Kernel Functions for Difference Operators of Ruijsenaars Type and Their Applications

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.


Introduction
Let Φ(x; y) be a meromorphic function on C m × C n in the variables x = (x 1 , . . . , x m ) and y = (y 1 , . . . , y n ), and consider two operators A x , B y which act on meromorphic functions in x and y, respectively. We say that Φ(x; y) is a kernel function for the pair (A x , B y ) if it satisfies a functional equation of the form A x Φ(x; y) = B y 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 (A n−1 , A n−1 ) and (BC n , BC n ) 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 (BC m , BC n ) 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 BC n 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 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 BC m interpolation polynomials of Okounkov [18].

Kernel functions for Ruijsenaars operators 2.1 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: associated with the period lattice Ω = Zω 1 ⊕ Zω 2 , generated by ω 1 , ω 2 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 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. (1) Trigonometric case: We set z=e(u/ω 1 ) and q=e(δ/ω 1 ), and suppose that Im(δ/ω 1 )>0 so that |q| < 1. We now consider the function For this [u] two meromorphic functions (1 − q i z), satisfy the difference equations respectively. Namely, for ǫ = ±, G ǫ (u|δ) is a gamma function for ǫ [u]. (Note that the quadratic function u 2 = 1 2 u(u − 1) satisfies u+1 (2) Elliptic case: Let p, q be nonzero complex numbers with |p| < 1, |q| < 1. Then the Ruijsenaars elliptic gamma function satisfies the q-difference equation Γ(qz; p, q) = θ(z; p)Γ(z; p, q), θ(z; p) = (z; p) ∞ (p/z; p) ∞ .
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]

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 in m variables x = (x 1 , . . . , x m ), where δ, κ ∈ C are complex parameters / ∈ Ω, and T δ x i stands for the δ-shift operator 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 Φ A (x; y|δ, κ) by with an extra parameter v. We also consider the function These two functions Φ A (x; y|δ, κ) and Ψ A (x; y) are kernel functions of Cauchy type and of dual Cauchy type for this case, respectively. (1) If m = n, then the function Φ A (x; y|δ, κ) defined as (2.5) satisfies the functional equation (2) Under the balancing condition mκ+ nδ = 0, the function Ψ A (x; y) defined as (2.6) satisfies the functional equation Statement (1) of Theorem 2.1 is due to Ruijsenaars [22,23]. (See Appendix B.1 for an explicit comparison between Φ A (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 Φ A (x; y|δ, κ) and Ψ A (x; y) satisfy more general functional equations without balancing conditions. Theorem 2.2. Suppose that [u] is a constant multiple of sin(πu/ω 1 ) or u.
(1) For arbitrary m and n, the function Φ A (x; y|δ, κ) satisfies the functional equation (2) The function Ψ A (x; y) satisfies the functional equation 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.

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 including 2ρ parameters µ = (µ 1 , . . . , µ 2ρ ) 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/ω 1 ) (ρ = 2), e au 2 σ(u; Ω) (ρ = 4). (2.8) In each case we define ω 1 , . . . , ω ρ ∈ Ω as (0) rational case: (1) trigonometric case: elliptic case: Then the quasi-periodicity of [u] is described as for some η r ∈ C and ǫ r = ±1. In the trigonometric and rational cases (without exponential factors), one can simply take η r = 0 (r = 1, . . . , ρ). Note also that [u] admits the duplication formula of the form .
(2.9) (In the trigonometric and rational cases, these formulas fail for [u] containing nontrivial exponential factors e(au 2 ).) In relation to the parameters µ = (µ 1 , . . . , µ 2ρ ), we introduce This parameter is related to quasi-periodicity of the coefficients of the Ruijsenaars operator, and plays a crucial role in various places of our argument. Note that the last term ρ s=1 ω s is nontrivial only in the trigonometric case: it is ω 1 when ρ = 2, and 0 when ρ = 1, 4.
In the case of 0 variables, the Ruijsenaars operator E (µ|δ,κ) x with m = 0 reduces to the multiplication operator by the constant .
(1) The function Φ BC (x; y|δ, κ) satisfies the functional equation The function Ψ BC (x; y) satisfies the functional equation As a special case n = 0 of this theorem, we see that the constant function 1 is a eigenfunction of E (µ|δ,κ) x 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 with the same coefficients A ǫ i (x; µ|δ, κ) (i = 1, . . . , m; ǫ = ±) as those of the Ruijsenaars operator E (µ|δ,κ) x . In the trigonometric case, this operator D (µ|δ,κ) x is a constant multiple of Koornwinder's q-difference operator expressed in terms of additive variables. By using the relation (1) The function Φ BC (x; y|δ, κ) satisfies the functional equation where ν r = 1 2 (δ + κ) − µ r (r = 1, . . . , 2ρ). (2) For arbitrary m, n, the function Ψ BC (x; y) satisfies the functional equation  [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.

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, (x 1 , . . . , x N ), and parameters (c 1 , . . . , c N ), we have This identity can be proved by the induction on the number of factors by using the Riemann relation for [u]. Note that, by setting y j = x j − c j , this identity can also be rewritten as From this proposition, we obtain the following lemma, which provides key identities for our proof of kernel relations.
(2) Suppose that [u] is a constant multiple of sin(πu/ω 1 ) or u. Then for any as a meromorphic function in (x 1 , . . . , x N ).

Case of type A
We apply the key identities (3.2) and (3.3) for studying kernel functions. For two sets of variables x = (x 1 , . . . , x m ) and y = (y 1 , . . . , y n ), we consider the following meromorphic function: where κ, λ and v are complex parameters. Then by Proposition 3.1 this function F (z) is expanded as where c = mκ+nλ, with the coefficients A i (x; κ), A k (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 Also, when [u] is a constant multiple of sin(πu/ω 1 ) or u, we always have We now try to find a function Φ(x; y) satisfying the system of first-order difference equations 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 The compatibility condition for the system (3.4) of difference equations is given by 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: In the first case, the difference equation to be solved is: This system is solved by Hence we see that Φ − A (x; y|δ, κ) satisfies the functional equation is solved by Hence, under the balancing condition mκ + nδ = 0, we have This completes the proof of Theorem 2.1. When [u] is a constant multiple of sin(πu/ω 1 ) or u, for any solution Φ(x; y) of the system (3.4) of difference equations we have without imposing the balancing condition. This implies that the functions Φ A (x; y|δ, κ) and Ψ A (x; y) in these trigonometric and rational cases satisfy the functional equations respectively, as stated in Theorem 2.2.

Case of type BC
In this BC case, we assume that [u] is a constant multiple of one of the following functions: In order to discuss difference operators of type BC, we consider the meromorphic function (3.5) By Proposition 3.1 it can be expanded as into partial fractions, where Also by Lemma 3.1, we see that expression 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.
Proposition 3.2. When v = 1 2 (δ − λ), the function F (z) defined by (3.5) is expressed as follows in terms of the coefficients of Ruijsenaars operators: Proof . The expansion coefficients in (3.6) are determined from the residues at the corresponding poles. We first remark that , we obtain Note that P − i is obtained from P + i by replacing x j with −x j (j = 1, . . . , m). We next look at the coefficient Q + k : In view of by the quasi-periodicity of [u]. In this way y variables disappear from R r : .
Note that the exponential factor e(nλη r ) is nontrivial only in the elliptic case. In any case, from Finally, when v = 1 2 (δ − λ) and τ = κ + λ − δ, we can rewrite S r as .

Difference 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/ω 1 ) or u. We rewrite our results on kernel functions for these cases, in terms of difference operator of Koornwinder type. We remark that this operator has symmetry x with respect to the sign change, as in the case of E (2) The two difference operators D (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 with ν s = 1 2 (δ + κ) − µ s (s = 1, . . . , 2ρ), since Φ BC (x; y) in this case is the constant function 1. Also from the case m = n = 0 we have Combining these two formulas we obtain Let us rewrite the functional equations in Proposition 3.3 in terms of the operator D (µ|δ,κ) x .
In the notation of Proposition 3.3, (2) we have By taking the sum of the four formulas in (3.9) and (3.10), we obtain 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/ω 1 ) = e(u/2ω 1 ) − e(−u/2ω 1 ), we have a simple expression Hence, under the assumption of Proposition 3.3, (2), we have
In this convention of the trigonometric case, the Ruijsenaars difference operator D (δ,κ) x of type A is a constant multiple of the q-difference operator of Macdonald: where T q,z i denotes the q-shift operator with respect to z i : In what follows, we use the gamma function G − (u|δ) of Section 2, (2.3). Then our kernel function Φ A (x; y|δ, κ) with parameter v = κ is expressed as follows in terms of multiplicative variables: The functional equation of Theorem 2.2, (1) thus implies which can also be proved by the expansion formula 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 b λ (q, t).) On the other hand, kernel function Ψ A (x; y) with parameter v = 0 is expressed as Then the functional equation of Theorem 2.2, (2) implies This formula corresponds to the dual Cauchy formula 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 8], Kajihara-Noumi [5]) and integral representation , 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].
These four parameters are the Askey -Wilson parameters (a, b, c, d) for the Koornwinder polynomials P λ (z; a, b, c, d|q, t).
In this trigonometric BC case, the difference operator 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 in the multiplicative variables, where the coefficients A + i (z) = A + i (z; a, b, c, d|q, t) are given by . Note that this operator D (a,b,c,d|q,t) z is renormalized by dividing the one used in [10] by the factor (abcdq −1 ) In what follows, we simply suppress the dependence on the parameters (a, b, c, d|q, t) as D z = D (a,b,c,d|q,t) z , 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, and 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): . 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; w) = Ψ BC (x; y) of dual Cauchy type is given by which is precisely the kernel function introduced by Mimachi [15]. For the passage from additive variables to multiplicative variables, we introduce the multiplicative notation for the function [u]: For z = e(u/ω 1 ), we write z = [u]. Namely, we set with the square root z 1 2 regarded as the multiplicative notation for e(u/2ω 1 ). This function z is a natural object to be used in the case of BC type, because of the symmetry z −1 = − z . In this notation, the coefficients A + i (z) of the Koornwinder operator D z are expressed simply as It should be noted also that our parameter c (µ|δ,κ) = 4 s=1 µ s − (δ + κ) + ω 1 passes to multiplicative variables as with a minus sign. Then, Theorem 2.5 can be restated as follows.
(1) The function Φ(z; w|q, t) defined as above satisfies the functional equation where D w denotes the Koornwinder operator in w variables with parameters (a, b, c, d) (2) The function Ψ(z; w) defined as (4.5) satisfies the functional equation (−1) |λ * | P λ (z; a, b, c, d|q, t)P λ * (w; a, b, c, d|t, q) for Koornwinder polynomials, where the summation is taken over all partitions λ = (λ 1 , . . . , λ m ) contained in the m × n rectangle, and λ * = (m − λ ′ n , . . . , m − λ ′ 1 ). By this formula, he also constructed an integral representation of Selberg type for Koornwinder polynomials attached to rectangles (n m ) (n = 0, 1, 2, . . .). We expect that our kernel function Φ(z; w|q, t) of Cauchy type could be applied as well to the study of eigenfunctions of the q-difference operators of Koornwinder. As a first step of such applications, in Section 5 we construct explicit formulas for Koornwinder polynomials attached to single columns and single rows.

Application to Koornwinder polynomials
In this section, we apply our results on the kernel functions for Koornwinder operators to the study of Koornwinder polynomials. In particular, we present new explicit formulas for Koornwinder polynomials attached to single columns and single rows.
To be more precise, we make use of the kernel functions to express Koornwinder polynomials P (1 r ) (z; a, b, c, d|q, t) (r = 0, 1, . . . , m) and P (l) (z; a, b, c, d|q, t) (l = 0, 1, 2, . . .) in terms of certain explicitly defined Laurent polynomials E r (z; a|t) and H l (z; a|q, t), respectively (Theorems 5.1 and 5.2). We remark that these Laurent polynomials E r (z; a|t) are H l (z; a|q, t) are in fact constant multiples of the BC m interpolation polynomials of Okounkov [18] attached to the partitions (1 r ) and (l), respectively. (This fact will be proved in Appendix C.) Namely, Theorems 5.1 and 5.2 provide with two special cases of the binomial expansion of the Koornwinder polynomials in terms of BC m interpolation polynomials as is discussed in Okounkov [18] and Rains [19].
Once we establish the fact that E r (z; a|t) and H l (z; a|q, t) are interpolation polynomials, Theorems 5.1 and 5.2 can also be obtained from Okounkov's binomial formula [18], together with Rains' explicit evaluation of the binomial coefficients for the cases of (1 r ) and (l) [19] 1 .
Before starting the discussion of Koornwinder polynomials, we introduce a notation in additive variables. This expression, which appears frequently in the discussion of type BC, deserves a special attention. Note that as clearly seen by the definition. Also, the Riemann relation for [u] can be written as

Koornwinder polynomials
We briefly recall some basic facts about Koornwinder polynomials; for details, see Stokman [24] for example. Let K = Q a where ≤ stands for the dominance ordering of partitions.
(2) P λ (z) is an eigenfunction of Koornwinder's q-difference operator D z : These polynomials P λ (z), indexed by partitions λ, form a K-basis of the ring of W -invariants K[z ±1 ] W . Also, the eigenvalues d λ are given by We give below new explicit formulas for Koornwinder polynomials P (1 r ) (z) attached to single columns (1 r ) (r = 0, 1, . . . , m), and P (r) (z) attached to single rows (r = 0, 1, 2 . . .). As we already mentioned, our explicit formulas provide with the two special cases of Okounkov's binomial expansion of the Koornwinder polynomials in terms of BC m interpolation polynomials. We also remark that, in the cases of type B, C, D, some conjectures have been proposed by Lassalle [11] on explicit formulas for Macdonald polynomials attached to single rows. The relationship between his conjectures and our approach will be discussed in a separate paper.
In order to formulate our results, we define a set of W -invariant Laurent polynomials E r (z; a) with reference point a (r = 0, 1, . . . , m) by As we will see below, these Laurent polynomials are W -invariant in spite of their appearance, and they can be considered as a variation of the orbit sums m (1 r ) (z) (r = 0, 1, . . . , m) attached to the fundamental weights. In fact, these Laurent polynomials E r (z; a|t) (r = 0, 1, . . . , m) are essentially the BC m interpolation polynomials of Okounkov attached to single columns (1 r ) (for a proof, see Appendix C). We remark that these polynomials had appeared already in the work of van Diejen [3] in relation to the eigenvalues of his commuting family q-difference operators for this BC m case. They are also used effectively by a recent work of Aomoto-Ito [1] in their study of Jackson integrals of type BC. t m−r+1 , t m−r ab, t m−r ac, t m−r ad t,l t, t 2(m−r) abcd t,l E r−l (z; a|t), where a t,l = a ta · · · t l−1 a , and a 1 , . . . , a r t,l = a 1 t,l · · · a r t,l .
By using a t,l = (−1) l t − 1 2 l 2 a − l 2 (a; t) l , formula (5.2) can be rewritten as follows in terms of ordinary t-shifted factorials of [4]: (t m−r+1 , t m−r ab, t m−r ac, t m−r ad; t) l t l 2 +(m−r)l a l (t, t 2(m−r) abcd; t) l E r−l (z; a|t).
Note that z; a q,l is a monic Laurent polynomial in z of degree l. (The W -invariance of H l (z; a|q, t) will be proved in Lemma 5.4 below.) These Laurent polynomials H l (z; a|q, t) can be regarded as a BC m analogue of the A m−1 Macdonald polynomials attached to single rows: Also, they are special cases of BC m interpolation polynomials attached to single rows (l) (see Appendix C). t q,r q q,r P (r) (z; a, b, c, d|q, t) = t m , t m−1 ab, t m−1 ac, t m−1 ad q,r q, t 2(m−1) abcdq r−1 q,r × r l=0 (−1) l q −r , t 2(m−1) abcdq r−1 q,l t m , t m−1 ab, t m−1 ac, t m−1 ad q,l H l (z; a|q, t).

Case of a single column
We first explain some properties of the elementary Laurent polynomials E r (z; a|t) (r = 0, 1, . . . , m). where w; a t,l = w; a w; ta · · · w; t l−1 a . In particular, E r (z; a|t) is W -invariant for each r = 0, 1, . . . , m.
We omit the proof of this lemma, since it can be derived as a special case of the connection formula for Askey-Wilson polynomials with different parameters (see [4]). Note that, if we set d = t 1−l /a in (5.8), then p l (w; a, b, c, t 1−l /a|t) = w; a t,l .
Comparing this formula with (5.6) we obtain t m−r+1 , t m−r ab, t m−r ac, t m−r ad r−l t, abcdt 2(m−r) t,r−l E l (z; a|t), as desired.

Case of a single row
Recall that the kernel function of Cauchy type defined in (4.4), satisfies the difference equation where D w denotes the Koornwinder operator in w variables with parameters (a, b, c, d) replaced by ( √ qt/a, √ qt/b, √ qt/c, √ qt/d|q, t). We set hereafter Also, for any Laurent polynomial f (z) ∈ K[z ±1 ], we denote by f (z) ∈ K[z ±1 ] the Laurent polynomial obtained from f (z) by replacing the parameters (a, b, c, d) with ( a, b, c, d).
Let us consider the special case where t = q −k (k = 0, 1, 2, . . .). Then the kernel function Φ(z; w|q, q −k ) reduces to a Laurent polynomial in (z, w): Ignoring the sign factor, we set Note that In what follows, we analyze the case where n = 1 and t = q −k (k = 0, 1, 2, . . .). In this case, the kernel function is a symmetric Laurent polynomial in w of degree km. Also, this kernel function satisfies the functional equation With α = (abcdq −1 ) 1 2 , this formula can be written as Noting that Φ −k (z; w) is a symmetric Laurent polynomial, we expand this kernel in terms of the monic Askey-Wilson polynomials p l (w|q) = p l (w; a, b, c, d|q) in w with the twisted parameters, so that The Laurent polynomials G l (z) (0 ≤ l ≤ km) are uniquely determined by this expansion, and hence W -invariant.
As we have seen above, each G l (z) (l = 0, 1, . . . , km) is an eigenfunction of D z with precisely the same eigenvalue as the one for the Koornwinder polynomial P (l) (z) attached to the single row of length l. At this moment, however, we cannot conclude this G l (z) is indeed a constant multiple of the Koornwinder polynomial P (l) (z) specialized to the case t = q −k . This is because different partitions λ may give the same eigenvalue under this specialization. This point will be discussed later after we determine an explicit formula for G l (z).
Since we already know the relationship between the Askey-Wilson polynomials p l (w|q) and the Laurent polynomials w; a q,l , we consider to expand the kernel function Φ −k (z; w) in terms of w; a q,l .
Proof . Statement (2) follows from the expansion formula (5.11) of statement (1). Since Φ −k (z; w) is W -invariant in the z variables, formula (5.11) implies that H l (z; a|q, q −k ) is Winvariant for k ≥ l/m. Hence we see that H l (z; a|q, t) itself is W -invariant as a Laurent polynomial in K[z ± ] for each l = 0, 1, 2, . . .. In the following proof of statement (1), we omit the base q, and write w; a l = w; a q,l . We first present a connection formula for the Laurent polynomials w; a l and w; b l with different reference point a, b: w; b l = l r=0 (−1) r l r q l−r ab, b/a r w; a l−r , l r = (−1) r q −l r q r , which is equivalent to the q-Saalschütz sum [4] (bw, b/w; q) l (ba, b/a; q) l = 3 φ 2 q −l , aw, a/w ab, q 1−l a/b ; q, q . This implies Then in each term, we expand w; q By repeating this procedure, we finally obtain m j=1 w; q /a ν j w; a |µ|−|ν| for any µ = (µ 1 , . . . , µ m ), where the sum is taken over all multi-indices ν = (ν 1 , . . . , ν m ) such that ν i ≤ µ i (i = 1, . . . , m). As a special case of this formula where µ 1 = · · · = µ m = k, |µ| = km, we get the expansion formula Here the coefficients are determined as Replacing the parameters a by a = √ qt/a, we obtain We now have two expansions of the kernel function Φ −k (z; w): Also, from Lemma 5.2 we see w; a q,l = l r=0 (−1) l−r q r+1 , q r ab, q r ac, q r ad q,l−r q, abcdq 2r q,l−r p r (w|q), and hence, w; √ qt/a q,l = l r=0 (−1) l−r q r+1 , tq r+1 /ab, tq r+1 /ac, tq r+1 /ad q,l−r q, t 2 q 2(r+1) /abcd q,l−r p r (w|q) for l = 0, 1, 2, . . .. Substituting this into (5.12), we obtain an expression of G r (z) in term of H l (z) as follows: (−1) r−l q km−r+1 , tq km−r+1 /ab, tq km−r+1 /ac, tq km−r+1 /ad q,r−l q, t 2 q 2(km−r+1) /abcd q,r−l H l (z) From the expression obtained above, it is clear that H l (z) = t q,l q q,l m (l) (z) + terms lower than (l) with respect to ≤, and G r (z) = t q,r q q,r m (r) (z) + terms lower than (r) with respect to ≤.
Note that t q,r = q −k q,r = 0 for 0 ≤ r ≤ k. Also, we already know that each G r (z) (r = 0, 1, . . . , km) satisfies the difference equation Suppose in general that a partition λ = (λ 1 , . . . , λ m ) satisfies the condition λ i − λ i+1 ≤ k (i = 1, . . . , m − 1). Since in this case, it turns out that the eigenvalue d µ for any partition µ < λ is distinct from d λ when the square roots of a, b, c, d, q are regarded as indeterminates. For such a partition λ, the Koornwinder polynomial P λ (z) = P λ (z; a, b, c, d|q, t) can be specialized to t = q −k , and any eigenfunction having the nontrivial leading term m λ (z) must be a constant multiple of P λ (z; a, b, c, d|q, q −k ). This implies that, for each r with 0 ≤ r ≤ k, G r (z) is a constant multiple of P (r) (z) specialized to t = q −k : For each r = 0, 1, 2, . . ., consider the Laurent polynomial in K[z ±1 ] defined by right-hand side of the explicit formula (5.4) of Theorem 5.2. Then the both sides of (5.4) are regular at t = q −k (k ≥ r), and they coincide with each other for t = q −k (k = r, r + 1, . . .). Hence the both sides must be identical as rational functions in t
In the works of Ruijsenaars [23], the two periods ω 1 , ω 2 and the scaling constant δ are parametrized as in view of the symmetry between ω 2 and δ (or p and q). In terms of the R-function R(x) = "R(r, a + ; x)" = θ p Note that R(−x) = R(x). The elliptic gamma function G(x) = "G(r, a + , a − ; x)" = Γ p The function G(x) is symmetric with respect to ω 2 and δ (p and q), and satisfies the functional equation B.1 Case of type A As in (A.1), we consider the commuting family of difference operators D (δ,κ) r,x (r = 1, . . . , m) of type A in the variables x = (x 1 , . . . , x m ). These operators are in fact identical to the difference operator A r,+ (−x) of [23, I, (2.1)], up to multiplication by constants: under the identification of the parameter "µ" = κ; we denote below this operator by A r,−x = "A r,+ (−x)". In the multiplicative variables z i = e(x i /ω 1 ) (i = 1, . . . , m), t = e(κ/ω 1 ), the same operator is expressed as The kernel function Φ A (x; y|δ, κ) of (2.5) for the m = n case can be expressed in terms of G(x) as Γ(pcz j w l /t; p, q) Γ(pcz j w l ; p, q) (c = e(v/ω 1 )).
The constants in front of the both sides simplifies by (B.6), to imply the functional equation C E r (z; a|t) and H l (z; a|q, t) as interpolation polynomials In Section 5 we presented some explicit expansion formulas for Koornwinder polynomials attached to single columns and to single rows, in terms of invariant Laurent polynomials E r (z; a|t) and H l (z; a|q, t), respectively. These Laurent polynomials E r (z; a|t) and H l (z; a|q, t) are essentially the same objects as the BC m -interpolation polynomials of Okounkov [18] attached to single columns and single rows. Our Theorems 5.1 and 5.2 provides with explicit expressions for the corresponding special cases of the binomial expansion of Koornwinder polynomials in terms of BC m -interpolation polynomials as discussed in Okounkov [18] and Rains [19].
In this section we show that the Laurent polynomials E r (zt δ a; a|t) and H l (zt δ a; a|q, t) coincide, up to constant multiplication, with the interpolation polynomials P * (1 r ) (z; q, t, a) and P * (l) (z; q, t, a), respectively. We prove that these polynomials actually have the interpolation properties as mentioned above.