sigma09-054. 456KB Jun 04 2011 12:10:27 AM

Symmetry, Integrability and Geometry: Methods and Applications

SIGMA 5 (2009), 054, 40 pages

Kernel Functions for Dif ference Operators
of Ruijsenaars Type and Their Applications⋆
Yasushi KOMORI † , Masatoshi NOUMI

‡

§

and Jun’ichi SHIRAISHI

†

Graduate School of Mathematics, Nagoya University, Chikusa-Ku, Nagoya 464-8602, Japan
E-mail: komori@math.nagoya-u.ac.jp

‡

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

§

Graduate School of Mathematical Sciences, University of Tokyo,
Komaba, Tokyo 153-8914, Japan
E-mail: shiraish@ms.u-tokyo.ac.jp

Received December 01, 2008, in final form April 30, 2009; Published online May 12, 2009
doi:10.3842/SIGMA.2009.054
Abstract. A unified approach is given to kernel functions which intertwine Ruijsenaars
difference operators of type A and of type BC. As an application of the trigonometric
cases, new explicit formulas for Koornwinder polynomials attached to single columns and
single rows are derived.
Key words: kernel function; Ruijsenaars operator; Koornwinder polynomial
2000 Mathematics Subject Classification: 81R12; 33D67

Contents
1 Introduction

2

2 Kernel functions for Ruijsenaars operators
2.1 Variations of the gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Kernel functions of type A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Kernel functions of type BC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3
3
5
6

3 Derivation of kernel functions
3.1 Key identities . . . . . . . . . . . . . . . .
3.2 Case of type A . . . . . . . . . . . . . . .
3.3 Case of type BC . . . . . . . . . . . . . .
3.4 Difference operators of Koornwinder type

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4 Kernel functions for q-dif ference operators
4.1 Kernel functions for Macdonald operators . . . . . . . . . . . . . . . . . . . . . .
4.2 Kernel functions for Koornwinder operators . . . . . . . . . . . . . . . . . . . . .

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⋆
This paper is a contribution to the Proceedings of the Workshop “Elliptic Integrable Systems, Isomonodromy
Problems, and Hypergeometric Functions” (July 21–25, 2008, MPIM, Bonn, Germany). The full collection is
available at http://www.emis.de/journals/SIGMA/Elliptic-Integrable-Systems.html

2

Y. Komori, M. Noumi and J. Shiraishi

5 Application to Koornwinder polynomials
5.1 Koornwinder polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Case of a single column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Case of a single row . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22
23
25

27

A Remarks on higher order dif ference operators

32

B Comparison with [23]
B.1 Case of type A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.2 Case of type BC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33
34
34

C Er (z; a|t) and Hl (z; a|q, t) as interpolation polynomials

38

References

40

1

Introduction

Let Φ(x; y) be a meromorphic function on Cm × Cn in the variables x = (x1 , . . . , xm ) and
y = (y1 , . . . , yn ), and consider two operators Ax , By which act on meromorphic functions in x
and y, respectively. We say that Φ(x; y) is a kernel function for the pair (Ax , By ) if it satisfies
a functional equation of the form
Ax Φ(x; y) = By Φ(x; y).
In the theory of Jack and Macdonald polynomials [14], certain explicit kernel functions play
crucial roles in eigenfunction expansions and integral representations. Recently, kernel functions
in this sense have been studied systematically by Langmann [12, 13] in the analysis of eigenfunctions for the elliptic quantum integrable systems of Calogero–Moser type, and by Ruijsenaars
[21, 22, 23] for the relativistic elliptic quantum integrable systems of Ruijsenaars–Schneider type.
In this paper we investigate two kinds of kernel functions, of Cauchy type and of dual Cauchy
type, which intertwine pairs of Ruijsenaars difference operators. In the cases of elliptic difference
operators, kernel functions of Cauchy type for the (An−1 , An−1 ) and (BCn , BCn ) cases were
found by Ruijsenaars [22, 23]. Extending his result, we present kernel functions of Cauchy type,
as well as those of dual Cauchy type, for the (BCm , BCn ) cases with arbitrary m, n (under certain
balancing conditions on the parameters in the elliptic cases). For the trigonometric difference
operators of type A, kernel functions both of Cauchy and dual Cauchy types were already
discussed by Macdonald [14]. Kernel functions of dual Cauchy type for the trigonometric BCn
cases are due to Mimachi [15]. In this paper we develop a unified approach to kernel functions
for Ruijsenaars operators of type A and of type BC, with rational, trigonometric and elliptic
coefficients, so as to cover all these known examples in the difference cases. We expect that our
framework could be effectively applied to the study of eigenfunctions for difference operators of
Ruijsenaars type.
As such an application of kernel functions in the trigonometric BC cases, we derive new
explicit formulas for Koornwinder polynomials attached to single columns and single rows. This
provides with a direct construction of those special cases of the binomial expansion formula for
the Koornwinder polynomials as studied by Okounkov [18] and Rains [19]. We also remark that,
regarding explicit formulas for Macdonald polynomials attached to single rows of type B, C, D,
some conjectures have been proposed by Lassalle [11]. The relationship between his conjectures
and our kernel functions will be discussed in a separate paper.
Our main results on the kernel functions for Ruijsenaars difference operators of type A and
of type BC will be formulated in Section 2. In Section 3 we give a unified proof for them on the

Kernel Functions for Difference Operators of Ruijsenaars Type

3

basis of two key identities. After giving remarks in Section 4 on the passage to the q-difference
operators of Macdonald and Koornwinder, as an application of our approach we present in
Section 5 explicit formulas for Koornwinder polynomials attached to single columns and single
rows. We also include three sections in Appendix. We will give some remarks in Appendix A
on higher order difference operators, and in Appendix B make an explicit comparison of our
kernel functions in the elliptic cases with those constructed by Ruijsenaars [23]. In Appendix C,
we will give a proof of the fact that certain Laurent polynomials, which appear in our explicit
formulas for Koornwinder polynomials attached to single columns and single rows, are special
cases of the BCm interpolation polynomials of Okounkov [18].

2
2.1

Kernel functions for Ruijsenaars operators
Variations of the gamma function

In order to specify the class of Ruijsenaars operators which we shall discuss below, by the
symbol [u] we denote a nonzero entire function in one variable u, satisfying the following Riemann
relation:
[x + u][x − u][y + v][y − v] − [x + v][x − v][y + u][y − u]
= [x + y][x − y][u + v][u − v]

(2.1)

for any x, y, u, v ∈ C. Under this condition, it is known that [u] must be an odd function
([−u] = −[u], and hence [0] = 0), and the set Ω of all zeros of [u] forms an additive subgroup
of C. Furthermore such functions are classified into the three categories, rational, trigonometric
and elliptic, according to the rank of the additive subgroup Ω ⊂ C of all zeros of [u]. In fact,
up to multiplication by nonzero constants, [u] coincides with one of the following three types of
functions:


(0) rational case:
e au2 u
(Ω = 0),


(1) trigonometric case:
e au2 sin(πu/ω1 )
(Ω = Zω1 ),


2
(2) elliptic case:
e au σ(u; Ω)
(Ω = Zω1 ⊕ Zω2 ),
√
where a ∈ C and e(u) = exp(2π −1u). Also, σ(x; Ω) denotes the Weierstrass sigma function
σ(u; Ω) = u

Y

ω∈Ω; ω6=0

1−

u
ω




u

u2

e ω + 2ω2

associated with the period lattice Ω = Zω1 ⊕ Zω2 , generated by ω1 , ω2 which are linearly
independent over R.
We start with some remarks on gamma functions associated with the function [u]. Fixing
a nonzero scaling constant δ ∈ C, suppose that a nonzero meromorphic function G(u|δ) on C
satisfies the difference equation
G(u + δ|δ) = [u]G(u|δ)

(u ∈ C).

(2.2)

Such a function G(u|δ), determined up to multiplication by δ-periodic functions, will be called
a gamma function for [u]. We give typical examples of gamma functions in this sense for the
rational, trigonometric and elliptic cases.
(0) Rational case: For any δ ∈ C (δ 6= 0), the meromorphic function
G(u|δ) = δ u/δ Γ(u/δ),
defined with the Euler gamma function Γ(u), is a gamma function for [u] = u.

4

Y. Komori, M. Noumi and J. Shiraishi

(1) Trigonometric case: We set z=e(u/ω1 ) and q=e(δ/ω1 ), and suppose that Im(δ/ω1 )>0
so that |q| < 1. We now consider the function
√
1
1
[u] = z 2 − z − 2 = 2 −1 sin(πu/ω1 ).
For this [u] two meromorphic functions



e − 2ωδ 1 u/δ
2
G− (u|δ) =
,
G+ (u|δ) = e
(z; q)∞
where (z; q)∞ =

∞
Q

i=0

δ u/δ
2ω1 2





(q/z; q)∞ ,

(2.3)

(1 − q i z), satisfy the difference equations

G− (u + δ|δ) = −[u]G− (u|δ),

G+ (u + δ|δ) = [u]G+ (u|δ),

respectively.
Namely, for ǫ = ±, Gǫ (u|δ)

is au
gamma function for ǫ[u]. (Note that the quadratic
function u2 = 21 u(u − 1) satisfies u+1
= 2 + u.) Another set of gamma functions for ǫ[u] of
2
this case is given by
1
1
1

1
1
1

q 2 z− 2 ; q 2 ∞
− q 2 z− 2 ; q 2 ∞
G+ (u|δ) =
,
G− (u|δ) =
1

1

1
1
− z2;q2 ∞
z2;q2 ∞
1

by means of the infinite products with base q 2 .
(2) Elliptic case: Let p, q be nonzero complex numbers with |p| < 1, |q| < 1. Then the
Ruijsenaars elliptic gamma function
(pq/z; p, q)∞
Γ(z; p, q) =
,
(z; p, q)∞

(z; p, q)∞ =

∞
Y

i,j=0

1 − pi q j z

satisfies the q-difference equation
Γ(qz; p, q) = θ(z; p)Γ(z; p, q),




θ(z; p) = (z; p)∞ (p/z; p)∞ .

Note also Γ(pq/z; p, q) = Γ(z; p, q)−1 . We set p = e(ω2 /ω1 ), q = e(δ/ω1 ) and z = e(u/ω1 ), and
suppose that Im(ω2 /ω1 ) > 0, Im(δ/ω1 ) > 0 so that |p| < 1, |q| < 1. Instead of θ(z; p) above, we
consider the function
√
1
[u] = −z − 2 θ(z; p) = 2 −1 sin(πu/ω1 )(pz; p)∞ (p/z; p)∞
which is a constant multiple of the odd Jacobi theta function with modulus ω2 /ω1 , so that [u]
should satisfy the Riemann relation. Then the two meromorphic functions
G− (u|δ) = e −
G+ (u|δ) = e

δ u/δ
2ω1 2

δ u/δ
2ω1 2









Γ(z; p, q) = e −

Γ(pz; p, q) = e

satisfy the difference equations
G− (u + δ|δ) = −[u]G− (u|δ),

δ
2ω1



(pq/z; p, q)∞
,
(z; p, q)∞



u/δ (q/z; p, q)∞
.
2
(pz; p, q)∞
δ u/δ
2ω1 2

(2.4)

G+ (u + δ|δ) = [u]G+ (u|δ),

respectively. Another set of gamma functions for ǫ[u] is given by
1

1
1

1
1

1 1
1
1
Gǫ (u|δ) = θ ǫz 2 ; q 2 Γ(z; p, q) = Γ ǫp 2 z 2 ; p 2 , q 2 Γ − ǫz 2 ; p 2 , q 2

(ǫ = ±).

In the limit as p → 0, these examples recover the previous ones in the trigonometric case.
We remark that, if G(u|δ) is a gamma function for [u], then G(δ − u|δ)−1 is a gamma function
for −[u]. Also, when we transform [u] to [u]′ = ce(au2 )[u], a gamma function for [u]′ is obtained
1 3
for instance as G′ (u|δ) = cu/δ e(ap(u))G(u|δ), where p(u) = 3δ
u − 21 u2 + 6δ u.

Kernel Functions for Difference Operators of Ruijsenaars Type

2.2

5

Kernel functions of type A

The elliptic difference operator which we will discuss below was introduced by Ruijsenaars [20]
together with the commuting family of higher order difference operators. In order to deal with
rational, trigonometric and elliptic cases in a unified manner, we formulate our results for this
class of (first order) difference operators in terms of an arbitrary function [u] satisfying the
Riemann relation. (As to the commuting family of higher order difference operators, we will
give some remarks later in Appendix A.)
Fix a nonzero entire function [u] satisfying the Riemann relation (2.1). For type A, we
consider the difference operator
Dx(δ,κ)

=

m
X

Ai (x; κ)Txδi ,

Ai (x; κ) =

i=1

Y

1≤j≤m; j6=i

[xi − xj + κ]
[xi − xj ]

in m variables x = (x1 , . . . , xm ), where δ, κ ∈ C are complex parameters ∈
/ Ω, and Txδi stands for
the δ-shift operator
Txδi f (x1 , . . . , xm ) = f (x1 , . . . , xi + δ, . . . , xm )

(i = 1, . . . , m).

Note that this operator remains invariant if one replaces the function [u] with its multiple by
any nonzero constant.
By taking a gamma function G(u|δ) for (any constant multiple of) [u] as in (2.2), we define
a function ΦA (x; y|δ, κ) by
ΦA (x; y|δ, κ) =

m Y
n
Y
G(xj + yl + v − κ|δ)
G(xj + yl + v|δ)

(2.5)

j=1 l=1

with an extra parameter v. We also consider the function
m Y
n
Y
ΨA (x; y) =
[xj − yl + v].

(2.6)

j=1 l=1

These two functions ΦA (x; y|δ, κ) and ΨA (x; y) are kernel functions of Cauchy type and of dual
Cauchy type for this case, respectively.
Theorem 2.1. Let [u] be any nonzero entire function satisfying the Riemann relation.
(1) If m = n, then the function ΦA (x; y|δ, κ) defined as (2.5) satisfies the functional equation
Dx(δ,κ) ΦA (x; y|δ, κ) = Dy(δ,κ) ΦA (x; y|δ, κ).
(2) Under the balancing condition mκ+nδ = 0, the function ΨA (x; y) defined as (2.6) satisfies
the functional equation
[κ]Dx(δ,κ) ΨA (x; y) + [δ]Dy(κ,δ) ΨA (x; y) = 0.
Statement (1) of Theorem 2.1 is due to Ruijsenaars [22, 23]. (See Appendix B.1 for an
explicit comparison between ΦA (x; y) and Ruijsenaars’ kernel function of [23].) In the scope of
the present paper, the balancing conditions (m = n in (1), and mκ + nδ = 0 in (2)) seem to be
essential in the elliptic cases. In the context of elliptic differential operators of Calogero–Moser
type, however, Langmann [13] has found a natural generalization of the kernel identities of
Cauchy type, which include the differentiation with respect to the elliptic modulus, to arbitrary
pair (m, n). It would be a intriguing problem to find a generalization of this direction for elliptic
difference operators of Ruijsenaars type.
In trigonometric and rational cases, these functions ΦA (x; y|δ, κ) and ΨA (x; y) satisfy more
general functional equations without balancing conditions.

6

Y. Komori, M. Noumi and J. Shiraishi

Theorem 2.2. Suppose that [u] is a constant multiple of sin(πu/ω1 ) or u.
(1) For arbitrary m and n, the function ΦA (x; y|δ, κ) satisfies the functional equation
Dx(δ,κ) ΦA (x; y|δ, κ) − Dy(δ,κ) ΦA (x; y|δ, κ) =

[(m − n)κ]
ΦA (x; y|δ, κ).
[κ]

(2) The function ΨA (x; y) satisfies the functional equation
[κ]Dx(δ,κ) ΨA (x; y) + [δ]Dy(κ,δ) ΨA (x; y) = [mκ + nδ]ΨA (x; y).
These results for the trigonometric (and rational) cases are essentially contained in the discussion of Macdonald [14].
A unified proof of Theorems 2.1 and 2.2 will be given in Section 3. We will also explain in
Section 4 how Theorem 2.2 is related with the theory of Macdonald polynomials.

2.3

Kernel functions of type BC

The (first-order) elliptic difference operator of type BC was first proposed by van Diejen [2]. It
is also known by Komori–Hikami [9] that it admits a commuting family of higher order difference
operators. In the following we use the expression of the first-order difference operator due to [9],
with modification in terms of [u]. (In Appendix B, we will give some remarks on the comparison
of our difference operator with other expressions in the literature.)
For type BC, we consider difference operators of the form
Ex(µ|δ,κ)

=

m
X

δ
A+
i (x; µ|δ, κ)Txi

i=1

+

m
X

0
−δ
A−
i (x; µ|δ, κ)Txi + A (x; µ|δ, κ)

(2.7)

i=1

including 2ρ parameters µ = (µ1 , . . . , µ2ρ ) besides (δ, κ), where ρ = 1, 2 or 4 according as
rank Ω = 0, 1, or 2. In the trigonometric and rational cases of type BC, we assume that the
function [u] does not contain exponential factors. Namely, we assume that [u] is a constant
multiple of one of the functions


u (ρ = 1),
sin(πu/ω1 ) (ρ = 2),
e au2 σ(u; Ω) (ρ = 4).
(2.8)
In each case we define ω1 , . . . , ωρ ∈ Ω as
(0) rational case:

Ω = 0,

ω1 = 0,

(1) trigonometric case:

Ω = Zω1 ,

ω2 = 0,

(2) elliptic case:

Ω = Zω1 ⊕ Zω2 ,

ω3 = −ω1 − ω2 ,

ω4 = 0.

Then the quasi-periodicity of [u] is described as



[u + ωr ] = ǫr e ηr u + 21 ωr [u]
(r = 1, . . . , ρ)

for some ηr ∈ C and ǫr = ±1. In the trigonometric and rational cases (without exponential
factors), one can simply take ηr = 0 (r = 1, . . . , ρ). Note also that [u] admits the duplication
formula of the form
ρ−1
Y

[u − 21 ωs ]
,
[2u] = 2[u]
[− 21 ωs ]
s=1

ρ
[2u + c] Y [u + 21 (c − ωs )]
=
.
[2u]
[u − 12 ωs ]
s=1

(2.9)

Kernel Functions for Difference Operators of Ruijsenaars Type

7

(In the trigonometric and rational cases, these formulas fail for [u] containing nontrivial exponential factors e(au2 ).) In relation to the parameters µ = (µ1 , . . . , µ2ρ ), we introduce
(µ|δ,κ)

c

=

2ρ
X
s=1

µs −

ρ
2 (δ

+ κ) +

ρ
X

ωs .

s=1

This parameter is related to quasi-periodicity of the coefficients of the Ruijsenaars operator,
ρ
P
and plays a crucial role in various places of our argument. Note that the last term
ωs is
s=1

nontrivial only in the trigonometric case: it is ω1 when ρ = 2, and 0 when ρ = 1, 4.
(µ|δ,κ)
With these data, we define the coefficients of the Ruijsenaars operator Ex
of type BC
as follows:
2ρ
Q
[xi + µs ]
Y
[xi ± xj + κ]
s=1
+
Ai (x; µ|δ, κ) = ρ
,
Q
[xi ± xj ]
1
1
1≤j≤m;
j6
=
i
[xi − 2 ωs ][xi + 2 (δ − ωs )]
A−
i (x; µ|δ, κ)

=

s=1
A+
i (−x; µ|δ, κ)

(i = 1, . . . , m),

(2.10)

with an abbreviated notation [x ± y] = [x + y][x − y] of products, and
A0 (x; µ|δ, κ) =

ρ
X

A0r (x; µ|δ, κ),

r=1

A0r (x; µ|δ, κ) =

e(−(mκ + 12 c(µ|δ,κ) )ηr )
[κ]

Q

s6=r
(µ|δ,κ)

[ 12 (ωr − ωs )]

ρ
Q

s=1

2ρ
Q

s=1

[ 12 (ωr − δ) + µs ] Y
m

[ 21 (ωr − ωs + κ − δ)]

[ 21 (ωr − δ) ± xj + κ]
. (2.11)
[ 21 (ωr − δ) ± xj ]
j=1

This operator Ex
is one of the expressions for the first-order Ruijsenaars operator of type
(µ|δ,κ)
BCm due to Komori–Hikami [9, (4.21)]. We remark that this difference operator Ex
has
symmetry
Ex(µ|−δ,−κ) = Ex(−µ|δ,κ)

(2.12)

with respect to the sign change of parameters. (This property can be verified directly by using
ρ
Q
ǫs = 1, −1, −1 according as ρ = 1, 2, 4.) Note
the quasi-periodicity of [u] and the fact that
s=1

also that it remains invariant if one replace the function [u] with its multiple by a nonzero
(µ|δ,κ)
in the
constant. By the duplication formula (2.9), one can also rewrite the operator Ex
form
2ρ
Q
ρ−1
m
Y
Y
X s=1[xi + µs ]
[xi ± xj + κ] δ
2 (µ|δ,κ)
1
1
T xi
ω
]
E
=
[
s
x
4
2
[2xi ][2xi + δ]
[xi ± xj ]
s=1

i=1

+

m
X
i=1

+

ρ
X

1≤j≤m; j6=i

2ρ
Q

[−xi + µs ]

s=1

[−2xi ][−2xi + δ]
2ρ
Q

Kr s=1

1≤j≤m; j6=i

[ 12 (ωr − δ) + µs ] Y
m

2[κ][κ − δ]


where Kr = e − (ωr + (m + 1)κ − δ + 12 c(µ|δ,κ) )ηr .
r=1

Y

[−xi ± xj + κ] −δ
T xi
[−xi ± xj ]

[ 21 (ωr − δ) ± xj + κ]
,
[ 21 (ωr − δ) ± xj ]
j=1

(2.13)

8

Y. Komori, M. Noumi and J. Shiraishi

By using any gamma function G(u|δ) for (a constant multiple of) [u], we define a function
ΦBC (x; y|δ, κ) of Cauchy type, either by
ΦBC (x; y|δ, κ) =

m Y
n
Y
G(xj ± yl + 21 (δ − κ)|δ)
,
1
(δ
+
κ)|δ)
G(x
±
y
+
j
l
2
j=1 l=1

(2.14)

ΦBC (x; y|δ, κ) =

m Y
n
Y
Y

(2.15)

or by

j=1 l=1 ǫ1 ,ǫ2 =±

G(ǫ1 xj + ǫ2 yl + 21 (δ − κ)|δ).

Note that the function (2.15) differs from (2.14), only by a multiplicative factor which is δperiodic in all the variables xj (j = 1, . . . , m) and yl (l = 1, . . . , n). We also introduce the
function
ΨBC (x; y) =

m Y
n
Y
[xj ± yl ]

(2.16)

j=1 l=1

of dual Cauchy type.
Theorem 2.3. Suppose that [u] is a constant multiple of any function in (2.8).
(1) Under the balancing condition
2(m − n)κ + c(µ|δ,κ) = 0,
the function ΦBC (x; y|δ, κ) defined as (2.14) or (2.15) satisfies the functional equation
Ex(µ|δ,κ) ΦBC (x; y|δ, κ) = Ey(ν|δ,κ) ΦBC (x; y|δ, κ),
where the parameters ν = (ν1 , . . . , ν2ρ ) for the y variables are defined by
νr = 12 (δ + κ) − µr

(r = 1, . . . , 2ρ).

(2) Under the balancing condition
2mκ + 2nδ + c(µ|δ,κ) = 0,
the function ΨBC (x; y) defined as (2.16) satisfies the functional equation
[κ]Ex(µ|δ,κ) ΨBC (x; y) + [δ]Ey(µ|κ,δ) ΨBC (x; y) = 0.
Statement (1) for the cases m = n is due to Ruijsenaars [21, 23]. An explicit comparison
will be made in Appendix B.2 between our ΦBC (x; y|δ, κ) and Ruijsenaars’ kernel function of
type BC in [23].
(µ|δ,κ)
In the case of 0 variables, the Ruijsenaars operator Ex
with m = 0 reduces to the
multiplication operator by the constant

C (µ|δ,κ) =

ρ
X

r=1 [κ]

e(− 21 c(µ|δ,κ) ηr )
Q

s6=r

[ 21 (ωr

− ωs )]

2ρ
Q

s=1
ρ
Q

s=1

[ 21 (ωr − δ) + µs ]

[ 12 (ωr

− ωs + κ − δ)]

.

Kernel Functions for Difference Operators of Ruijsenaars Type

9

We remark that Theorem 2.3, (2) for n = 0 implies that, when 2mκ + c(µ|δ,κ) = 0, the constant
(µ|δ,κ)
function 1 is an eigenfunction for Ex
:
[κ]Ex(µ|δ,κ) (1) = −[δ]C (µ|κ,δ) .
(Statement (1) for n = 0 gives the same formula since C (ν|δ,κ) = −[δ]C (µ|κ,δ) /[κ].)
In the trigonometric and rational cases, these functions ΦBC (x; y|δ, κ) and ΨBC (x; y) satisfy
the following functional equations without balancing conditions.
Theorem 2.4. Suppose that [u] is a constant multiple of sin(πu/ω1 ) (ρ = 2) or u (ρ = 1).
(1) The function ΦBC (x; y|δ, κ) satisfies the functional equation
[κ]Ex(µ|δ,κ) ΦBC (x; y|δ, κ) − [κ]Ey(ν|δ,κ) ΦBC (x; y|δ, κ)
= [2(m − n)κ + c(µ|δ,κ) ]ΦBC (x; y|δ, κ),

where νr = 21 (δ + κ) − µr (r = 1, . . . , 2ρ).

(2) The function ΨBC (x; y) satisfies the functional equation
[κ]Ex(µ|δ,κ) ΨBC (x; y) + [δ]Ey(µ|κ,δ) ΨBC (x; y) = [2mκ + 2nδ + c(µ|δ,κ) ]ΨBC (x; y).
As a special case n = 0 of this theorem, we see that the constant function 1 is a eigenfunction
(µ|δ,κ)
of Ex
for arbitrary values of the parameters. Theorems 2.3 and 2.4 will be proved in
Section 3.
In the trigonometric and rational BC cases, it is convenient to introduce another difference
operator
Dx(µ|δ,κ)

=

m
X
i=1

A+
i (x; µ|δ, κ)

Txδi




−1 +

m
X
i=1

−δ
A−
i (x; µ|δ, κ) Txi − 1




with the same coefficients Aǫi (x; µ|δ, κ) (i = 1, . . . , m; ǫ = ±) as those of the Ruijsenaars opera(µ|δ,κ)
(µ|δ,κ)
tor Ex
. In the trigonometric case, this operator Dx
is a constant multiple of Koornwinder’s q-difference operator expressed in terms of additive variables. By using the relation
Dx(µ|δ,κ) = Ex(µ|δ,κ) − Ex(µ|δ,κ) (1),
(µ|δ,κ)

.
one can rewrite the functional equations in Theorem 2.4 into those for Dx
√
Theorem 2.5. Suppose that [u] = 2 −1 sin(πu/ω1 ), ρ = 2 (resp. [u] = u, ρ = 1).
(1) The function ΦBC (x; y|δ, κ) satisfies the functional equation
[κ]Dx(µ|δ,κ) ΦBC (x; y|δ, κ) − [κ]Dy(ν|δ,κ) ΦBC (x; y|δ, κ)

= [mκ][−nκ][(m − n)κ + c(µ|δ,κ) ]ΦBC (x; y|δ, κ)

(resp. = 0),

where νr = 21 (δ + κ) − µr (r = 1, . . . , 2ρ).

(2) For arbitrary m, n, the function ΨBC (x; y) satisfies the functional equation
[κ]Dx(µ|δ,κ) ΨBC (x; y) + [δ]Dy(µ|κ,δ) ΨBC (x; y)
= [mκ][nδ][mκ + nδ + c(µ|δ,κ) ]ΨBC (x; y)

(resp. = 0).

Statement (2) of Theorem 2.5 recovers a main result of Mimachi [15]. A proof of Theorem 2.5
will be given in Section 3.4. Also, we will explain in Section 4 how Theorem 2.5 is related to
the theory of Koornwinder polynomials.

10

3
3.1

Y. Komori, M. Noumi and J. Shiraishi

Derivation of kernel functions
Key identities

We start with a functional identity which decomposes a product of functions expressed by [u]
into partial fractions.
Proposition 3.1. Let [u] be any nonzero entire function satisfying the Riemann relation. Then,
for variables z, (x1 , . . . , xN ), and parameters (c1 , . . . , cN ), we have
N
N
Y
[z − xj + cj ] X [z − xi + c]
=
[ci ]
[c]
[z − xj ]
[z − xi ]
i=1

j=1

Y

1≤j≤N ; j6=i

[xi − xj + cj ]
,
[xi − xj ]

(3.1)

where c = c1 + · · · + cN .
This identity can be proved by the induction on the number of factors by using the Riemann
relation for [u]. Note that, by setting yj = xj − cj , this identity can also be rewritten as
[c]

N
Y
[z − yj ]

j=1

where c =

[z − xj ]

N
P

j=1

=

N
X
i=1

[xi − yi ]

[z − xi + c]
[z − xi ]

Y

1≤j≤N ; j6=i

[xi − yj ]
,
[xi − xj ]

(xj − yj ).

From this proposition, we obtain the following lemma, which provides key identities for our
proof of kernel relations.
Lemma 3.1. Consider N variables x1 , . . . , xN and N complex parameters c1 , . . . , cN .
(1) Let [u] be any nonzero function satisfying the Riemann relation. Then under the balancing
N
P
cj = 0, we have an identity
condition
j=1

N
X
i=1

[ci ]

Y

1≤j≤N ; j6=i

[xi − xj + cj ]
= 0.
[xi − xj ]

(3.2)

as a meromorphic function in (x1 , . . . , xN ).
(2) Suppose that [u] is a constant multiple of sin(πu/ω1 ) or u. Then for any c1 , . . . , cN ∈ C,
we have an identity
"N #
N
X
X
Y
[xi − xj + cj ]
=
ci
(3.3)
[ci ]
[xi − xj ]
i=1

1≤j≤N ; j6=i

i=1

as a meromorphic function in (x1 , . . . , xN ).
When [z] = sin(πz/ω1 ), we have [z + a]/[z + b] → e(−(a − b)/2ω1 ) as Im(z/ω1 ) → ∞. This
implies in (3.1)
N
Y
[z − xj + cj ]
→ e(−c/2ω1 ),
[z − xj ]

j=1

[z − xi + c]
→ e(−c/2ω1 )
[z − xi ]

(i = 1, . . . , m),

as Im(z/ω1 ) → ∞. Hence we obtain (3.3) from (3.1). In the rational case [z] = z, formula (3.3)
is derived by a simple limiting procedure z → ∞.

Kernel Functions for Difference Operators of Ruijsenaars Type

3.2

11

Case of type A

We apply the key identities (3.2) and (3.3) for studying kernel functions. For two sets of variables
x = (x1 , . . . , xm ) and y = (y1 , . . . , yn ), we consider the following meromorphic function:
F (z) =

m
n
Y
[z − xj + κ] Y [z − yl + v + λ]
[z − xj ]
[z − yl + v]

j=1

l=1

where κ, λ and v are complex parameters. Then by Proposition 3.1 this function F (z) is
expanded as
m
X
[z − xi + c]
[c]F (z) = [κ]
[z − xi ]
i=1

n
X

+ [λ]

k=1
m
X

= [κ]

i=1

n

Y

1≤j≤m; j6=i

[z − yk + v + c]
[z − yk + v]

[xi − xj + κ] Y [xi − yl + v + λ]
[xi − xj ]
[xi − yl + v]
l=1

m

Y

[yk − yl + λ] Y [yk − xj − v + κ]
[yk − yl ]
[yk − xj − v]
j=1

1≤l≤n; l6=k
n
Y
[xi − yl

[z − xi + c]
Ai (x; κ)
[z − xi ]

+ v + λ]
[xi − yl + v]

l=1

m
n
Y
X
[yk − xj − v + κ]
[z − yk + v + c]
Ak (y; λ)
,
+ [λ]
[z − yk + v]
[yk − xj − v]
j=1

k=1

where c = mκ+nλ, with the coefficients Ai (x; κ), Ak (y; λ) of Ruijsenaars operators in x variables
and y variables. By Lemma 3.1, under the balancing condition c = mκ + nλ = 0, we have
n
m
m
n
X
Y
X
Y
[yk − xj − v + κ]
[xi − yl + v + λ]
+ [λ] Ak (y; λ)
= 0.
[κ] Ai (x; κ)
[xi − yl + v]
[yk − xj − v]
i=1

l=1

k=1

j=1

Also, when [u] is a constant multiple of sin(πu/ω1 ) or u, we always have
n
m
m
n
X
Y
X
Y
[yk − xj − v + κ]
[xi − yl + v + λ]
+ [λ] Ak (y; λ)
= [mκ + nλ].
[κ] Ai (x; κ)
[xi − yl + v]
[yk − xj − v]
i=1

k=1

l=1

j=1

We now try to find a function Φ(x; y) satisfying the system of first-order difference equations
Txδi Φ(x; y) =
Tyτk Φ(x; y) =

n
Y
[xi − yl + v + λ]
l=1
m
Y

j=1

[xi − yl + v]

Φ(x; y)

[yk − xj − v + κ]
Φ(x; y)
[yk − xj − v]

(i = 1, . . . , m),
(k = 1, . . . , n),

(3.4)

where τ stands for the unit scale of difference operators for y variables. In order to fix the idea,
assume that the balancing condition mκ + nλ = 0 is satisfied. Then, any solution of this system
should satisfy the functional equation
m
n
X
X
δ
[κ] Ai (x; κ)Txi Φ(x; y) + [λ] Ak (y; λ)Tyτk Φ(x; y) = 0,
i=1

k=1

namely,
[κ]Dx(δ,κ) Φ(x; y) + [λ]Dy(τ,λ) Φ(x; y) = 0.

12

Y. Komori, M. Noumi and J. Shiraishi
The compatibility condition for the system (3.4) of difference equations is given by
[xi − yk − τ + v + λ][xi − yk + v]
[yk − xi − δ − v + κ][yk − xi − v]
=
[xi − yk − τ + v][xi − yk + v + λ]
[yk − xi − δ − v][yk − xi − v + κ]

for i = 1, . . . , m and k = 1, . . . , n. From this we see there are (at least) two cases where the
difference equation (3.4) becomes compatible:
(case 1) :

τ = −δ,

(case 2) :

τ = κ,

λ = −κ,

λ = δ,

v : arbitrary,
v : arbitrary.

In the first case, the difference equation to be solved is:
Txδi Φ(x; y)

=

n
Y
[xi − yl + v − κ]
l=1
m
Y

Φ(x; y) =
Ty−δ
k

j=1

[xi − yl + v]

Φ(x; y)

[yk − xj − v + κ]
Φ(x; y)
[yk − xj − v]

(i = 1, . . . , m),
(k = 1, . . . , n).

This system is solved by
Φ−
A (x; y|δ, κ)

m Y
n
Y
G(xj − yl + v − κ|δ)
.
=
G(xj − yl + v|δ)
j=1 l=1

Hence we see that Φ−
A (x; y|δ, κ) satisfies the functional equation
(−δ,−κ) −
Dx(δ,κ) Φ−
ΦA (x; y|δ, κ) = 0,
A (x; y|δ, κ) − Dy

under the balancing condition (m − n)κ = 0, namely, m = n. By setting ΦA (x; y) = Φ−
A (x; −y),
we obtain
Dx(δ,κ) ΦA (x; y|δ, κ) − Dy(δ,κ) ΦA (x; y|δ, κ) = 0
as in Theorem 2.1. The second case
Txδi Φ(x; y) =
Tyκk Φ(x; y) =

n
Y
[xi − yl + v + δ]
l=1
m
Y

j=1

[xi − yl + v]

Φ(x; y)

[yk − xj − v + κ]
Φ(x; y)
[yk − xj − v]

(i = 1, . . . , m),
(k = 1, . . . , n)

is solved by
ΨA (x; y) =

m Y
n
Y
[xj − yl + v].

j=1 l=1

Hence, under the balancing condition mκ + nδ = 0, we have
[κ]Dx(δ,κ) ΨA (x; y) + [δ]Dy(κ,δ) ΨA (x; y) = 0.
This completes the proof of Theorem 2.1.
When [u] is a constant multiple of sin(πu/ω1 ) or u, for any solution Φ(x; y) of the system (3.4)
of difference equations we have
m
n
X
X
δ
[κ] Ai (x; κ)Txi Φ(x; y) + [λ] Ak (y; λ)Tyκk Φ(x; y) = [mκ + nλ]Φ(x; y),
i=1

k=1

Kernel Functions for Difference Operators of Ruijsenaars Type

13

namely,
[κ]Dx(δ,κ) Φ(x; y) + [λ]Dy(τ,λ) Φ(x; y) = [mκ + nλ]Φ(x; y)
without imposing the balancing condition. This implies that the functions ΦA (x; y|δ, κ) and
ΨA (x; y) in these trigonometric and rational cases satisfy the functional equations
Dx(δ,κ) ΦA (x; y|δ, κ) − Dy(δ,κ) ΦA (x; y|δ, κ) =

[(m − n)κ]
ΦA (x; y|δ, κ)
[κ]

and
[κ]Dx(δ,κ) ΨA (x; y) + [δ]Dy(κ,δ) ΨA (x; y) = [mκ + nδ]ΨA (x; y).
respectively, as stated in Theorem 2.2.

3.3

Case of type BC

In this BC case, we assume that [u] is a constant multiple of one of the following functions:


u (ρ = 1),
sin(πu/ω1 ) (ρ = 2),
e au2 σ(u; Ω) (ρ = 4).
In order to discuss difference operators of type BC, we consider the meromorphic function

F (z) =

2ρ
Q

[z + µs ]

n
m
Y
[z ± xj + κ] Y [z ± yl + v + λ]
.
[z ± xj ]
[z ± yl + v]

s=1
ρ
Q

1
2 (δ

[z +

s=1

− ωs )][z +

1
2 (κ

ωs )] j=1

−

(3.5)

l=1

By Proposition 3.1 it can be expanded as
[c]F (z) =

m
X
[z − xi + c]
i=1

+

[z − xi ]

n
X
k=1

Pi+ +

m
X
[z + xi + c]
i=1

[z + xi ]

Pi−

n

[z − yk + v + c] + X [z + yk + v + c] −
Qk +
Qk
[z − yk + v]
[z + yk + v]
k=1

ρ
ρ
X
X
[z + 21 (κ − ωr ) + c]
[z + 12 (δ − ωr ) + c]
R
+
Sr
+
r
[z + 21 (δ − ωr )]
[z + 21 (κ − ωr )]
r=1
r=1

(3.6)

into partial fractions, where
c = 2mκ + 2nλ +

2ρ
X
s=1

µs − ρ2 (δ + κ) +

ρ
X

ωs = 2mκ + 2nλ + c(µ|δ,κ) .

s=1

Also by Lemma 3.1, we see that expression
X

1≤i≤m; ǫ=±

Piǫ +

X

1≤k≤n; ǫ=±

Qǫk +

X

1≤r≤ρ

Rr +

X

Sr

1≤r≤ρ

reduces to 0 when the balancing condition c = 0 is satisfied, or to [c] when [u] is trigonometric
or rational. A remarkable fact is that, if the parameter v is chosen appropriately, then the
expansion coefficients of F (z) are expressed in terms of the coefficients of Ruijsenaars operators
of type BC.

14

Y. Komori, M. Noumi and J. Shiraishi

Proposition 3.2. When v = 12 (δ − λ), the function F (z) defined by (3.5) is expressed as follows
in terms of the coefficients of Ruijsenaars operators:
[c]

2ρ
Q

[z + µs ]

m
n
Y
[z ± xj + κ] Y [z ± yl + v + λ]
[z ± xj ]
[z ± yl + v]

s=1

[c]F (z) =

ρ
Q

s=1

[z + 21 (δ − ωs )][z + 21 (κ − ωs )] j=1

l=1

n
Y
[ǫxi ± yl + 21 (δ + λ)]
[z − ǫxi + c] ǫ
Ai (x; µ|δ, κ)
[z − ǫxi ]
[ǫxi ± yl + 21 (δ − λ)]
1≤i≤m; ǫ=±
l=1

= [κ]

X

ρ
X
+ [κ] e

1
2 cηr

ρ
X
+ [λ] e

1
2 cηr

r=1


[z + 21 (δ − ωr ) + c] 0
Ar (x; µ|δ, κ)
[z + 21 (δ − ωr )]
m

Y [ǫyk ± xj + 1 (τ + κ)]
[z − ǫyk + v + c] ǫ
2
+ [λ]
Ak (y; ν|τ, λ)
1
[z − ǫyk + v]
[ǫy
±
x
+
j
k
2 (τ − κ)]
j=1
1≤k≤n; ǫ=±
X

r=1


[z + 12 (κ − ωr ) + c] 0
Ar (y; ν|τ, λ),
[z + 21 (κ − ωr )]

where
c = 2mκ + 2nλ + c(µ|δ,κ) ,

τ = κ + λ − δ,

ν = (µ1 − v, . . . , µ2ρ − v).

Proof . The expansion coefficients in (3.6) are determined from the residues at the corresponding poles. We first remark that
[κ]
Pi+ =

[xi + µs ]

s=1
ρ
Q

s=1

Since

2ρ
Q

[xi + 12 (δ − ωs )][xi + 21 (κ − ωs )]

m

n

j6=i

l=1

[2xi + κ] Y [xi ± xj + κ] Y [xi ± yl + v + λ]
.
[2xi ]
[xi ± xj ]
[xi ± yl + v]

ρ
[2xi + κ] Y [xi + 21 (κ − ωs )]
,
=
[2xi ]
[xi − 12 ωs ]
s=1

we obtain
[κ]
Pi+

=

2ρ
Q

[xi + µs ]

s=1
ρ
Q

s=1

m
n
Y
[xi ± xj + κ] Y [xi ± yl + v + λ]
[xi ± xj ]
[xi ± yl + v]

[xi − 12 ωs ][xi + 21 (δ − ωs )] j6=i

l=1

n
Y
[xi ± yl + v + λ]
.
= [κ]A+
(x;
µ|δ,
κ)
i
[xi ± yl + v]
l=1

Note that Pi− is obtained from Pi+ by replacing xj with −xj (j = 1, . . . , m). We next look at
the coefficient Q+
k:
[λ]
Q+
k

=

2ρ
Q

s=1
ρ
Q

s=1

[yk + µs − v]

[yk + 21 (δ − ωs ) − v][yk + 21 (κ − ωs ) − v]

[2yk + λ]
[2yk ]

Kernel Functions for Difference Operators of Ruijsenaars Type
×

15

m
Y [yk ± yl + λ] Y
[yk ± xj + κ − v]
.
[yk ± yl ]
[yk ± xj − v]
j=1

l6=k

In view of
ρ
[2yk + λ] Y [yk + 21 (λ − ωs )]
=
[2yk ]
[yk − 12 ωs ]
s=1

we set 21 δ − v = 12 λ, namely, v = 21 (δ − λ). Then we have
[λ]
Q+
k

=

s=1
ρ
Q

s=1

=

2ρ
Q

[yk + µs − v]

m
Y [yk ± yl + λ] Y
[yk ± xj + κ − v]
[yk ± yl ]
[yk ± xj − v]

[yk − 21 ωs ][yk + 12 (τ − ωs )] l6=k

j=1

m
Y
[yk ± xj + κ − v]
+
[λ]Ak (y; ν|τ, λ)
,
[yk ± xj − v]
j=1

v = 21 (δ − λ),

τ = κ + λ − δ,

ν = (µ1 − v, . . . , µ2ρ − v).

+
The coefficient Q−
k is obtained from Qk by the sign change yl → −yl . The coefficient Rr is
given by

Rr =

2ρ
Q

s=1

Q

s6=r

[ 21 (ωr − δ) + µs ]

[ 21 ωrs ]

ρ
Q

s=1

[ 12 ωrs + 12 (κ − δ)]

m 1
n
Y
[ 2 (ωr − δ) ± xj + κ] Y [ 21 (ωr − δ) ± yl + v + λ]
,
1
1
(ω
−
δ)
±
x
]
(ω
−
δ)
±
y
+
v]
[
[
r
j
r
l
2
2
j=1
l=1

where ωrs = ωr − ωs . When v = 12 (δ − λ), we have
[ 12 (ωr − δ) ± yl + v + λ]
[ 21 (ωr + λ) ± yl ]
=
= e(ληr )
[ 21 (ωr − δ) ± yl + v]
[ 21 (ωr − λ) ± yl ]
by the quasi-periodicity of [u]. In this way y variables disappear from Rr :
e(nληr )
Rr =

Q

s6=r

[ 21 ωrs ]

2ρ
Q

s=1
ρ
Q

[ 12 (ωr − δ) + µs ] Y
m 1
[ 2 (ωr − δ) ± xj + κ]
.
1
(ω
−
δ)
±
x
]
[
r
j
1
1
2
[ 2 ωrs + 2 (κ − δ)] j=1

s=1

Note that the exponential factor e(nληr ) is nontrivial only in the elliptic case. In any case, from
2nλ = c − 2mκ − c(µ|δ,κ) we have
Rr = [κ]e

1
2 cηr




A0r (x; µ|δ, κ).

Finally, when v = 12 (δ − λ) and τ = κ + λ − δ, we can rewrite Sr as

Sr =

2ρ
Q

s=1

Q

s6=r

[ 12 (ωr − κ) + µs ]

[ 21 ωrs ]

ρ
Q

s=1

[ 21 ωrs

+

1
2 (δ

m 1
n
Y
[ (ωr + κ) ± xj ] Y [ 1 (ωr − κ) ± yl + v + λ]
2

− κ)]

[ 1 (ωr − κ) ± xj ] l=1
j=1 2

2

[ 12 (ωr − κ) ± yl + v]

16

Y. Komori, M. Noumi and J. Shiraishi
e(mκηr )
=

Q

s6=r

8
Q

[ 12 (ωr − τ ) + µs − v] Y
n

s=1
4
Q

[ 21 ωrs ]

s=1

[ 12 ωrs + 12 (λ − τ )]

[ 21 (ωr − τ ) ± yl + λ]
.
[ 21 (ωr − τ ) ± yl ]
l=1

Note that x variables have disappeared again by the quasi-periodicity of [u]. Since 2mκ =
c − 2nλ − c(µ|δ,κ) and c(µ|δ,κ) = c(ν|τ,λ) , we have


Sr = [λ]e 12 cηr A0r (y; ν|τ, λ).
This completes the proof of proposition.



In what follows we set v = 12 (δ − λ) and τ = κ + λ − δ. Then Proposition 3.2, combined with
Lemma 3.1, implies
[κ]

X

1≤i≤m; ǫ=±

+

ρ
n
Y
X
[ǫxi ± yl + 21 (δ + λ)]
+ [κ] A0r (x; µ|δ, κ)
1
(δ
−
λ)]
[ǫx
±
y
+
i
l
2
s=1
l=1

Aǫi (x; µ|δ, κ)

m
Y
[ǫyk
ǫ
[λ]
Ak (y; ν|τ, λ)
[ǫyk
j=1
1≤k≤n; ǫ=±

= 0,

X

ρ
X
± xj + 21 (τ + κ)]
+ [λ] A0r (y; ν|τ, λ)
1
± xj + 2 (τ − κ)]
r=1

(resp. = [2mκ + 2nλ + c(µ|δ,κ) ]),

when the balancing condition 2mκ+2nλ+c(µ|δ,κ) = 0 is satisfied (resp. when [u] is trigonometric
or rational.) Hence we have
Proposition 3.3. Suppose that the parameters δ, κ, τ , λ satisfy the relation δ + τ = κ + λ, and
define ν = (ν1 , . . . , ν2ρ ) by
νs = µs − 12 (δ − λ) = µs + 12 (τ − κ)

(s = 1, . . . , 2ρ).

Let Φ(x; y) any meromorphic function in the variables x = (x1 , . . . , xm ) and y = (y1 , . . . , yn )
satisfying the system of first-order difference equations
Txδi Φ(x; y) =
Tyτk Φ(x; y)

n
Y
[xi ± yl + 21 (δ + λ)]
Φ(x; y)
1
(δ
−
λ)]
[x
±
y
+
i
l
2
l=1

m
Y
[yk ± xj + 12 (τ + κ)]
=
Φ(x; y)
[y ± xj + 12 (τ − κ)]
j=1 k

(i = 1, . . . , m),
(k = 1, . . . , n).

(3.7)

(1) If the balancing condition 2mκ + 2nλ + c(µ|δ,κ) = 0 is satisfied, then Φ(x; y) satisfies the
functional equation
[κ]Ex(µ|δ,κ) Φ(x; y) + [λ]Ey(ν|τ,λ) Φ(x; y) = 0.
(2) If [u] is a constant multiple of sin(πu/ω1 ) or u, then Φ(x; y) satisfies the functional equation
[κ]Ex(µ|δ,κ) Φ(x; y) + [λ]Ey(ν|τ,λ) Φ(x; y) = [2mκ + 2nλ + c(µ|δ,κ) ]Φ(x; y).
In fact there are essentially two cases where the system of first-order linear difference equations (3.7) become compatible:
(case 1):
(case 2):

τ = −δ,

τ = κ,

λ = −κ,

λ = δ,

νs = µs − 12 (δ + κ)

νs = µs

(s = 1, . . . , 2ρ);
(s = 1, . . . , 2ρ).

(3.8)

Kernel Functions for Difference Operators of Ruijsenaars Type

17

In the first case, the system (3.7) of difference equations to be solved is:
Txδi Φ(x; y) =
Tyδk Φ(x; y)

n
Y
[xi ± yl + 21 (δ − κ)]
Φ(x; y)
1
(δ
+
κ)]
[x
±
y
+
i
l
2
l=1

(i = 1, . . . , m),

m
Y
[yk ± xj + 12 (δ − κ)]
=
Φ(x; y)
[y ± xj + 12 (δ + κ)]
j=1 k

(k = 1, . . . , n).

It is solved either by the function
m Y
n
Y
G(xj ± yl + 21 (δ − κ)|δ)
ΦBC (x; y|δ, κ) =
,
G(xj ± yl + 21 (δ + κ)|δ)
j=1 l=1

or by
ΦBC (x; y|δ, κ) =

m Y
n
Y
Y

j=1 l=1 ǫ1 ,ǫ2 =±

G(ǫ1 xj + ǫ2 yl + 12 (δ − κ)|δ).

Hence we see that, under the balancing condition 2(m−n)κ+c(µ|δ,κ) = 0, ΦBC (x; y|δ, κ) satisfies
the functional equation
Ex(µ|δ,κ) ΦBC (x; y|δ, κ) − Ey(ν|−δ,−κ) ΦBC (x; y|δ, κ) = 0,
for νs = µs − 21 (δ + κ) (s = 1, . . . , 2ρ). By symmetry (2.12) with respect to the sign change of
parameters, we obtain
Ex(µ|δ,κ) ΦBC (x; y|δ, κ) − Ey(ν|δ,κ) ΦBC (x; y|δ, κ) = 0,
for νs = 21 (δ + κ) − µs (s = 1, . . . , 2ρ). The second case
Txδi Φ(x; y) =
Tyκk Φ(x; y) =

n
Y
[xi ± yl + δ]
l=1
m
Y

j=1

[xi ± yl ]

Φ(x; y)

[yk ± xj + κ]
Φ(x; y)
[yk ± xj ]

(i = 1, . . . , m),
(k = 1, . . . , n)

is solved by
n
m Y
Y
ΨBC (x; y) =
[xj ± yl ].
j=1 l=1

Hence, under the balancing condition 2mκ + 2nδ + c(µ|δ,κ) = 0, this function satisfies the functional equation
[κ]Ex(µ|δ,κ) ΨBC (x; y) + [δ]Ey(µ|κ,δ) ΨBC (x; y) = 0.
This completes the proof of Theorem 2.3. When [u] is trigonometric or rational, Proposition 3.3, (2) implies that these functions ΦBC (x; y|δ, κ) and ΨBC (x; y) satisfy the functional
equations as stated in Theorem 2.4.

18

3.4

Y. Komori, M. Noumi and J. Shiraishi

Dif ference operators of Koornwinder type

In the rest of this section, we confine ourselves to the trigonometric and rational BC cases and
suppose that [u] is a constant multiple of sin(πu/ω1 ) or u. We rewrite our results on kernel
functions for these cases, in terms of difference operator
Dx(µ|δ,κ)

=

m
X

A+
i (x; µ|δ, κ)

Txδi

i=1




−1 +

m
X
i=1

−δ
A−
i (x; µ|δ, κ) Txi − 1




of Koornwinder type. We remark that this operator has symmetry
Dx(µ|−δ,−κ) = Dx(−µ|δ,κ)
(µ|δ,κ)

(µ|δ,κ)

with respect to the sign change, as in the case of Ex
. We show first that Dx
(µ|δ,κ)
from Ex
only by an additive constant in the 0th order term.

differs

Lemma 3.2.
(µ|δ,κ)

(1) The constant function 1 is an eigenfunction of Ex
Ex(µ|δ,κ) (1) = C (µ|δ,κ) +



1
[2mκ + c(µ|δ,κ) ] − [c(µ|δ,κ) ] .
[κ]
(µ|δ,κ)

(2) The two difference operators Dx

Ex(µ|δ,κ) = Dx(µ|δ,κ) + C (µ|δ,κ) +
(µ|κ)

(µ|δ,κ)

:

(µ|δ,κ)

and Ex

are related as



1
[2mκ + c(µ|δ,κ) ] − [c(µ|δ,κ) ] .
[κ]

(µ|δ,κ)

Proof . Since Dx
= Ex
− Ex
(1), statement (2) follows from statement (1). For the
proof of (1), we make use of our Theorem 2.4. This theorem is valid even in the case where the
dimension m or n reduces to zero. When n = 0, Theorem 2.4, (1) implies
[κ]Ex(µ|δ,κ) (1) − [κ]C (ν|δ,κ) = [2mκ + c(µ|δ,κ) ],
with νs = 21 (δ + κ) − µs (s = 1, . . . , 2ρ), since ΦBC (x; y) in this case is the constant function 1.
Also from the case m = n = 0 we have
[κ]C (µ|δ,κ) − [κ]C (ν|δ,κ) = [c(µ|δ,κ) ].
Combining these two formulas we obtain
[κ]Ex(µ|δ,κ) (1) = [κ]C (µ|δ,κ) + [2mκ + c(µ|δ,κ) ] − [c(µ|δ,κ) ].


(µ|δ,κ)

Let us rewrite the functional equations in Proposition 3.3 in terms of the operator Dx
In the notation of Proposition 3.3, (2) we have
[κ]Ex(µ|δ,κ) Φ(x; y) + [λ]Ey(ν|τ,λ) Φ(x; y) = [2mκ + 2nλ + c(µ|δ,κ) ]Φ(x; y).
As special cases where (m, n) = (m, 0), (0, n) and (0, 0), we have
[κ]Ex(µ|δ,κ) (1) + [λ]C (ν|τ,λ) = [2mκ + c(µ|δ,κ) ],
[κ]C (µ|δ,κ) + [λ]Ey(ν|τ,λ) (1) = [2nλ + c(µ|δ,κ) ],
[κ]C (µ|δ,κ) + [λ]C (ν|τ,λ) = [c(µ|δ,κ) ],

.

(3.9)

Kernel Functions for Difference Operators of Ruijsenaars Type

19

and hence
−[κ]Ex(µ|δ,κ) (1)Φ(x; y) − [λ]C (ν|τ,λ) Φ(x; y) = −[2mκ + c(µ|δ,κ) ]Φ(x; y),

−[κ]C (µ|δ,κ) Φ(x; y) − [λ]Ey(ν|τ,λ) (1)Φ(x; y) = −[2nλ + c(µ|δ,κ) ]Φ(x; y),

[κ]C (µ|δ,κ) Φ(x; y) + [λ]C (ν|τ,λ) Φ(x; y) = [c(µ|δ,κ) ]Φ(x; y),

(3.10)

By taking the sum of the four formulas in (3.9) and (3.10), we obtain
[κ]Dx(µ|δ,κ) Φ(x; y) + [λ]Dy(ν|τ,λ) Φ(x; y) = CΦ(x; y),
C = [2mκ + 2nλ + c(µ|δ,κ) ] − [2mκ + c(µ|δ,κ) ] − [2nλ + c(µ|δ,κ) ] + [c(µ|δ,κ) ].
In the rational case, it is clear that C = 0.
√ In the trigonometric case, this constant C can
be factorized. In fact, if we choose [u] = 2 −1 sin(πu/ω1 ) = e(u/2ω1 ) − e(−u/2ω1 ), we have
a simple expression
C = [mκ][nλ][mκ + nλ + c(µ|δ,κ) ].
Hence, under the assumption of Proposition 3.3, (2), we have
[κ]Dx(µ|δ,κ) Φ(x; y) + [λ]Dy(ν|τ,λ) Φ(x; y) = [mκ][nλ][mκ + nλ + c(µ|δ,κ) ]Φ(x; y)

(resp. = 0),

when [u] = e(u/2ω1 ) − e(−u/2ω1 ) (resp. when [u] = u). Applying this to the two cases of (3.8),
one can easily derive functional equations as in Theorem 2.5.

4

Kernel functions for q-dif ference operators

In the trigonometric case, it is also important √
to consider q-difference operators passing to
multiplicative variables. Assuming that [u] = 2 −1 sin(πu/ω1 ) = e(u/2ω1 ) − e(−u/2ω1 ), we
define by z = e(u/ω1 ) the multiplicative variable associated with the additive variable u. When
1
1
1
1
we write [u] = z 2 −z − 2 = −z − 2 (1−z), we regard the square root z 2 as a multiplicative notation
for e(u/2ω1 ). We set q = e(δ/ω1 ) and t = e(κ/ω1 ), assuming that Im(δ/ω1 ) > 0, namely, |q| < 1.

4.1

Kernel functions for Macdonald operators

We introduce the multiplicative variables z = (z1 , . . . , zm ) and w = (w1 , . . . , wn ) corresponding
to the additive variables x = (x1 , . . . , xm ) and y = (y1 , . . . , yn ), by zi = e(xi /ω1 ) (i = 1, . . . , m)
and wk = e(yk /ω1 ) (k = 1, . . . , n), respectively.
(δ,κ)
In this convention of the trigonometric case, the Ruijsenaars difference operator Dx
of
type A is a constant multiple of the q-difference operator of Macdonald:
Dx(δ,κ)

=t

− 21 (m−1)

Dz(q,t)

Dz(q,t) ,

=

m
X

Y

i=1 1≤j≤m; j6=i

tzi − zj
Tq,zi ,
zi − zj

where Tq,zi denotes the q-shift operator with respect to zi :
Tq,zi f (z1 , . . . , zm ) = f (z1 , . . . , qzi , . . . , zm )

(i = 1, . . . , m).

In what follows, we use the gamma function G− (u|δ) of Section 2, (2.3). Then our kernel function
ΦA (x; y|δ, κ) with parameter v = κ is expressed as follows in terms of multiplicative variables:
ΦA (x; y|δ, κ) = e

mnδ κ/δ
2
2ω1





nκ

mκ

(z1 · · · zm ) 2δ (w1 · · · wn ) 2δ Π(z; w|q, t),

20

Y. Komori, M. Noumi and J. Shiraishi
Π(z; w|q, t) =

m Y
n
Y
(tzj wl ; q)∞

j=1 l=1

(zj wl ; q)∞

.

(4.1)

The functional equation of Theorem 2.2, (1) thus implies
(q,t)
Dz(q,t) Π(z; w|q, t) − tm−n Dw
Π(z; w|q, t) =

1 − tm−n
Π(z; w|q, t),
1−t

(4.2)

which can also be proved by the expansion formula
Π(z; w|q, t) =

X

bλ (q, t)Pλ (z|q, t)Pλ (w|q, t)

(4.3)

l(λ)≤min{m,n}

of Cauchy type for Macdonald polynomials [14]. (Formula (4.2) already implies that Π(z; w|q, t)
has an expansion of this form, apart from the problem of determining the coefficients bλ (q, t).)
On the other hand, kernel function ΨA (x; y) with parameter v = 0 is expressed as
n
−2

ΨA (x; y) = (z1 · · · zm )

m n
m YY
−2
(w1 · · · wn )
(zj
j=1 l=1

− wl ).

Then the functional equation of Theorem 2.2, (2) implies
m Y
n

Y
(t,q)
− (1 − tm q n )
(1 − t)Dz(q,t) − (1 − q)Dw
(zj − wl ) = 0.
j=1 l=1

This formula corresponds to the dual Cauchy formula
m Y
n
Y
X
∗
(zj − wl ) =
(−1)|λ | Pλ (z|q, t)Pλ∗ (w|t, q),

j=1 l=1

λ⊂(nm )

where λ∗ = (m − λ′n , . . . , n − λ′1 ) is the partition representing the complement of λ in the m × n
rectangle.
These kernel functions for Macdonald operators have been applied to the studies of raising
and lowering operators (Kirillov–Noumi [7, 8], Kajihara–Noumi [5]) and integral representation
(Mimachi—Noumi [16], for instance). We also remark that, in this A type case, a kernel function
of Cauchy type for q-Dunkl operators has been constructed by Mimachi—Noumi [17].

4.2

Kernel functions for Koornwinder operators

We now consider the trigonometric BC case. Instead of the additive parameters (µ1 , µ2 , µ3 , µ4 )
(ρ = 2), we use the multiplicative parameters
(a, b, c, d) = (e(µ1 /ω1 ), e(µ2 /ω1 ), e(µ3 /ω1 ), e(µ4 /ω1 )).
These four parameters are the Askey–Wilson parameters (a, b, c, d) for the Koornwinder polynomials Pλ (z; a, b, c, d|q, t).
In this trigonometric BC case, the difference operator
Dx(µ|δ,κ) =

m
X
i=1

δ
A+
i (x; µ|δ, κ)(Txi − 1) +

m
X
i=1

−δ
A−
i (x; µ|δ, κ)(Txi − 1),

Kernel Functions for Difference Operators of Ruijsenaars Type

21

which we have discussed in Section 3.4, is a constant multiple of Koornwinder’s q-difference
operator [10]. Let us consider the Koornwinder operator
Dz(a,b,c,d|q,t)

=

m
X
i=1

A+
i (z)(Tq,zi

− 1) +

m
X
i=1

−1
A−
i (z) Tq,zi − 1




+
in the multiplicative variables, where the coefficients A+
i (z) = Ai (z; a, b, c, d|q, t) are given by

A+
i (z)

=

1

(abcdq −1 ) 2 tm−1 (1 − zi2 )(1 − qzi2 )

1 − tzi zj±1

Y

(1 − azi )(1 − bzi )(1 − czi )(1 − dzi )

1≤j≤m; j6=i

1 − zi zj±1

(µ|δ,κ)

(a,b,c,d|q,t)

+ −1
and A−
= −Dz
. Note that this
i (z) = Ai (z ) for i = 1, . . . , m. Then we have Dx
1
(a,b,c,d|q,t)
operator Dz
is renormalized by dividing the one used in [10] by the factor (abcdq −1 ) 2 tm−1 .
In what follows, we simply suppress the dependence on the parameters (a, b, c, d|q, t) as Dz =
(a,b,c,d|q,t)
Dz
, when we refer to operators or functions associated with these standard parameters.
In Section 2, we described two types of kernel functions (2.14) and (2.15) of Cauchy type.
Also, depending on the choice of G(u|δ) we obtain several kernel functions for each type. From
the gamma functions G∓ (u|δ) of (2.3), we obtain two kernel functions of type (2.14); in the
multiplicative variables,


1 1
m Y
n
Y
q 2 t 2 zj wl±1 ; q ∞
nβ
(4.4)
Φ0 (z; w|q, t) = (z1 · · · zm )

,
1
− 12
2
zj wl±1 ; q ∞
j=1 l=1 q t

and

1

−nβ

Φ∞ (z; w|q, t) = (z1 · · · zm )

1

m Y
n
Y
q 2 t 2 zj−1 wl±1 ; q

j=1 l=1

q

1
2




∞

,
− 12 −1 ±1
t zj wl ; q ∞

respectively, where we put β = κ/δ so that t = q β . Similarly, we obtain two kernel functions of
type (2.15) from G± (u|δ):
Φ+ (z; w|q, t) = e

f (x;y)

ω1 δ

Φ− (z; w|q, t) = e −
where f (x; y) = n

m
P

j=1

m Y
n
Y
Y

1

j=1 l=1 ǫ1 ,ǫ2 =±
m Y
n
Y
Y
f (x;y)

ω1 δ
j=1 l=1 ǫ1 ,ǫ2 =±

x2j + m

n
P

l=1

yl2 +

1

q 2 t 2 zjǫ1 wlǫ2 ; q

mn 2
4 (κ

1

1




∞

,

q 2 t− 2 zjǫ1 wlǫ2 ; q


−1
∞

,

− δ 2 ). Each of these four functions differs from

the others by multiplicative factors which are δ-periodic in all the x variables and y variables,
It should be noted, however, that they have different analytic properties. In the following we
denote simply by Φ(z; w|q, t) one of these functions.
The kernel function Ψ(z