instead of
proving Lemma 4.1 it thus suffices to prove Lemma 5.3.
q-Difference Systems for the Jackson Integral of Symmetric Selberg Type 17
Before proving Lemma 5.3 we show it holds for the following specific cases.
Lemma 5.5. If i + k ≤ n, then the equation (5.5) holds for the points z = ζj
(
x, b−12
)
(j =
0, 1, . . . , n), while if i + k ≥ n, then the equation (5.7) holds for the points z = ζj(a1, y) (j =
0, 1, . . . , n).
Proof. Suppose i+ k ≤ n. If z = ζj
(
x, b−12
)
, then the right-hand side of (5.5) with coefficients
given by (5.6a)–(5.6c) can be written as
ck,iẼ
(n)
k,i
(
ζj
(
x, b−12
))
+ ck−1,iẼ
(n)
k−1,i
(
ζj
(
x, b−12
))
+ ck−1,i+1Ẽ
(n)
k−1,i+1
(
ζj
(
x, b−12
))
=
(
ck,ixt
n−j−k + ck−1,i
)
Ẽ
(n)
k−1,i
(
ζj
(
x, b−12
))
+ ck−1,i+1Ẽ
(n)
k−1,i+1
(
ζj
(
x, b−12
))
=
[(
1− a−11 xti−j
)
a−12 b−12 tn−i−1 − qαt2n−k−1
(
1− xb1tn−j−1
)]
Ẽ
(n)
k−1,i
(
ζj
(
x, b−12
))
+ tn−1
(
1− a−12 b−12 t−i
)
Ẽ
(n)
k−1,i+1
(
ζj
(
x, b−12
))
=
[(
1− xb2ti
)(
1− ti−j+1
)
a2b2ti
(
1− ti+1
) +
(
1− a−12 b−12 t−i
)]
tn−1Ẽ
(n)
k−1,i+1
(
ζj
(
x, b−12
))
− qαt2n−k−1
(
1− xb1tn−j−1
)
Ẽ
(n)
k−1,i
(
ζj
(
x, b−12
))
. (5.9)
The final equality follows from the relation(
1− a−11 xti−j
)(
1− ti+1
)
Ẽ
(n)
k−1,i
(
ζj
(
x, b−12
))
=
(
1− xb2ti
)(
1− ti−j+1
)
Ẽ
(n)
k−1,i+1
(
ζj
(
x, b−12
))
,
which follows from (3.10) and (3.11). On the other hand, using (5.1) and (5.2a)–(5.2c), the
left-hand side of (5.5) at z = ζj
(
x, b−12
)
can be written as
ϕ̃k,i
(
ζj
(
x, b−1
))
∆(n)
(
ζ
(n)
j
(
x, b−1
)) = F1
(
ζ
(n)
j
(
x, b−1
))
Ẽ
(n−1)
k−1,i
(
ζ
(n−1)
j−1
(
x, b−1
))∆(n−1)(ζ(n−1)j−1
(
x, b−1
))
∆(n)
(
ζ
(n)
j
(
x, b−1
))
+ (−1)jFj+1
(
ζ
(n)
j
(
x, b−1
))
Ẽ
(n−1)
k−1,i
(
ζ
(n−1)
j
(
xt, b−1
))∆(n−1)(ζ(n−1)j
(
xt, b−1
))
∆(n)
(
ζ
(n)
j
(
x, b−1
))
+ (−1)nGn
(
ζ
(n)
j
(
x, b−1
))
Ẽ
(n−1)
k−1,i
(
ζ
(n−1)
j
(
x, b−1
))∆(n−1)(ζ(n−1)j
(
x, b−1
))
∆(n)
(
ζ
(n)
j
(
x, b−1
)) . (5.10)
Since we can compute
Ẽ
(n−1)
k−1,i
(
ζ
(n−1)
j−1
(
x, b−1
))
=
tn−1(1− t)(1− xbti)Ẽ(n)
k−1,i+1
(
ζ
(n)
j
(
x, b−1
))(
1− xbtn−1
)(
1− a−1b−1t−(j−1)
)(
1− ti+1
) , (5.11a)
Ẽ
(n−1)
k−1,i
(
ζ
(n−1)
j
(
xt, b−1
))
=
tn−1(1− t)(1− ti−j+1)Ẽ
(n)
k−1,i+1
(
ζ
(n)
j
(
x, b−1
))(
1− xa−1
)(
1− ti+1
)(
1− tn−j
) , (5.11b)
Ẽ
(n−1)
k−1,i
(
ζ
(n−1)
j
(
x, b−1
))
=
tn−k(1− t)Ẽ(n)
k−1,i
(
ζ
(n)
j
(
x, b−1
))(
1− xbtn−1
)(
1− tn−j
) , (5.11c)
and
∆(n−1)(ζ(n−1)j−1
(
x, b−1
))
∆(n)
(
ζ
(n)
j
(
x, b−1
)) =
(
b2t
j−1)n−1
(t; t)j−1
(
xbtj−1; t
)
n−j
, (5.11d)
18 M. Ito
∆(n−1)(ζ(n−1)j
(
xt, b−1
))
∆(n)
(
ζ
(n)
j
(
x, b−1
)) =
x−(n−j−1)bj2t
(j2)
(t; t)n−j−1(xb; t)j
(5.11e)
∆(n−1)(ζ(n−1)j
(
xb−1
))
∆(n)
(
ζ
(n)
j
(
x, b−1
)) =
x−(n−j−1)bj2t
(j2)−(n−j−1
2 )
(t; t)n−j−1
(
xbtn−j−1; t
)
j
, (5.11f)
applying (5.11a)–(5.11f) to (5.10), the left-hand side of (5.5) at z = ζj
(
x, b−12
)
can be expressed
as
ϕ̃k,i
(
ζj
(
x, b−12
))
∆(n)
(
ζ
(n)
j
(
x, b−12
)) =
[(
1− a−12 b−12 t−(j−1)
)(
1− tj
)(
1− xb2ti
)(
1− xb2tj−1
)(
1− ti+1
)
+
(
1− a−12 x
)(
1− xb2t−1
)(
1− ti−j+1
)(
1− xb2tj−1
)(
1− ti+1
) tj
]
tn−1Ẽ
(n)
k−1,i+1
(
ζ
(n)
j
(
x, b−12
))
− qαt2n−k−1
(
1− xb1tn−j−1
)
Ẽ
(n)
k−1,i
(
ζ
(n)
j
(
x, b−12
))
. (5.12)
Comparing (5.9) with (5.12), the claim of the lemma is proved if we can check the identity(
1− xb2ti
)(
1− ti−j+1
)
a2b2ti(1− ti+1)
+
(
1− a−12 b−12 t−i
)
=
(
1− a−12 b−12 t−(j−1)
)(
1− tj
)(
1− xb2ti
)(
1− xb2tj−1
)(
1− ti+1
) +
(
1− a−12 x
)(
1− xb2t−1
)(
1− ti−j+1
)(
1− xb2tj−1
)(
1− ti+1
) tj ,
which is confirmed by direct computation.
Next suppose i + k ≥ n. If z = ζj(a1, y), then the right-hand side of (5.7) with coefficients
given by (5.8a)–(5.8c) can be written as
dk,iẼ
(n)
k,i (ζj(a1, y)) + dk,i+1Ẽ
(n)
k,i+1(ζj(a1, y)) + dk−1,i+1Ẽ
(n)
k−1,i+1(ζj(a1, y))
= dk,iẼ
(n)
k,i (ζj(a1, y)) +
[
dk,i+1yt
−(k+j−n−1) + dk−1,i+1
]
Ẽ
(n)
k−1,i+1(ζj(a1, y))
= −qαa−11 tn+i−1
(
1− a1b1tn−i−1
)
Ẽ
(n)
k,i (ζj(a1, y))
+
[
tn−1
(
1− ya−12 t−(j−1)
)
− qαt2n−k−1
(
1− yb2t−(j−i−1)
)]
Ẽ
(n)
k−1,i+1(ζj(a1, y))
= −qαtn+i−1
[
a−11
(
1− a1b1tn−i−1
)
+
(
1− ya−11 t−(n−i−1)
)(
1− t−(j−i)
)
yt−(j−i−1)
(
1− t−(n−i)
) ]
Ẽ
(n)
k,i (ζj(a1, y))
+ tn−1
(
1− ya−12 t−(j−1)
)
Ẽ
(n)
k−1,i+1(ζj(a1, y))
= −qαtn+j−2
[
1− a1b1tn−i−1
a1tj−i−1
+
(
1− ya−11 t−(n−i−1)
)(
1− t−(j−i)
)
y
(
1− t−(n−i)
) ]
Ẽ
(n)
k,i (ζj(a1, y))
+ tn−1
(
1− ya−12 t−(j−1)
)
Ẽ
(n)
k−1,i+1(ζj(a1, y)). (5.13)
On the other hand, using (5.1) and (5.3a)–(5.3c), the left-hand side of (5.7) at z = ζj(a1, y) can
be written as
ϕ̃k,i(ζj(a1, y))
∆(n)
(
ζ
(n)
j (a1, y)
) = F1(ζ
(n)
j (a1, y))Ẽ
(n−1)
k−1,i
(
ζ
(n−1)
j−1 (a1, y)
)∆(n−1)(ζ(n−1)j−1 (a1, y)
)
∆(n)
(
ζ
(n)
j (a1, y)
)
− (−1)j−1Gj
(
ζ
(n)
j (a1, y)
)
Ẽ
(n−1)
k−1,i
(
ζ
(n−1)
j−1
(
a1, yt
−1))∆(n−1)(ζ(n−1)j−1
(
a1, yt
−1))
∆(n)
(
ζ
(n)
j (a1, y)
)
q-Difference Systems for the Jackson Integral of Symmetric Selberg Type 19
− (−1)n−1Gn
(
ζ
(n)
j (a1, y)
)
Ẽ
(n−1)
k−1,i
(
ζ
(n−1)
j (a1, y)
)∆(n−1)(ζ(n−1)j (a1, y)
)
∆(n)
(
ζ
(n)
j (a1, y)
) . (5.14)
Since we can compute
Ẽ
(n−1)
k−1,i
(
ζ
(n−1)
j−1 (a, y)
)
=
(
1− t−1
)
Ẽ
(n)
k−1,i+1
(
ζ
(n)
j (a, y)
)(
1− t−j
)(
1− ya−1t−(n−1)
) , (5.15a)
Ẽ
(n−1)
k−1,i
(
ζ
(n−1)
j−1
(
a, yt−1
))
=
(
1− t−1
)(
1− t−(j−i)
)
Ẽ
(n)
k,i
(
ζ
(n)
j (a, y)
)(
1− t−j
)(
1− t−(n−i)
)
(1− yb)
, (5.15b)
Ẽ
(n−1)
k−1,i
(
ζ
(n−1)
j (a, y)
)
=
(
1− t−1
)(
1− ya−1t−(n−i−1)
)
Ẽ
(n)
k,i
(
ζ
(n)
j (a, y)
)
atn−j−1
(
1− t−(n−i)
)(
1− ya−1t−(n−1)
)(
1− abtn−j−1
) , (5.15c)
and
∆(n−1)(ζ(n−1)j−1 (a1, y)
)
∆(n)
(
ζ
(n)
j (a1, y)
) =
(−1)n−j
yj−1an−j1 t(
n−j
2 )−(j−1
2 )(ya−11 t−(n−2); t
)
n−j
(
t−1; t−1
)
j−1
, (5.15d)
∆(n−1)(ζ(n−1)j−1 (a1, yt
−1)
)
∆(n)
(
ζ
(n)
j (a1, y)
) =
(−1)n−1
yj−1an−j1 t(
n−j
2 )(ya−11 t−(n−j−1); t
)
n−j
(
t−1; t−1
)
j−1
, (5.15e)
∆(n−1)(ζ(n−1)j (a1, y)
)
∆(n)
(
ζ
(n)
j (a1, y)
) =
(−1)n−1(
a1tn−j−1
)n−1(
ya−11 t−(n−2); t
)
j
(
t−1; t−1
)
n−j−1
, (5.15f)
applying (5.15a)–(5.15f) to (5.14), the left-hand side of (5.7) at z = ζj(a, y) can be expressed as
ϕ̃k,i(ζj(a1, y))
∆(n)
(
ζ
(n)
j (a1, y)
) = tn−1
(
1− ya−12 t−(j−1)
)
Ẽ
(n)
k−1,i+1(ζj(a1, y))
− qαtn+j−2
[
(1− yb1)
(
1− ya−1t
)(
1− t−(j−i)
)
y
(
1− ya−1t−(n−j−1)
)(
1− t−(n−i)
)
+
(
1− a1b1tn−j−1
) t(1− ya−1t−(n−i−1))(1− t−(n−j))
a
(
1− ya−1t−(n−j−1)
)(
1− t−(n−i)
)]Ẽ(n)
k,i (ζj(a1, y)). (5.16)
Comparing with (5.13) and (5.16), the claim of the lemma is proved if we can check the identity
1− a1b1tn−i−1
a1tj−i−1
+
(
1− ya−11 t−(n−i−1)
)(
1− t−(j−i)
)
y
(
1− t−(n−i)
) =
(1− yb1)
(
1− ya−1t
)(
1− t−(j−i)
)
y
(
1− ya−1t−(n−j−1)
)(
1− t−(n−i)
)
+
(
1− a1b1tn−j−1
) t(1− ya−1t−(n−i−1))(1− t−(n−j))
a
(
1− ya−1t−(n−j−1)
)(
1− t−(n−i)
) ,
which follows from direct computation. �
Proof of Lemma 5.3. Set Dj =
{
(l, i) ∈ Z2 | j ≤ i, 0 ≤ l, i + l ≤ n
}
, which satisfies D0 ⊃
D1 ⊃ · · · ⊃ Dn = {(0, n)}. The set
{
Ẽk,i(z) | (k, i) ∈ D0
}
forms a basis for the linear space
spanned by {mλ(z) |λ ≤ (2n)}. If we put
ψ(z) :=
ϕ̃k,i(z)
∆(z)
−
(
ck,iẼk,i(z) + ck−1,iẼk−1,i(z) + ck−1,i+1Ẽk−1,i+1(z)
)
,
20 M. Ito
where ck,i, ck−1,i and ck−1,i+1 are specified by (5.6a)–(5.6c), then the symmetric polynomial ψ(z)
is expressed as a linear combination of Ẽk,i(z), (k, i) ∈ D0, i.e.,
ψ(z) =
∑
(l,m)∈D0
c′lmẼl,m(z), (5.17)
where the coefficients c′lm are some constants. We now prove ψ(z) = 0 identically, i.e., c′lm = 0
for all (l,m) ∈ D0 inductively. Namely, we prove that, if c′lm = 0 for (l,m) ∈ Dj+1, then c′lm = 0
for (l,m) ∈ Dj .
First we show that c′0n = 0 as the starting point of induction. Using Lemma 3.4 for (5.17)
at z = ζn
(
x, b−12
)
we have ψ
(
ζn
(
x, b−12
))
= c′0nẼ0,n
(
ζn
(
x, b−12
))
. From Lemma 5.5 we have
ψ
(
ζn(x, b−12
))
= 0, while Ẽ0,n
(
ζn
(
x, b−12
))
6= 0. Therefore c′0n = 0.
Next suppose that c′lm = 0 for (l,m) ∈ Dj+1. Then using Lemma 3.4 for (5.17) at z =
ζj
(
x, b−12
)
we have
ψ
(
ζj
(
x, b−12
))
=
n−j∑
l=0
c′ljẼl,j
(
ζj
(
x, b−12
))
=
(
n−j∑
l=0
c′ljx
ltl(n−j)−(l+1
2 )
)
Ẽ0,j
(
ζj
(
x, b−12
))
.
From Lemma 5.5 ψ
(
ζj
(
x, b−12
))
vanishes as a function of x, while Ẽ0,j
(
ζj
(
x, b−12
))
6= 0. Thus,∑n−j
l=0 c
′
ljx
ltl(n−j)−(l+1
2 ) = 0, i.e., the coefficient c′ljt
l(n−j)−(l+1
2 ) of xl vanishes for 0 ≤ l ≤ n − j.
Therefore c′lj = 0 for 0 ≤ l ≤ n− j. This implies c′lm = 0 for (l,m) ∈ Dj .
On the other hand, we prove (5.7) of Lemma 5.3. Set D′j =
{
(l, i) ∈ Z2 |n ≤ i+ l, 0 ≤ l ≤ n,
0 ≤ i ≤ j
}
, which satisfies {(n, 0)} = D′0 ⊂ D′1 ⊂ · · · ⊂ D′n. The set {Ẽk,i(z) | (k, i) ∈ D′n} also
forms a basis for the linear space spanned by {mλ(z) |λ ≤ (2n)}. If we put
ψ′(z) :=
ϕ̃k,i(z)
∆(z)
−
(
dk,iẼk,i(z) + dk,i+1Ẽk,i+1(z) + dk−1,i+1Ẽk−1,i+1(z)
)
,
where dk,i, dk,i+1 and dk−1,i+1 are specified by (5.8a)–(5.8c), then the symmetric polynomial
ψ′(z) is expressed as a linear combination of Ẽk,i(z), (k, i) ∈ D′n, i.e.,
ψ′(z) =
∑
(l,m)∈D′n
d′lmẼl,m(z), (5.18)
where the coefficients d′lm are some constants. We now prove ψ′(z) = 0 identically, i.e., d′lm = 0
for all (l,m) ∈ D′n inductively. Namely, we prove that, if d′lm = 0 for (l,m) ∈ D′j−1, then d′lm = 0
for (l,m) ∈ D′j .
First we show that d′n0 = 0 as the starting point of induction. Using Lemma 3.5 for (5.18) at
z = ζ0((a1, y)) we have ψ′(ζ0(a1, y)) = d′n0Ẽn,0(ζ0(a1, y)). From Lemma 5.5 we have ψ′(ζ0(a1, y))
= 0, while Ẽn,0(ζ0(a1, y)) 6= 0. Therefore d′n0 = 0.
Next suppose that d′lm = 0 for (l,m) ∈ D′j−1. Then using Lemma 3.5 for (5.18) at z = ζj(a1, y)
we have
ψ′(ζj(a1, y)) =
n∑
l=n−j
d′ljẼl,j(ζj(a1, y))
=
n∑
l=n−j
d′ljy
l+j−nan−j1 t(
n−j
2 )−(l+j−n
2 )
Ẽ0,j(ζj(a1, y)).
From Lemma 5.5 ψ′(ζj(a1, y)) vanishes as a function of y, while Ẽ0,j(ζj(a1, y)) 6= 0. Thus,∑n
l=n−j d
′
ljy
l+j−nan−j1 t(
n−j
2 )−(l+j−n
2 ) = 0, i.e., the coefficient d′lja
n−j
1 t(
n−j
2 )−(l+j−n
2 ) of yl+j−n va-
nishes for n − j ≤ l ≤ n. Therefore d′lj = 0 for n − j ≤ l ≤ n. This implies d′lm = 0 for
(l,m) ∈ D′j . �
q-Difference Systems for the Jackson Integral of Symmetric Selberg Type 21
6 The transition matrix R
In this section we give a proof of Theorem 1.4. Before proving Theorem 1.4, we will show the
results deduced from Theorem 1.4. By the definition (1.12) of the transition matrix R, we have
R−1 = JR̄J,
where the symbol R̄ is the matrix R after the interchange (a1, b1)↔ (a2, b2) and J is the matrix
specified by
J =
1
1
· · ·
1
.
The explicit form of the inverse matrix of R is given by
Corollary 6.1. The inverse matrix R−1 is written as Gauss matrix decomposition
R−1 = U−1R D−1R L−1R = L′−1R D′−1R U ′−1R ,
where the inverse matrices L−1R =
(
lR∗ij
)
0≤i,j≤n, D−1R =
(
dR∗
j δij
)
0≤i,j≤n, U−1R =
(
uR∗
ij
)
0≤i,j≤n are
lower triangular, diagonal, upper triangular, respectively, given by
lR∗ij = uR
n−i,n−j =
[
n− j
n− i
]
t−1
(
a2b2t
j ; t
)
i−j(
a2a
−1
1 ti+j−n; t
)
i−j
, (6.1a)
dR∗
j = dR
n−j =
(
a2a
−1
1 t−(n−j); t
)
j
(a1b2; t)n−j
(a2b1; t)j
(
a−12 a1t−j ; t
)
n−j
, (6.1b)
uR∗
ij = lRn−i,n−j = (−1)j−it−(j−i
2 )
[
j
i
]
t−1
(
a1b1t
n−j ; t
)
j−i(
a−12 a1tn−2j+1; t
)
j−i
, (6.1c)
and the inverse matrices U ′−1R =
(
uR ′∗
ij
)
0≤i,j≤n, D′−1R =
(
dR ′∗
j δij
)
0≤i,j≤n, L′−1R =
(
lR ′∗ij
)
0≤i,j≤n
are upper triangular, diagonal, lower triangular, respectively, given by
uR ′∗
ij = lR ′n−i,n−j =
[
j
i
]
t
(
a−11 b−11 t−(n−i−1); t
)
j−i(
b2b
−1
1 t−(n−2i−1); t
)
j−i
, (6.2a)
dR ′∗
j = dR ′
n−j =
(
b2b
−1
1 t−(n−2j−1); t
)
n−j
(
a−11 b−12 t−(j−1); t
)
j(
a−12 b−11 t−(n−j−1); t
)
n−j
(
b−12 b1tn−2j+1; t
)
j
, (6.2b)
lR ′∗ij = uR ′
n−i,n−j = (−1)i−jt(
i−j
2 )
[
n− j
n− i
]
t
(
a−12 b−12 t−(i−1); t
)
i−j(
b−12 b1tn−i−j ; t
)
i−j
. (6.2c)
Proof. Since R−1 = JR̄J , we have R−1 = U−1R D−1R L−1R , where L−1R = JURJ , D−1R = JDRJ
and U−1R = JLRJ . Thus we immediately have the expressions lR∗ij = uR
n−i,n−j , d
R∗
j = dR
n−j and
uR∗
ij = lRn−i,n−j . From Theorem 1.4 this gives the explicit forms (6.1a), (6.1b) and (6.1c). On
the other hand, we also have R−1 = L′−1R D′−1R U ′−1R , where U ′−1R = JL′RJ , D′−1R = JD′RJ and
L′−1R = JU ′RJ . Therefore uR ′∗
ij = lR ′n−i,n−j , d
R ′∗
j = dR ′
n−j , and lR ′∗ij = uR ′
n−i,n−j . Thus we obtain the
expressions (6.2a), (6.2b) and (6.2c). �
22 M. Ito
The rest of this section is devoted to the proof of Theorem 1.4. For this purpose we introduce
another set of symmetric polynomials different from Matsuo’s polynomials.
For 0 ≤ r ≤ n, let fr(a1, a2; t; z) be (symmetric) polynomials specified by
fr(a1, a2; t; z) :=
∑
I⊆{1,...,n}
|I|=r
r∏
k=1
zik − a2tik−k
a1tk−1 − a2tik−k
n−r∏
l=1
zjl − a1tjl−l
a2tl−1 − a1tjl−l
, (6.3)
where the summation is over all r-subsets I of {1, . . . , n}, and I = {i1 < · · · < ir}, J =
{1, . . . , n}\I = {j1 < · · · < jn−r}. In particular,
f0(a1, a2; t; z) =
n∏
i=1
zi − a1
a2ti−1 − a1
, fn(a1, a2; t; z) =
n∏
i=1
zi − a2
a1ti−1 − a2
.
We remark that the polynomials fr(a1, a2; t; z) are called the Lagrange interpolation polynomials
of type A and their properties are discussed in [19, Appendix B]. By definition the polynomial
fi(a1, a2; t; z) satisfies
fi(a1, a2; t; z) = fn−i(a2, a1; t; z). (6.4)
When we need to specify the number of variables z1, . . . , zn, we use the notation f
(n)
i (a1, a2; t; z)
= fi(a1, a2; t; z).
Lemma 6.2 (recurrence relation). The polynomials (6.3) satisfy the following recurrence rela-
tions:
f
(n)
i (a1, a2; t; z) =
zn − a2tn−i
a1ti−1 − a2tn−i
f
(n−1)
i−1 (a1, a2; t; ẑn) +
zn − a1ti
a2tn−i−1 − a1ti
f
(n−1)
i (a1, a2; t; ẑn)
for i = 0, 1, . . . , n, where ẑn = (z1, . . . , zn−1) ∈ (C∗)n−1.
Proof. The lemma follows from a direct computation and we omit the detail. �
For arbitrary x, y ∈ C∗ we define
ξj(x, y; t) :=
(
x, xt, . . . , xtj−1︸ ︷︷ ︸
j
, y, yt, . . . , ytn−j−1︸ ︷︷ ︸
n−j
)
∈ (C∗)n (6.5)
for j = 0, 1, . . . , n.
Proposition 6.3. The polynomial fi(a1, a2; t; z) is symmetric in the variables z = (z1, . . . , zn).
The leading term of fi(a1, a2; t; z) is m(1n)(z) up to a multiplicative constant. The functions
fi(a1, a2; t; z) (i = 0, 1, . . . , n) satisfy
fi(a1, a2; t; ξj(a1, a2; t)) = δij . (6.6)
Proof. See [19, Example 4.3 and equation (4.7)]. Otherwise, using Lemma 6.2 we can also
prove this proposition directly by induction on n. �
Remark 6.4. The set of symmetric polynomials {fi(a1, a2; t; z) | i = 0, 1, . . . , n} forms a basis
of the linear space spanned by {mλ(z) |λ ≤ (1n)}. Conversely such basis satisfying the con-
dition (6.6) is uniquely determined. Thus we can take Proposition 6.3 as a definition of the
polynomials fi(a1, a2; t; z), instead of (6.3).
q-Difference Systems for the Jackson Integral of Symmetric Selberg Type 23
Lemma 6.5 (triangularity). Suppose that
ξj(a1) :=
(
a1, a1t, . . . , a1t
j−1︸ ︷︷ ︸
j
, z1, z2, . . . , zn−j
)
∈ (C∗)n.
If i < j, then
fi(a1, a2; t; ξj(a1)) = 0. (6.7)
Moreover, fi(a1, a2; t; ξi(a1)) evaluates as
fi(a1, a2; t; ξi(a1)) =
n−i∏
l=1
zl − a1ti
a2tl−1 − a1ti
=
∏n−i
l=1
(
1− zla−11 t−i
)(
a2a
−1
1 t−i; t
)
n−i
. (6.8)
On the other hand, suppose that
ηj(a2) :=
(
z1, z2, . . . , zj , a2, a2t, . . . , a2t
n−j−1︸ ︷︷ ︸
n−j
)
∈ (C∗)n.
If i > j, then
fi(a1, a2; t; ηj(a2)) = 0. (6.9)
Moreover, fi(a1, a2; t; ηi(a2)) evaluates as
fi(a1, a2; t; ηi(a2)) =
i∏
l=1
zl − a2tn−i
a1tl−1 − a2tn−i
=
∏i
l=1
(
1− zla−12 t−(n−i)
)(
a1a
−1
2 t−(n−i); t
)
i
. (6.10)
Proof. First we show (6.9) by induction on n. For simplicity we write ηi(a2) as ηi. Suppose
i > j. Using Lemma 6.2 we have
f
(n)
i (a1, a2; t; ηj) =
a2t
n−j−1 − a2tn−i
a1ti−1 − a2tn−i
f
(n−1)
i−1
(
a1, a2; t; η
(n−1)
j
)
+
a2t
n−j−1 − a1ti
a2tn−i−1 − a1ti
f
(n−1)
i
(
a1, a2; t; η
(n−1)
j
)
,
where η
(n−1)
j =
(
z1, z2, . . . , zj , a2, a2t, . . . , a2t
n−j−2) ∈ (C∗)n−1. Since f
(n−1)
i
(
a1, a2; t; η
(n−1)
i
)
=
0 by the induction hypothesis, we have
f
(n)
i (a1, a2; t; ηj) =
a2t
n−j−1 − a2tn−i
a1ti−1 − a2tn−i
f
(n−1)
i−1
(
a1, a2; t; η
(n−1)
j
)
.
If i−1 > j, then f
(n−1)
i−1
(
a1, a2; t; η
(n−1)
j
)
= 0 by the induction hypothesis, while if i−1 = j, then
a2t
n−j−1 − a2tn−i = 0. In any case we obtain f
(n)
i (a1, a2; t; ηi) = 0, which is the claim of (6.9).
Next we show (6.10). If we put zl = a2t
n−i for l ∈ {1, . . . , i} in the polynomial fi(a1, a2; t;
ηi(a2)) of z1, . . . , zi, then we have fi(a1, a2; t; ηi(a2)) = 0 because fi(a1, a2; t; ηi(a2))|zk=a2tn−i
satisfies the condition of (6.9). This implies fi(a1, a2; t; ηi(a2)) is divisible by
∏
l=1
(
zl−a2tn−i
)
,
so that we have fi(a1, a2; t; ηi(a2)) = c
∏i
l=1
(
zl − a2tn−i
)
, where c is some constant. Thus
fi(a1, a2; t; ηi(a2))
∣∣∣
(z1,...,zi)=(a1,a1t,...,a1ti−1)
= c
i∏
l=1
(
a1t
l−1 − a2tn−i
)
.
24 M. Ito
On the other hand, (6.6) implies that
fi(a1, a2; t; ηi(a2))
∣∣∣
(z1,...,zi)=(a1,a1t,...,a1ti−1)
= fi(a1, a2; t; ξi(a1, a2; t)) = 1.
We therefore obtain c = 1/
∏i
l=1
(
a1t
l−1 − a2tn−i
)
, which implies (6.10).
Finally we show (6.7) and (6.8). From (6.4) we have
fi(a1, a2; t; ξj(a1)) = fn−i(a2, a1; t; ξj(a1)) = fn−i(a2, a1; t; ηn−j(a1)).
If i < j (i.e., n − i > n − j), then using (6.9) we see that the right-hand side of the above is
equal to zero. Moreover, using (6.10) we obtain
fi(a1, a2; t; ξi(a1)) = fn−i(a2, a1; t; ηn−i(a1)) =
n−i∏
l=1
zl − a1ti
a2tl−1 − a1ti
=
∏n−i
l=1
(
1− zla−11 t−i
)(
a2a
−1
1 t−i; t
)
n−i
,
which completes the proof. �
Corollary 6.6. Let ξj(x, a2; t) ∈ (C∗)n be the point specified by (6.5) with y = a2. Then
fi(a1, a2; t; ξj(x, a2; t)) evaluates as
fi(a1, a2; t; ξj(x, a2; t)) =
[
j
i
]
t−1
(
xa−11 ; t
)
j−i
(
xa−12 t−(n−j); t
)
i(
a−11 a2tn−j−i; t
)
j−i
(
a1a
−1
2 t−(n−i); t
)
i
. (6.11)
Proof. If i > j, then fi(a1, a2; t; ξj(x, a2; t)) = 0 is a special case of (6.9) in Lemma 6.5. Now
we assume that i ≤ j. If we put x = a2t
n−j−k (k = 0, 1, . . . , i − 1), then from (6.9) we have
fi(a1, a2; t; ξj(x, a2; t)) = 0. If we put x = a1t
−k (k = 0, 1, . . . , j − i− 1), then from (6.7) we also
have fi(a1, a2; t; ξj(x, a2; t)) = 0. This implies that fi(a1, a2; t; ξj(x, a2; t)) as a polynomial of x
is divisible by (xa1)j−i
(
xa−12 t−(n−j)
)
i
. Since the degree of fi(a1, a2; t; ξj(x, a2; t)) as a function
of x is equal to j, the function fi(a1, a2; t; ξj(x, a2; t)) can be expressed as
fi(a1, a2; t; ξj(x, a2; t)) = c
(
xa−11 ; t
)
j−i
(
xa−12 t−(n−j); t
)
i
,
where c is some constant. In order to determine the constant c, we put x = a2t
n−j−i in the
above equation. Then
fi(a1, a2; t; ξj(x, a2; t))
∣∣∣
x=a2tn−j−i
= c
(
a−11 a2t
n−j−i; t
)
j−i
(
t−i; t
)
i
,
while, from (6.10) we have
fi(a1, a2; t; ξj(x, a2; t))
∣∣∣
x=a2tn−j−i
= fi(a1, a2; t; ξi(x, a2; t))
∣∣∣
x=a2tn−j−i
=
(
t−j ; t
)
i(
a1a
−1
2 t−(n−i); t
)
i
.
The constant c can be explicitly computed as
c =
(
t−j ; t
)
i(
a−11 a2tn−j−i; t
)
j−i
(
a1a
−1
2 t−(n−i); t
)
i
(
t−i; t
)
i
=
(
t−1; t−1
)
j(
a−11 a2tn−j−i; t
)
j−i
(
a1a
−1
2 t−(n−i); t
)
i
(
t−1; t−1
)
j−i
(
t−1; t−1
)
i
.
We therefore obtain (6.11). �
q-Difference Systems for the Jackson Integral of Symmetric Selberg Type 25
Lemma 6.7. Suppose that ŨR is the (n+ 1)× (n+ 1) matrix satisfying(
en(a2, b1; z), en−1(a2, b1; z), . . . , e0(a2, b1; z)
)
=
(
fn(a1, a2; t; z), fn−1(a1, a2; t; z), . . . , f0(a1, a2; t; z)
)
ŨR. (6.12)
Then ŨR =
(
ũR
ij
)
0≤i,j≤n is an upper triangular matrix with entries given by
ũR
ij =
(
a1b1t
n−j ; t
)
j−i
(
a1a
−1
2 t−i; t
)
n−j(a2b1; t)i
(
t−1; t−1
)
n(
1− t−1
)n
[
j
i
]
t−1[
n
i
]
t−1
. (6.13)
Suppose that L̃R is the (n+ 1)× (n+ 1) matrix satisfying(
fn(a1, a2; t; z), fn−1(a1, a2; t; z), . . . , f0(a1, a2; t; z)
)
=
(
e0(a1, b2; z), e1(a1, b2; z), . . . , en(a1, b2; z)
)
L̃R. (6.14)
Then L̃R =
(
l̃Rij
)
0≤i,j≤n is a lower triangular matrix with entries given by
l̃Rij =
(−1)i−jt−(i−j
2 )(a2b2tj ; t)i−j(1− t−1)n(
a−11 a2t−(n−2j−1); t
)
i−j(a1b2; t)n−j
(
a−11 a2t−(n−j); t
)
j
(
t−1; t−1
)
n
[
n
i
]
t−1
[
i
j
]
t−1
. (6.15)
Proof. Since both {ei(a2, b1; z) | i = 0, 1, . . . , n} and {fi(a1, a2; t; z) | i = 0, 1, . . . , n} form bases
of the linear space spanned by {mλ(z) |λ < (1n)}, the polynomial ei(a2, b1; z) is expressed as
a linear combination of fi(a1, a2; t; z) (i = 0, 1, . . . , n), i.e.,
en−j(a2, b1; z) =
n∑
i=0
ũR
ijfn−i(a1, a2; t; z),
where ũR
ij are some constants. From the vanishing property (6.6), the coefficient ũR
ij is given by
ũR
ij = en−j(a2, b1; ξn−i(a1, a2; t)) = en−j(a2, b1; ζn−i(a2, y))
∣∣∣
y=a1tn−i−1
. (6.16)
From (3.15) in Lemma 3.5, en−j(a2, b1; ζn−i(a2, y)) evaluates as
en−j(a2, b1; ζn−i(a2, y))
=
(
yb1t
−(j−i−1); t
)
j−i
(
ya−12 t−(n−1); t
)
n−j(a2b1; t)i
(
t−1; t−1
)
n−i
(
t−1; t−1
)
j(
t−1; t−1
)
j−i
(
1− t−1
)n .
Combining this and (6.16), we obtain the expression (6.13).
Since {ei(a1, b2; z) | i = 0, 1, . . . , n} is also a basis of the linear space spanned by {mλ(z) |λ <
(1n)}, the polynomial fi(a1, a2; t; z) is expressed as a linear combination of ei(a1, b2; z) (i =
0, 1, . . . , n), i.e.,
fn−j(a1, a2; t; z) =
n∑
i=0
l̃Rijei(a1, b2; z),
where l̃Rij are some constants. From (3.4), the coefficient l̃Rij is written as
l̃Rij =
fn−j
(
a1, a2; t; ζi
(
a1, b
−1
2
))
ci
= c−1i fn−j(a1, a2; t; ξn−i(a1, x))
∣∣∣
x=b−1
2 t−(i−1)
, (6.17)
26 M. Ito
where ci is given explicitly in (3.5b) as
ci = (a1b2; t)n−i
(
a−11 b−12 t−(n−1); t
)
i
(
t−1; t−1
)
i
(
t−1; t−1
)
n−i(
1− t−1
)n . (6.18)
Using (6.11) in Corollary 6.6, we have
fn−j(a1, a2; t; ξn−i(a1, x)) = fj(a2, a1; t; ξn−i(a1, x)) = fj(a2, a1; t; ξi(x, a1))
=
[
i
j
]
t−1
(
xa−12 ; t
)
i−j
(
xa−11 t−(n−i); t
)
j(
a−12 a1tn−i−j ; t
)
i−j
(
a2a
−1
1 t−(n−j); t
)
j
.
Combining this, (6.17) and (6.18), we therefore obtain the expression (6.15). �
Lemma 6.8. Suppose that L̃′R is the (n+ 1)× (n+ 1) matrix satisfying(
en(a2, b1; z), en−1(a2, b1; z), . . . , e0(a2, b1; z)
)
=
(
fn
(
b−11 , b−12 ; t−1; z
)
, fn−1
(
b−11 , b−12 ; t−1; z
)
, . . . , f0
(
b−11 , b−12 ; t−1; z
))
L̃′R. (6.19)
Then L̃′R =
(
l̃R ′ij
)
0≤i,j≤n is a lower triangular matrix with entries given by
l̃R ′ij =
(
a−12 b−12 t−(i−1); t
)
i−j
(
a−12 b−11 t−(n−i−1); t
)
n−i
(
b1b
−1
2 tn−i−j+1; t
)
j
(t; t)n
t(
n
2)(1− t)n
[
i
j
]
t[
n
j
]
t
. (6.20)
Suppose that Ũ ′R is the (n+ 1)× (n+ 1) matrix satisfying(
fn
(
b−11 , b−12 ; t−1; z
)
, fn−1
(
b−11 , b−12 ; t−1; z
)
, . . . , f0
(
b−11 , b−12 ; t−1; z
))
=
(
e0(a1, b2; z), e1(a1, b2; z), . . . , en(a1, b2; z)
)
Ũ ′R, (6.21)
then Ũ ′R =
(
ũR ′
ij
)
0≤i,j≤n is an upper triangular matrix with entries given by
ũR ′
ij =
(−1)j−it(
j−i
2 )+(n2)
(
a−11 b−11 t−(n−i−1); t
)
j−i(1− t)
n(
b−11 b2t−(n−i−j); t
)
j−i
(
b−11 b2t−(n−2j−1); t
)
n−j
(
a−11 b−12 t−(j−1); t
)
j
(t; t)n
×
[
n
j
]
t
[
j
i
]
t
. (6.22)
Proof. Since both {ei(a2, b1; z) | i = 0, 1, . . . , n} and
{
fi
(
b−11 , b−12 ; t−1; z
)
| i = 0, 1, . . . , n
}
form
bases of the linear space spanned by {mλ(z) |λ < (1n)}, the polynomial ei(a2, b1; z) is expressed
as a linear combination of fi
(
b−11 , b−12 ; t−1; z
)
(i = 0, 1, . . . , n), i.e.,
en−j(a2, b1; z) =
n∑
i=0
l̃R ′ij fn−i
(
b−11 , b−12 ; t−1; z
)
,
where l̃R ′ij are some constants. From (6.6) we have
l̃R ′ij = en−j
(
a2, b1; ξn−i
(
b−11 , b−12 ; t−1
))
= en−j
(
a2, b1; ζn−i
(
x, b−11
))∣∣∣
x=b−1
2 t−(i−1)
. (6.23)
Using (3.10) in Lemma 3.4 we have
en−j
(
a2, b1; ζn−i
(
x, b−11
))
=
(
xb1t
n−j ; t
)
j
(
xa−12 ; t
)
i−j
(
a−12 b−11 t−(n−i−1); t
)
n−i
q-Difference Systems for the Jackson Integral of Symmetric Selberg Type 27
× (t; t)n−j(t; t)i
t(
n
2)(1− t)n(t; t)i−j
.
Combining this and (6.23), we obtain (6.20).
On the other hand, since {ei(a1, b2; z) | i = 0, 1, . . . , n} is also a basis of the linear space
spanned by {mλ(z) |λ < (1n)}, the polynomial fi
(
b−11 , b−12 ; t−1; z
)
is expressed as a linear com-
bination of ei(a1, b2; z) (i = 0, 1, . . . , n), i.e.,
fn−j
(
b−11 , b−12 ; t−1; z
)
=
n∑
i=0
ũR ′
ij ei(a1, b2; z),
where ũR ′
ij are some constants. From (3.4) we have
ũR ′
ij =
fn−j
(
b−11 , b−12 ; t−1; ζi
(
a1, b
−1
2
))
ci
=
fn−j
(
b−11 , b−12 ; t−1; ξn−i
(
x, b−12 ; t−1
))
ci
∣∣∣
x=a1tn−i−1
, (6.24)
where ci is the constant given in (3.5a) as
ci =
(
a1b2t
i; t
)
n−i
(
a−11 b−12 t−(i−1); t
)
i
(t; t)i(t; t)n−i
t(
n
2)(1− t)n
. (6.25)
From (6.11) in Corollary 6.6 we have
fn−j
(
b−11 , b−12 ; t−1; ξn−i
(
x, b−12 ; t−1
))
=
(
xb1; t
−1)
j−i
(
xb2t
i; t−1
)
n−j(t; t)n−i(
b1b
−1
2 t−(i+j−n); t−1
)
j−i
(
b−11 b2tj ; t−1
)
n−j(t; t)j−i(t; t)n−j
.
Combining this, (6.24) and (6.25), we therefore obtain the expression (6.22). �
Proof of Theorem 1.4. From (6.12) and (6.14) in Lemma 6.7, we have(
en(a2, b1; z), en−1(a2, b1; z), . . . , e0(a2, b1; z)
)
=
(
e0(a1, b2; z), e1(a1, b2; z), . . . , en(a1, b2; z)
)
L̃RŨR, (6.26)
where L̃R =
(
l̃Rij
)
0≤i,j≤n and ŨR =
(
ũR
ij
)
0≤i,j≤n are the matrices given by (6.13) and (6.15),
respectively. Comparing (6.26) with (1.14), we obtain R = L̃RŨR = LRDRUR, i.e.,
lRij =
l̃Rij
l̃Rjj
, dR
j = l̃Rjj ũ
R
jj , uR
ij =
ũR
ij
ũR
ii
.
Lemma 6.7 implies that lRij , d
R
j and uR
ij above coincide with (1.15a), (1.15b) and (1.15c), respec-
tively. On the other hand, from (6.19) and (6.21) in Lemma 6.8, we have(
en(a2, b1; z), en−1(a2, b1; z), . . . , e0(a2, b1; z)
)
=
(
e0(a1, b2; z), e1(a1, b2; z), . . . , en(a1, b2; z)
)
Ũ ′RL̃
′
R, (6.27)
where Ũ ′R =
(
ũR ′
ij
)
0≤i,j≤n and L̃′R =
(
l̃R ′ij
)
0≤i,j≤n are the matrices given by (6.20) and (6.22),
respectively. Comparing (6.27) with (1.14), we obtain R = Ũ ′RL̃
′
R = U ′RD
′
RL
′
R, i.e.,
uR ′
ij =
ũR ′
ij
ũR ′
jj
, dR ′
j = ũR ′
jj l̃
R ′
jj , lR ′ij =
l̃R ′ij
l̃R ′ii
.
Lemma 6.8 implies that uR ′
ij , dR ′
j and lR ′ij above coincide with (1.16a), (1.16b) and (1.16c),
respectively. �
28 M. Ito
A Appendix
In this appendix we consider the Gauss decomposition part A = U ′AD
′
AL
′
A of Theorem 1.7. Since
the method to compute A = U ′AD
′
AL
′
A is almost the same as that to compute A = LADAUA, we
only give the outline of the proof. For this purpose, we define another family of interpolation
polynomials Ẽ′k,i(a, b; z) slightly different from (3.2). Set
Ẽ′k,i(z) = Ẽ′k,i(a, b; z) := AE′k,i(a, b; z)/∆(z),
where
E′k,i(a, b; z) := zn−k+1zn−k+2 · · · zn︸ ︷︷ ︸
k
∆(t; z)
n−i∏
j=1
(1− bzj)
n∏
j=n−i+1
(
1− a−1zj
)
.
We now specify a = a1, b = b2, i.e., we set Ẽ′k,i(z) = Ẽ′k,i(a1, b2; z) throughout this section.
Lemma A.1 (three-term relations). Suppose k ≤ i. Then,
a−12
(
1− qαa1a2b1b2t2n−k−1
)〈
Ẽ′k,i
〉
= tk−1
(
1− qαa1b1t2n−k−i
)〈
Ẽ′k−1,i
〉
− qαtn−1
(
1− a1b1tn−i
)〈
Ẽ′k−1,i−1
〉
. (A.1)
On the other hand, if k ≥ i, then,
a−11
(
1− qαa1b1t2n−k−i
)〈
Ẽ′k,i−1
〉
= tk−i
(
1− qαtn−k
)〈
Ẽ′k−1,i−1
〉
− a−12 t−(i−1)
(
1− a2b2ti−1
)〈
Ẽ′k,i
〉
. (A.2)
Proof. Put ϕ̃′k,i−1(z) := A∇1ϕ′k,i−1(z), where
ϕ′k,i−1(z) :=
(
1− a−11 z1
)(
1− a−12 z1
) n∏
j=2
(z1 − tzj)× E(n−1)
k−1,i−1(z2, . . . , zn).
Then, by a similar argument to that used in the proof of Lemma 5.3, it follows that, if k ≤ i,
the polynomial ϕ̃′k,i−1(z) satisfies
ϕ̃′k,i−1(z)
∆(z)
= c′k,iẼ
′
k,i(z) + c′k−1,iẼ
′
k−1,i(z) + c′k−1,i−1Ẽ
′
k−1,i−1(z), (A.3)
where
c′k,i = −a−12 tn−1
(
1− qαa1a2b1b2t2n−k−1
)
,
c′k−1,i = tn+k−2
(
1− qαa1b1t2n−k−i
)
,
c′k−1,i−1 = −qαt2n−2
(
1− a1b1tn−i
)
,
while, if i ≤ k, then ϕ̃′k,i−1(z) satisfies
ϕ̃′k,i−1(z)
∆(z)
= d′k,iẼ
′
k,i(z) + d′k,i−1Ẽ
′
k,i−1(z) + d′k−1,i−1Ẽ
′
k−1,i−1(z), (A.4)
where
d′k,i = −a−12 tn−1
(
1− a2b2ti−1
)
,
d′k,i−1 = −a−11 tn+i−2
(
1− qαa1b1t2n−k−i
)
,
d′k−1,i−1 = tn+k−2
(
1− qαtn−k
)
.
Using the expressions (A.3) and (A.4) for ϕ̃k,i−1(z), (A.1) and (A.2) follow by application of
Lemma 5.1. �
q-Difference Systems for the Jackson Integral of Symmetric Selberg Type 29
By repeated use of the three-term relations (A.1) and (A.2), we obtain the following.
Lemma A.2. If k ≤ i, then
〈
Ẽ′k,i
〉
=
l∑
j=0
U ′k,ik−l,i−j
〈
Ẽ′k−l,i−j
〉
,
where
U ′k,ik−l,i−j =
(
−qαtn−k+l−1
)j(
a2t
k−l)lt(l−j
2 )
[
l
j
]
t
(
a1b1t
n−i; t
)
j
(
qαa1b1t
2n−k−i+j ; t
)
l−j(
qαa1a2b1b2t2n−k−1; t
)
l
,
while, if k ≥ i, then
〈
Ẽ′k,i
〉
=
l∑
j=0
L′k,ik−l+j,i+j
〈
Ẽ′k−l+j,i+j
〉
,
where
L′k,ik−l+j,i+j =
[
l
j
]
t
(
−a−12 t−(k−1)
)j(
a1t
k−i−1)lt−(l
2)
(
qαtn−k; t
)
l−j
(
a2b2t
i; t
)
j(
qαa1b1t2n−k−i−j−1; t
)
l−j
(
qαa1b1t2n−k−i−2j+l; t
)
j
.
As a special case of the above lemma we immediately have the following.
Lemma A.3. For 0 ≤ j ≤ n,
〈
Ẽ′j,j
〉
is expressed as
〈
Ẽ′j,j
〉
=
j∑
i=0
ũ′ij
〈
Ẽ′0,i
〉
, (A.5)
where
ũ′ij = U ′j,j0,i =
(
−qαtn−1
)j−i
aj2t
(i
2)
[
j
i
]
t
(
a1b1t
n−j ; t
)
j−i
(
qαa1b1t
2n−i−j ; t
)
i(
qαa1a2b1b2t2n−j−1; t
)
j
, (A.6)
while, for 0 ≤ j ≤ n,
〈
Ẽ′n,j
〉
is expressed as
〈
Ẽ′n,j
〉
=
n∑
i=j
l̃′ij
〈
Ẽ′i,i
〉
, (A.7)
where
l̃′ij = L′n,ji,i = (−1)i−j
[
n− j
n− i
]
t
an−j1 a
−(i−j)
2 t(
n−i
2 )+(j2)−(i
2)
(
qα; t
)
n−i
(
a2b2t
j ; t
)
i−j(
qαa1b1tn−i−1; t
)
n−i
(
qαa1b1t2(n−i); t
)
i−j
. (A.8)
Proof of (1.19a)–(1.19c) in Theorem 1.7. From (A.5), we have(〈
Ẽ′0,0
〉
,
〈
Ẽ′1,1
〉
, . . . ,
〈
Ẽ′n−1,n−1
〉
,
〈
Ẽ′n,n
〉)
=
(〈
Ẽ′0,0
〉
,
〈
Ẽ′0,1
〉
, . . . ,
〈
Ẽ′0,n−1
〉
,
〈
Ẽ′0,n
〉)
Ũ ′,
where the matrix Ũ ′ =
(
ũ′ij
)
0≤i,j≤n is defined by (A.6). Moreover, from (A.7) we have(〈
Ẽ′n,0
〉
,
〈
Ẽ′n,1
〉
, . . . ,
〈
Ẽ′n,n−1
〉
,
〈
Ẽ′n,n
〉)
=
(〈
Ẽ′0,0
〉
,
〈
Ẽ′1,1
〉
, . . . ,
〈
Ẽ′n−1,n−1
〉
,
〈
Ẽ′n,n
〉)
L̃′
=
(〈
Ẽ′0,0
〉
,
〈
Ẽ′0,1
〉
, . . . ,
〈
Ẽ′0,n−1
〉
,
〈
Ẽ′0,n
〉)
Ũ ′L̃′, (A.9)
30 M. Ito
where the matrix L̃′ =
(
l̃′ij
)
0≤i,j≤n is defined by (A.8). Since TαΦ(z) = z1z2 · · · znΦ(z) and
z1z2 · · · znẼ′0,i(z) = Ẽ′n,i(z), we have Tα
〈
Ẽ′0,i
〉
=
〈
Ẽ′n,i
〉
, i.e.,
Tα
(〈
Ẽ′0,0
〉
,
〈
Ẽ′0,1
〉
, . . . ,
〈
Ẽ′0,n−1
〉
,
〈
Ẽ′0,n
〉)
=
(〈
Ẽ′n,0
〉
,
〈
Ẽ′n,1
〉
, . . . ,
〈
Ẽ′n,n−1
〉
,
〈
Ẽ′n,n
〉)
. (A.10)
From (A.9) and (A.10), we obtain the difference system
Tα
(〈
Ẽ′0,0
〉
,
〈
Ẽ′0,1
〉
, . . . ,
〈
Ẽ′0,n−1
〉
,
〈
Ẽ′0,n
〉)
=
(〈
Ẽ′0,0
〉
,
〈
Ẽ′0,1
〉
, . . . ,
〈
Ẽ′0,n−1
〉
,
〈
Ẽ′0,n
〉)
Ũ ′L̃′.
Comparing this with (1.17), we therefore obtain A = Ũ ′L̃′ = U ′AD
′
AL
′
A, i.e.,
uA ′
ij =
ũ′ij
ũ′jj
, dA ′
j = ũ′jj l̃
′
jj , lA ′ij =
l̃′ij
l̃′ii
.
Lemma A.3 implies that uA ′
ij , dA ′
j and lA ′ij above coincide with (1.19a), (1.19b) and (1.19c),
respectively, which completes the proof. �
Finally we give an explicit forms for L′A
−1.
Proposition A.4. The inverse matrix L′−1A =
(
lA ′∗ij
)
0≤i,j≤n is lower triangular and is written
as
lA ′∗ij =
[
n− j
n− i
]
t
(
a1a
−1
2 t−j
)i−j(
a2b2t
j ; t
)
i−j(
qαa1b1t2n−i−j−1; t
)
i−j
.
Proof. Using (A.2) we can calculate the entries lA ′∗ij of the lower triangular matrix L′−1A by
completely the same way as Proposition 4.4. We omit the details. �
Acknowledgements
The author would like to thank the anonymous referees who kindly provided many careful
comments and suggestions for improving his manuscript. He would also like to express his
gratitude to Professor Peter J. Forrester for providing considerable encouragement from the
early stage of this research. The comments and suggestions by Professor Yasuhiko Yamada on
the preliminary version of this manuscript are most appreciated. This work was supported by
JSPS KAKENHI Grant Number (C)18K03339.
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1 Introduction
2 Notation
3 Interpolation polynomials
4 Three-term relations
5 Proof of Lemma 4.1
6 The transition matrix R
A Appendix
References
|
| id | nasplib_isofts_kiev_ua-123456789-211007 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-16T07:30:58Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Ito, Masahiko 2025-12-22T09:25:02Z 2020 q-Difference Systems for the Jackson Integral of Symmetric Selberg Type. Masahiko Ito. SIGMA 16 (2020), 113, 31 pages 1815-0659 2020 Mathematics Subject Classification: 33D60; 39A13 arXiv:1910.08393 https://nasplib.isofts.kiev.ua/handle/123456789/211007 https://doi.org/10.3842/SIGMA.2020.113 We provide an explicit expression for the first-order -difference system for the Jackson integral of symmetric Selberg type. The q-difference system gives a generalization of the -analog of contiguous relations for the Gauss hypergeometric function. As a basis of the system, we use a set of symmetric polynomials introduced by Matsuo in his study of the -KZ equation. Our main result is an explicit expression for the coefficient matrix of the -difference system in terms of its Gauss matrix decomposition. We introduce a class of symmetric polynomials called interpolation polynomials, which includes Matsuo's polynomials. By repeated use of three-term relations among the interpolation polynomials, we compute the coefficient matrix. The author would like to thank the anonymous referees who kindly provided many careful comments and suggestions for improving his manuscript. He would also like to express his gratitude to Professor Peter J. Forrester for providing considerable encouragement from the early stage of this research. The comments and suggestions by Professor Yasuhiko Yamada on the preliminary version of this manuscript are most appreciated. This work was supported by JSPS KAKENHI Grant Number (C)18K03339. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications q-Difference Systems for the Jackson Integral of Symmetric Selberg Type Article published earlier |
| spellingShingle | q-Difference Systems for the Jackson Integral of Symmetric Selberg Type Ito, Masahiko |
| title | q-Difference Systems for the Jackson Integral of Symmetric Selberg Type |
| title_full | q-Difference Systems for the Jackson Integral of Symmetric Selberg Type |
| title_fullStr | q-Difference Systems for the Jackson Integral of Symmetric Selberg Type |
| title_full_unstemmed | q-Difference Systems for the Jackson Integral of Symmetric Selberg Type |
| title_short | q-Difference Systems for the Jackson Integral of Symmetric Selberg Type |
| title_sort | q-difference systems for the jackson integral of symmetric selberg type |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211007 |
| work_keys_str_mv | AT itomasahiko qdifferencesystemsforthejacksonintegralofsymmetricselbergtype |