Séminaire Lotharingien de Combinatoire 72 (2015), Article B72a

OKOUNKOV’S BC-TYPE INTERPOLATION MACDONALD POLYNOMIALS AND THEIR q = 1 LIMIT

TOM H. KOORNWINDER

Abstract. This paper surveys eight classes of polynomials associated with A-type and BC-type root systems: Jack, Jacobi, Macdonald and Koornwinder polynomials and interpolation (or shifted) Jack and Macdonald polynomials and their BC-type extensions. Among these the BC-type interpolation Jack polynomials were probably unobserved until now. Much emphasis is put on combinatorial formulas and binomial formulas for (most of) these polynomials. Possibly new results derived from these formulas are a limit from Koornwinder to Macdonald polynomials, an explicit formula for Koornwinder polynomials in two variables, and a combinatorial expression for the coefficients of the expansion of BC-type Jacobi polynomials in terms of Jack polynomials which is different from Macdonald’s combinatorial expression. For these last coefficients in the two-variable case the explicit expression of Koornwinder and Sprinkhuizen [SIAM J. Math. Anal. 9 (1978), 457–483] is now obtained in a quite different way.

1. Introduction

In the past half century special functions associated with root systems became an active area of research with many interconnections and applications. The early results were strongly motivated by the notion of spherical functions on Riemannian symmetric spaces. An ambitious program, which still has not come to an end, started to do “zonal spherical analysis” without underlying group and for a wider parameter range than the discrete set of parameter values for which a group theoretic interpretation is possible. Another motivation came from applications in multivariate statistics. By the end of the eighties of the past century Heckman and Opdam consolidated the theory of Jacobi polynomials associated with root systems. In the same period Macdonald, in his annus mirabilis 1987, introduced the q-analogues of these Jacobi polynomials in several manuscripts which were circulated and eventually published: Macdonald polynomials P_{λ}(x; q, t) (associated with A-type root systems) in [17] and [18, Ch. VI], Macdonald polynomials associated with root systems in [19], and scratch notes about hypergeometric functions (associated with BC-type root systems) in [21]. Again in the same period Dunkl introduced his Dunkl operators, which inspired Heckman, Opdam and in particular Cherednik to consider the Weyl group invariant (W -invariant) special functions as part of a more general theory of non-symmetric special functions which are eigenfunctions of operators having a reflection term. Special representations of graded and double affine Hecke algebras (DAHA’s) were an important tool. This approach not only introduced new interesting special functions, but also greatly simplified the W -invariant theory.

E-mail: T.H.Koornwinder@uva.nl.

The author [13] introduced a 5-parameter class of q-polynomials, on the one hand extending the 3-parameter class of Macdonald polynomials associated with root system BC_{n} [19] and on the other hand providing the n-variable analogue of the Askey-Wilson polynomials [1]. These polynomials became known in the literature as Macdonald-Koornwinder or Koornwinder polynomials. Cherednik’s DAHA approach could also be used for these polynomials, see Sahi [34], [35] and Macdonald’s monograph [20]. A different approach started by work of Sahi, Knop, Okounkov and Olshanski ([32], [12], [11], [27], [24], [25], [26]). It used the so-called shifted or interpolation versions of Jack and Macdonald polynomials. These could be characterized very briefly by their vanishing property at a finite part of a (q-)lattice, they could be represented by combinatorial formulas (tableau sums) generalizing those for Jack and Macdonald polynomials, and they occurred in generalized binomial formulas. In particular, Okounkov’s [26] BC_{n} type interpolation Macdonald polynomials inspired Rains [29] to use these in the definition of Koornwinder polynomials, thus building the theory of these latter polynomials in a completely new way. An analogous approach then enabled Rains to develop a theory of elliptic analogues of Koornwinder polynomials, as surveyed in [30].

Jack and Macdonald polynomials in n variables play a double role, on the one hand as homogeneous orthogonal polynomials associated with root system A_{n−1}, on the other hand as generalized “monomials” (in the one-variable case ordinary monomials) in terms of which orthogonal polynomials associated with root system BC_{n} can be naturally expanded. This second role is emphasized in the approach using interpolation polynomials, in particular where it concerns binomial formulas.

The present paper surveys, mainly in Sections 4 and 5 and after some preliminaries in Section 3, the definition and properties of eight classes of polynomials: four associated with root system BC_{n} and four with root system A_{n−1}. Also four of these classes are for general q and four are for q = 1. Four of these classes can be considered as orthogonal polynomials while the other four (interpolation) classes only play a role as generalized monomials. There are many limit connections between these eight classes. For six of them (however, see [6] and Remark 6.1) combinatorial formulas are known, see such formulas mainly in Section 6. In a sense these combinatorial formulas are generalized hypergeometric series.

One of the eight classes, the BC_{n}-type interpolation Jack polynomials, seems to have been overlooked in the literature, although it occurs very naturally in the scheme formed by the limit connections. It will be defined in Section 7. All its properties will be obtained here as limit cases of properties of BC_{n}-type interpolation Macdonald polynomials, including the combinatorial formula for polynomials of this latter class.

OKOUNKOV’S BC-TYPE INTERPOLATION MACDONALD POLYNOMIALS 3

Binomial formulas as they were already known for three classes of polynomials are surveyed in Section 8. The probably new binomial formula for BC_{n}-type interpolation Jack polynomials is given in Section 9. It gives a new approach to coefficients of the expansion of BC_{n}-type Jacobi polynomials in terms of Jack polynomials. As a consequence of the binomial formulas a new limit formula (8.4) and a new proof of an already known limit formula (9.2) will follow.

All classes of polynomials and formulas for them become much more elementary and explicit in the one-variable case. This is the subject of the Prelude in Section 2. The two-variable case is already more challenging, but explicit formulas are feasible. This is the topic of Sections 10 and 11. In particular, in Subsection 11.2 we arrive at an explicit expression for BC_{2}-type Jacobi polynomials which was earlier obtained in a very different way by the author together with Sprinkhuizen in [14].

Acknowledgement The material of this paper was first presented in lectures given at the 72nd Séminaire Lotharingien de Combinatoire, Lyon, France, 24–26 March 2014. I thank the organizers for the invitation. I thank Siddhartha Sahi, Ole Warnaar, Genkai Zhang and an anonymous referee for helpful remarks. Thanks also to Masatoshi Noumi for making available to me his unpublished slides on interpolation functions of type BC.

Notation See [8]. Throughout we assume 0 < q < 1. (q)-shifted factorials are given by

(a)_{k} := a(a + 1) ⋯ (a + k − 1),  (a)_{0} := 1,  (a_{1}, …, a_{r})_{k}
:= (a_{1})_{k} ⋯ (a_{r})_{k};

(a; q)_{k} := (1 − a)(1 − aq) ⋯ (1 − aq^{k−1}), (a; q)_{0} := 1, (a_{1}, …, a_{r}; q)_{k} := (a_{1}; q)_{k} ⋯ (a_{r}; q)_{k} .

For n a nonnegative integer we have terminating (q-)hypergeometric series

rF_{s}({−n, a_{2}, …, a_{r} \atop b_{1}, …, b_{s}}; z) := ∑_{k=0}^{n} \frac{(−n)_{k}}{k!} \frac{(a_{2}, …, a_{r})_{k}}{(b_{1}, …, b_{s})_{k}} z^{k},

rϕ_{s}({−n, a_{2}, …, a_{r} \atop b_{1}, …, b_{s}}; q, z) := ∑_{k=0}^{n} \frac{(q^{−n}; q)_{k}}{(q; q)_{k}} \frac{(a_{2}, …, a_{r}; q)_{k}}{(b_{1}, …, b_{s}); q_{k}} ((−1)^{k}q^{\frac{1}{2} k(k−1)})^{r−s+1} z^{k}.

2. Prelude: the one-variable case

Let us explicitly consider the most simple situation, for polynomials in one variable (in this section n will denote the degree rather than the number of variables). Then both Jack and Macdonald polynomials are simple monomials x^{n}. Put

P_{n}(x) := x^{n}, P_{n}(x; q) := x^{n}, P_{k}^{ip}(x) := x(x − 1) ⋯ (x − k + 1) = (−1)^{k}(−x)_{k}.

P_{k}^{ip}(x) is the unique monic polynomial of degree k which vanishes at 0, 1, …, k − 1. A binomial formula is given by

(x + 1)^{n} = ∑_{k=0}^{n} \binom{n}{k}x^{k}, or P_{n}(x + 1) = ∑_{k=0}^{n} \frac{P_{k}^{ip}(n)}{P_{k}^{ip}(k)} P_{k}(x).  (2.1)

In the q-case put

P_{k}^{ip}(x; q) := (x − 1)(x − q) ⋯ (x − q^{k−1}) = x^{k}(x^{−1}; q)_{k}.

P_{k}^{ip}(x; q) is the unique monic polynomial of degree k which vanishes at 1, q, …, q^{k−1}. A q-binomial formula (see [8, Exercise 1.6(iii)]) is given by

x^{n} = _{2}ϕ_{0}({q^{−n}, x^{−1} \atop −}; q, q^{n}x) = ∑_{k=0}^{n} \frac{(q^{−n}, x^{−1}; q)_{k}}{(−1)^{k}q^{\frac{1}{2} k(k−1)}(q; q)_{k}} (q^{n}x)^{k}, or

P_{n}(x; q) = ∑_{k=0}^{n} \frac{P_{k}^{ip}(q^{n}; q)}{P_{k}^{ip}(q^{k}; q)} P_{k}^{ip}(x; q).  (2.2)

Identity (2.1) is the limit case for q ↑ 1 of (2.2). The polynomials P_{k}^{ip}(x) and P_{k}^{ip}(x; q) are the one-variable cases of the interpolation Jack and the interpolation Macdonald polynomials, respectively.

In the one-variable case BC_{n}-type Jacobi polynomials become classical Jacobi polynomials and Koornwinder polynomials become Askey-Wilson polynomials. Their standard expressions as (q-)hypergeometric series are:

\frac{P_{n}^{(α,β)}(1 − 2x)}{P_{n}^{(α,β)}(1)} = ∑_{k=0}^{n} \frac{(−n)_{k}(n + α + β + 1)_{k}}{(α + 1)_{k} k!} x^{k} = _{2}F_{1}({−n, n + α + β + 1 \atop α + 1}; x) (2.3)

and

\frac{p_{n}(\frac{1}{2}(x + x^{−1}); a_{1}, a_{2}, a_{3}, a_{4} | q)}{p_{n}(\frac{1}{2}(a_{1} + a_{1}^{−1}); a_{1}, a_{2}, a_{3}, a_{4} | q)} = ∑_{k=0}^{n} \frac{(q^{−n}, q^{n−1}a_{1}a_{2}a_{3}a_{4}, a_{1}x, a_{1}x^{−1}; q)_{k}}{(a_{1}a_{2}, a_{1}a_{3}, a_{1}a_{4}, q; q)_{k}} q^{k}

= _{4}ϕ_{3}({q^{−n}, q^{n−1}a_{1}a_{2}a_{3}a_{4}, a_{1}x, a_{1}x^{−1} \atop a_{1}a_{2}, a_{1}a_{3}, a_{1}a_{4}} ; q, q) . (2.4)

Note that (2.3) gives an expansion in terms of monomials P_{k}(x) = x^{k} (Jack polynomials in one variable), while (2.4) gives an expansion in terms of monic symmetric Laurent polynomials

P_{k}^{ip}(x; q, a_{1}) := ∏_{j=0}^{k−1}(x + x^{−1} − a_{1}q^{j} − a_{1}^{−1}q^{−j}) = \frac{(a_{1}x, a_{1}x^{−1}; q)_{k}}{(−1)^{k} q^{\frac{1}{2} k(k−1)} a_{1}^{k}} .  (2.5)

The monic symmetric Laurent polynomial (2.5) is characterized by its vanishing at a_{1}, a_{1}q, …, a_{1}q^{k−1}. It is the one-variable case of Okounkov’s BC-type interpolation Macdonald polynomial. If we consider (2.4) as an expansion of its left-hand side as a function of n, then we see that it is expanded in terms of functions P_{k}^{ip}(q^{n}a′_{1}; q, a′_{1}) (using the definition in (2.5)), where a′_{1} := (q^{−1}a_{1}a_{2}a_{3}a_{4})^{\frac{1}{2}} . Furthermore, if we replace x by a_{1}x in (2.5), divide by a_{1}^{k}, and let a_{1} → ∞, then we obtain the q-binomial formula (2.2). Therefore, Okounkov [26] calls (2.5), as well as its multi-variable analogue, also a binomial formula.

If we replace in (2.4) the parameters a_{1}, a_{2}, a_{3}, a_{4} by q^{α+1}, −q^{β+1}, 1, −1 and let q ↑ 1, then we arrive at (2.3), which therefore might also be called a binomial formula. If we consider (2.3) as an expansion of its left-hand side as a function of n, then we see that it is expanded in terms of functions P_{k}^{ip}(n + α′; α′), where α′ := \frac{1}{2}(α + β + 1) and

P_{k}^{ip}(x; α) := ∏_{j=0}^{k−1} (x^{2} − (α + j))^{2} = (−1)^{k} (α − x)_{k} (α + x)_{k} ,  (2.6)

a monic even polynomial of degree 2k in x which is characterized by its vanishing at α, α + 1, …, α + k − 1. This is the one-variable case of the BC-type interpolation Jack polynomial, which (possibly for the first time) will be defined in the present paper.

3. Preliminaries

3.1. Partitions. We recapitulate some notions about partitions, diagrams and tableaux from Macdonald [18, § I.1]. However, in contrast to [18], we fix an integer n ≥ 1 and always understand a partition λ to be of length ≤ n, i.e., λ = (λ_{1}, …, λ_{n}) ∈ ℤ^{n} with λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{n} ≥ 0. Write ℓ(λ) := |{j | λ_{j} > 0}| for the length of λ and |λ| := λ_{1} + ⋯ + λ_{n} for its weight. Also put

n(λ) := ∑_{i=1}^{n}(i − 1)λ_{i}.  (3.1)

We may abbreviate k parts of λ equal to m by m^{k} and we may omit 0^{k} at the end. For instance, (2, 2, 1, 0, 0, 0) = (2^{2}, 1, 0^{3}) = (2^{2}, 1). There is the special partition

δ := (n − 1, n − 2, …, 1, 0).  (3.2)

A partition λ can be displayed by a Young diagram, also notated by λ, which consists of boxes (i, j) with i = 1, …, ℓ(λ) and j = 1, …, λ_{i} for a given i. The conjugate partition λ′ has diagram such that (j, i) ∈ λ′ if and only if (i, j) ∈ λ The example below of the diagram of λ = (7, 5, 5, 2, 2) and its conjugate λ′ = (5, 5, 3, 3, 3, 1, 1) will make clear how a diagram is drawn:

For (i, j) a box of a partition λ, the arm-length a_{λ}(i, j) and leg-length l_{λ}(i, j) are defined by

a_{λ}(i, j) := λ_{i} − j, l_{λ}(i, j) := |{k > i | λ_{k} ≥ j}|.

Also the arm-colength a′_{λ}(i, j) and leg-colength l′_{λ}(i, j) are defined by

a′_{λ}(i, j) := j − 1, l′_{λ}(i, j) := i − 1.

The dominance partial ordering ≤ and the inclusion partial ordering ⊆ are defined by

μ ≤ λ if and only if μ_{1} + ⋯ + μ_{i} ≤ λ_{1} + ⋯ + λ_{i} (i = 1, …, n);
μ ⊆ λ if and only if μ_{i} ≤ λ_{i} (i = 1, …, n).

Clearly, if μ ⊆ λ then μ ≤ λ, while μ < λ implies that μ is less than λ in the lexicographic ordering. If μ ⊆ λ then we say that λ contains μ. Note that, for the dominance partial ordering, we do not make the usual requirement that |λ| = |μ|.

For μ ⊆ λ define the skew diagram λ − μ as the set of boxes {s ∈ λ | s ∉ μ}. A horizontal strip is a skew diagram with at most one box in each column.

For a horizontal strip λ − μ define (R\C)_{λ/μ} as the set of boxes which are in a row of λ intersecting with λ − μ but not in a column of λ intersecting with λ − μ. Then clearly (R\C)_{λ/μ} is completely contained in μ. For an example consider again λ = (7, 5, 5, 2, 2) and take μ = (5, 5, 3, 2, 1). In the following diagram the cells of λ−μ have black squares and the cells of (R\C)_{λ/μ} have black diamonds.

♦ ♦ ■ ■
♦ ♦ ■ ■
♦ ■

3.2. Tableaux. For λ a partition (of length ≤ n) we can fill the boxes s of λ by numbers T (s) ∈ {1, 2, …, n}. Then T is called a reverse tableau of shape λ with entries in {1, …, n} if T (i, j) is weakly decreasing in j and strongly decreasing in i. (Clearly, the number of different entries has to be ≥ ℓ(λ). In [18, § I.1] tableaux rather than reverse tableaux are defined.) For an example consider again λ = (7, 5, 5, 2, 2), so ℓ(λ) = 5. Let us take n = 6. Then an example of a reverse tableau T of shape λ is given by

6 6 6 4 3 1 1 5 5 5 2 2 4 4 2 1 1 3 2 2 1

For T of shape λ and for k = 0, 1, …, n let λ^{(k)} be the partition of which the Young tableau consists of all s ∈ λ such that T (s) > k. Thus

0^{n} = λ^{(n)} ⊆ λ^{n−1} ⊆ ⋯ ⊆ λ^{(1)} ⊆ λ^{(0)} = λ.  (3.3)

Then the skew diagram λ^{(k−1)} − λ^{(k)} is actually a horizontal strip and it consists of all boxes s with T (s) = k. We call the sequence (μ_{1}, …, μ_{n}) with μ_{k} := |λ^{(k−1)} − λ^{(k}| = |T^{−1}({k}) the weight of T. For λ and T as in the example the inclusion sequence (3.3) becomes

() ⊆ (3) ⊆ (3, 3) ⊆ (4, 3, 2) ⊆ (5, 3, 2, 1) ⊆ (5, 5, 3, 2, 1) ⊆ (7, 5, 5, 2, 2).

and T has weight (5, 5, 2, 3, 3, 3).

For a skew diagram λ − μ a standard tableau T of shape λ − μ puts T (s) in box s of λ − μ such that each number in {1, …, |λ − μ|} occurs and T (s) is strictly increasing in each row and in each column.

3.3. Symmetrized monomials. Write x^{μ} := x_{1}^{μ_{1}} ⋯ x_{n}^{μ_{n}} for μ ∈ ℤ^{n}. We say that x^{μ} has degree |μ| := μ_{1} + ⋯ + μ_{n}. By the degree of a Laurent polynomial p(x) we mean the highest degree of a monomial occurring in the Laurent expansion of p(x).

Let S_{n} be the symmetric group in n letters and W_{n} := S_{n} ⋉ (ℤ_{2})^{n}. For λ a partition and x = (x_{1}, …, x_{n}) ∈ ℂ^{n} put

m_{λ}(x) := ∑_{μ∈S_{n}λ} x^{μ},  m̃_{λ}(x) := ∑_{μ∈W_{n}λ} x^{μ}.  (3.4)

They form a basis of the space of S_{n}-invariant polynomials (respectively, W_{n}-invariant Laurent polynomials) in x_{1}, …, x_{n}. Call an S_{n}-invariant polynomial (respectively, W_{n}- invariant Laurent polynomial) of degree |λ| λ-monic if its coefficient of m_{λ} (respectively, of m̃_{λ}) is equal to 1.

4. Macdonald and Koornwinder polynomials and q = 1 limits

From now on n will be the number of variables and we will assume n > 1.

4.1. Macdonald polynomials. See Eqs. (4.7), (9.3), (9.5) and the Remark on p. 372 in Ch. VI in Macdonald [18].

Let 0 < t < 1. If x = (x_{1}, …, x_{n}) with x_{j} ≠ 0 for all j then write x^{−1} := (x_{1}^{−1}, …, x_{n}^{−1}). Put

Δ_{+}(x) = Δ_{+}(x; q, t) := ∏_{1≤i<j≤n} \frac{(x_{i}x_{j}^{−1}; q)_{∞}}{(tx_{i}x_{j}^{−1}; q)_{∞}} , Δ(x) := Δ_{+}(x)Δ_{+}(x^{−1}).

Macdonald polynomials (for root system A_{n−1}) are λ-monic S_{n}-invariant polynomials

P_{λ}(x; q, t) = P_{λ}(x) = ∑_{μ≤λ} u_{λ,μ}m_{μ}(x)  (4.1)

such that (with q, t-dependence of P_{λ} and Δ understood)

∫_{𝕋^{n}} P_{λ}(x) m_{μ}(x^{−1}) Δ(x) \frac{dx_{1}}{x_{1}} ⋯ \frac{dx_{n}}{x_{n}} = 0 if μ < λ.  (4.2)

Here 𝕋^{n} is the n-torus in ℂ^{n}. It follows from (4.2) that

∫_{𝕋^{n}} P_{λ}(x) P_{μ}(x^{−1}) Δ(x) \frac{dx_{1}}{x_{1}} ⋯ \frac{dx_{n}}{x_{n}} = 0  (4.3)

if μ < λ, and that P_{λ} is homogeneous of degree |λ|. In fact, it can be shown that the orthogonality (4.3) holds for λ ≠ μ. This deeper and very important result will also be met for the three other orthogonal families discussed below.

Macdonald polynomials can be explicitly evaluated in a special point (see [18, Ch. VI, Eq. (6.11)]):

P_{λ}(t^{δ}; q, t) = t^{n(λ)} \frac{Δ_{+}(q^{λ}t^{δ}; q, t)}{Δ_{+}(t^{δ}; q, t)} = t^{n(λ)} ∏_{1≤i<j≤n} \frac{(t^{j−i+1}; q)_{λ_{i}−λ_{j}}}{(t^{j−i}; q)_{λ_{i}−λ_{j}}} .  (4.4)

There is also the duality result (see [18, Ch. VI, Eq. (6.6)]):

\frac{P_{λ}(q^{ν}t^{δ}; q, t)}{P_{λ}(t^{δ}; q, t)} = \frac{P_{ν}(q^{λ}t^{δ}; q, t)}{P_{ν}(t^{δ}; q, t)} .  (4.5)

4.2. Jack polynomials. See Eqs. (10.13), (10.14), (10.35) and (10.36) in Ch. VI in Macdonald [18], and see Stanley [36].

Let τ > 0. Put

Δ_{+}(x) = Δ_{+}(x; τ) := ∏_{1≤i<j≤n}(1 − x_{i}x_{j}^{−1})^{τ},  Δ(x) := Δ_{+}(x)Δ_{+}(x^{−1}).  (4.6)

Jack polynomials are λ-monic S_{n}-invariant polynomials

P_{λ}(x; τ) = P_{λ}(x) = ∑_{μ≤λ} u_{λ,μ}m_{μ}(x)

satisfying (4.2) with Δ given by (4.6). Hence they satisfy (4.3) if μ < λ, and P_{λ} is homogeneous of degree |λ|. In fact, it can be shown that they satisfy (4.3) for λ ≠ μ.

Jack polynomials are limits of Macdonald polynomials:

lim_{q↑1} P_{λ}(x; q, q^{τ}) = P_{λ}(x; τ).  (4.7)

Our notation of Jack polynomials relates to Macdonald’s notation by P_{λ}(x; τ) = P_{λ}^{(τ^{−1})}(x). Alternatively, [18, Ch. VI, Eq. (10.22)] and [36, Theorem 1.1] work with J_{λ}^{(α)}(x) = J_{λ}(x; α), respectively. Then (Eqs. (10.22) and (10.21) in [18, Ch. VI])

J_{λ}^{(α)} = c_{λ}(α) P_{λ}^{(α)}, c_{λ}(α) = ∏_{s∈λ}(αa(s) + l(s) + 1).

We have the evaluation (see [36, Theorem 5.4])

P_{λ}(1^{n}; τ) = ∏_{1≤i<j≤n} \frac{((j − i + 1)τ)_{λ_{i}−λ_{j}}}{((j − i)τ)_{λ_{i}−λ_{j}}} .  (4.8)

The following limit is formally suggested by (4.7):

lim_{q↑1} P_{λ}(q^{τδ}; q, q^{τ}) = P_{λ}(1^{n}; τ).  (4.9)

It follows rigorously by comparing (4.4) and (4.8).

4.3. Koornwinder polynomials. See Koornwinder [13].

Let |a_{1}|, |a_{2}|, |a_{3}|, |a_{4}| ≤ 1 such that a_{i}a_{j} ≠ 1 if i ≠ j, and such that non-real a_{j} occur in complex conjugate pairs. Let 0 < t < 1. Put

Δ_{+}(x) = Δ_{+}(x; q, t; a_{1}, a_{2}, a_{3}, a_{4})

:= ∏_{j=1}^{n} \frac{(x_{j}^{2}; q)_{∞}}{(a_{1}x_{j}, a_{2}x_{j}, a_{3}x_{j}, a_{4}x_{j}; q)_{∞}} ∏_{1≤i<j≤n} \frac{(x_{i}x_{j}, x_{i}x_{j}^{−1}; q)_{∞}}{(tx_{i}x_{j}, tx_{i}x_{j}^{−1}; q)_{∞}} .

Put Δ(x) := Δ_{+}(x)Δ_{+}(x^{−1}). Koornwinder polynomials are λ-monic W_{n}-invariant Laurent polynomials

P_{λ}(x; q, t; a_{1}, a_{2}, a_{3}, a_{4}) = P_{λ}(x) = ∑_{μ≤λ} u_{λ,μ}m̃_{μ}(x)  (4.10)

such that

∫_{𝕋^{n}} P_{λ}(x) m̃_{μ}(x) Δ(x) \frac{dx_{1}}{x_{1}} ⋯ \frac{dx_{n}}{x_{n}} = 0 if μ < λ.  (4.11)

It follows from (4.11) that

∫_{𝕋^{n}} P_{λ}(x) P_{μ}(x) Δ(x) \frac{dx_{1}}{x_{1}} ⋯ \frac{dx_{n}}{x_{n}} = 0  (4.12)

if μ < λ, and that P_{λ} is symmetric in a_{1}, a_{2}, a_{3}, a_{4}. In fact, it can be shown that the orthogonality (4.12) holds for λ ≠ μ. Koornwinder polynomials are a 5-parameter generalization of Macdonald’s [19] 3-parameter q-polynomials associated with root system BC_{n}.

Van Diejen [3, § 5.2] showed that the Macdonald polynomial P_{λ}(x; q, t) is the term of highest degree |λ| of P_{λ}(x; q, t; a_{1}, a_{2}, a_{3}, a_{4}):

lim_{r→∞} r^{−|λ|}P_{λ}(rx; q, t; a_{1}, a_{2}, a_{3}, a_{4}) = P_{λ}(x; q, t).  (4.13)

For the following result we will need dual parameters a′_{1}, a′_{2}, a′_{3}, a′_{4}:

a′_{1} := (q^{−1}a_{1}a_{2}a_{3}a_{4})^{\frac{1}{2}} , a′_{1}a′_{2} = a_{1}a_{2}, a′_{1}a′_{3} = a_{1}a_{3}, a′_{1}a′_{4} = a_{1}a_{4}.  (4.14)

Below the ambiguity in taking a square root will cause no harm because Δ_{+} is invariant under the transformation (x, a_{1}, a_{2}, a_{3}, a_{4}) → (−x_{1}, −a_{1}, −a_{2}, −a_{3}, −a_{4}), by which P_{λ} will also have this invariance, up to a factor (−1)^{|λ|}.

An evaluation formula for Koornwinder polynomials was conjectured by Macdonald (1991, unpublished; see [4, Eq. (5.5)]). It reads

P_{λ}(t^{δ}a_{1}; q, t; a_{1},a_{2}, a_{3}, a_{4}) = t^{−⟨λ,δ⟩}a_{1}^{−|λ|} \frac{Δ_{+}(q^{λ}t^{δ}a′_{1}; q, t; a′_{1}, a′_{2}, a′_{3}, a′_{4})}{Δ_{+}(t^{δ}a′_{1}; q, t; a′_{1}, a′_{2}, a′_{3}, a′_{4})}

= t^{−⟨λ,δ⟩}a_{1}^{−|λ|} ∏_{j=1}^{n} \frac{(t^{n−j}a_{1}^{′2}; q)_{λ_{j}}}{(t^{2n−2j}a_{1}^{′2}; q)_{2λ_{j}}} (t^{n−j}a_{1}a_{2}, t^{n−j}a_{1}a_{3}, t^{n−j}a_{1}a_{4}; q)_{λ_{j}}

∏_{1≤i<j≤n} \frac{(t^{2n−i−j+1}a_{1}^{′2}; q)_{λ_{i}+λ_{j}}}{(t^{2n−i−j}a_{1}^{′2}; q)_{λ_{i}+λ_{j}}} \frac{(t^{j−i+1}; q)_{λ_{i}−λ_{j}}}{(t^{j−i}; q)_{λ_{i}−λ_{j}}} .  (4.15)

It was proved by van Diejen [4, Eq. (5.5)] in the self-dual case a_{1} = a′_{1}. In that case he also proved [4, Eq. (5.4)] Macdonald’s duality conjecture (1991):

\frac{P_{λ}(q^{ν}t^{δ}a_{1}; q, t; a_{1}, a_{2}, a_{3}, a_{4})}{P_{λ}(t^{δ}a_{1}; q, t; a_{1}, a_{2}, a_{3}, a_{4})} = \frac{P_{ν}(q^{λ}t^{δ}a′_{1}; q, t; a′_{1}, a′_{2}, a′_{3}, a′_{4})}{P_{ν}(t^{δ}a′_{1}; q, t; a′_{1}, a′_{2}, a′_{3}, a′_{4})} .  (4.16)

Sahi [34] proved (4.16) in the general case. As pointed out in [4, Section 7.2], this also implies (4.15) in the general case. Macdonald independently proved his conjectures in his book [20], see Eqs. (5.3.12) and (5.3.5), respectively, there.

4.4. BC_{n}-type Jacobi polynomials. See [10] and [37, Definition 3.5 and (3.18)]. Let α, β > −1 and τ > 0. Put

Δ(x) = Δ(x; τ; α, β) := ∏_{j=1}^{n} x_{j}^{α}(1 − x_{j})^{β} ∏_{1≤i<j≤n} |x_{i} − x_{j}|^{2τ}.

BC_{n}-type Jacobi polynomials are λ-monic S_{n}-invariant polynomials

P_{λ}(x; τ; α, β) = P_{λ}(x) = ∑_{μ≤λ} a_{λ,μ}m_{μ}(x)

such that

∫_{[0,1]^{n}} P_{λ}(x) m_{μ}(x) Δ(x) dx_{1} ⋯ dx_{n} = 0 if μ < λ.  (4.17)

It follows from (4.17) that

∫_{[0,1]^{n}} P_{λ}(x) P_{μ}(x) Δ(x) dx_{1} ⋯ dx_{n} = 0  (4.18)

if μ < λ. It can be shown, see [10, Corollary 3.12], that (4.18) holds more generally if λ ≠ μ.

The case c = 1, d = −1 of [37, Eq. (5.5)] says that

lim_{q↑1} P_{λ}(x; q, q^{τ}; q^{α+1}, −q^{β+1}, 1, −1) = (−4)^{|λ|} P_{λ}(\frac{1}{4}(2 − x − x^{−1}); τ; α, β).  (4.19)

Furthermore, it was pointed out in [31, Eq. (4.8)] that the Jack polynomial P_{λ}(x; τ) is the term of highest degree |λ| of the BC_{n}-type Jacobi polynomial P_{λ}(x; τ; α, β):

lim_{r→∞} r^{−|λ|}P_{λ}(rx; τ; α, β) = P_{λ}(x; τ).  (4.20)

This is the q = 1 analogue of the limit (4.13).

An evaluation formula for Jacobi polynomials associated with root systems, including BC_{n}, was given by Opdam [28, Corollary 5.2]. See reformulations of this result in the BC_{n} case by van Diejen [5, Eq. (6.43d)] and by Hallnäs [9, p. 1594]. The formula can be given very explicitly as follows:

P_{λ}(0; τ; α, β) = (−1)^{|λ|} ∏_{j=1}^{n} \frac{((n − j)τ + 2α′)_{λ_{j}} ((n − j)τ + α + 1)_{λ_{j}}}{((2n − 2j)τ + 2α′)_{2λ_{j}}}

× ∏_{1≤i<j≤n} \frac{((2n − i − j + 1)τ + 2α′)_{λ_{i}+λ_{j}}}{((2n − i − j)τ + 2α′)_{λ_{i}+λ_{j}}} \frac{((j − i + 1)τ)_{λ_{i}−λ_{j}}}{((j − i)τ)_{λ_{i}−λ_{j}}} . (4.21)

Here

α′ := \frac{1}{2}(α + β + 1).  (4.22)

The following limit is formally suggested by (4.19):

lim_{q↑1} P_{λ}(q^{τδ+α+1}; q, q^{τ}; q^{α+1}, −q^{β+1}, 1, −1) = (−4)^{|λ|} P_{λ}(0; τ; α, β).  (4.23)

It follows rigorously by comparing (4.15) and (4.21).

5. Interpolation polynomials

5.1. Interpolation Macdonald polynomials. See Sahi [33, Theorem 1.1], Knop [11, Theorem 2.4(b)], and Okounkov [25, Eqs. (4.2), (4.3)].

Let 0 < t < 1. The interpolation Macdonald polynomial (or shifted Macdonald polynomial) P_{λ}^{ip}(x; q, t) is the unique λ-monic S_{n}-invariant polynomial of degree |λ| such that P_{λ}^{ip}(q^{μ}t^{δ}; q, t) = 0 for each partition μ ≠ λ with |μ| ≤ |λ|. Here q^{μ}t^{δ} = (q^{μ_{1}}t^{n−1}, q^{μ_{2}}t^{n−2}, …, q^{μ_{n}}).

Our P_{λ}^{ip} is related to Sahi’s R_{λ}, Knop’s P_{λ} (use [11, Theorem 3.11]) and Okounkov’s P_{λ}^{∗} (use [25, Eq. (4.11)]), respectively, as follows:

P_{λ}^{ip}(x; q, t) = R_{λ}(x; q^{−1}, t^{−1}) = t^{(n−1)|λ|} P_{λ}(t^{−(n−1)}x) = t^{(n−1)|λ|} P_{λ}^{∗}(xt^{−δ}).

Okounkov [25] speaks about shifted polynomials because in his notation the polynomials are only symmetric after a (multiplicative) shift.

P_{λ}^{ip} has the extra vanishing property (see [11, p. 93] or [25, Eq. (4.12)])

P_{λ}^{ip}(q^{μ}t^{δ}; q, t) = 0 if μ is a partition not containing λ.

By [25, Eq. (4.11)], P_{λ}^{ip} can be expanded in terms of Macdonald polynomials as follows:

P_{λ}^{ip}(x; q, t) = ∑_{μ⊆λ} b_{λ,μ} P_{μ}(x; q, t)  (5.1)

for certain coefficients b_{λ,μ}, where b_{λ,λ} = 1 by λ-monicity. This has several consequences. By combination with (4.1) we see that

P_{λ}^{ip}(x; q, t) = ∑_{μ≤λ} c_{λ,μ} m_{μ}(x)  (5.2)

for certain coefficients c_{λ,μ}, and c_{λ,λ} = 1. Furthermore, by (5.1), P_{λ}(x; q, t) is the term of highest degree |λ| of the polynomial P_{λ}^{ip}(x; q, t):

lim_{r→∞} r^{−|λ|}P_{λ}^{ip}(rx; q, t)) = P_{λ}(x; q, t).  (5.3)

Although Knop [11] did not give (5.1), he did give (5.2) and (5.3), proved differently (see Theorem 3.11 and Theorem 3.9 in [11]). The result (5.3) is also proved by Sahi [33, Theorem 1.1].

5.2. Interpolation Jack polynomials. See Sahi [32, Theorem 1], Knop and Sahi [12, pp. 475, 478], Okounkov and Olshanski [27, p. 70], and Okounkov [25, Section 7].

Let τ > 0. The interpolation Jack polynomial (or shifted Jack polynomial) P_{λ}^{ip}(x; τ) is the unique λ-monic S_{n}-invariant polynomial of degree |λ| such that P_{λ}^{ip}(μ+τδ; τ) = 0 for each partition μ ≠ λ with |μ| ≤ |λ|. It can be expressed in terms of P_{λ}^{τδ} from [12] and in terms of P_{λ}^{∗}( .; τ) from [27], [25] as follows:

P_{λ}^{ip}(x; τ) = P_{λ}^{τδ}(x) = P_{λ}^{∗}(x − τδ; τ).

It has the extra vanishing property (see [12, Theorem 5.2])

P_{λ}^{ip}(μ + τδ; τ) = 0 if μ is a partition not containing λ.

It has an expansion of the form

P_{λ}^{ip}(x; τ) = ∑_{μ≤λ} c_{λ,μ} m_{μ}(x)

for certain coefficients c_{λ,μ}, and c_{λ,λ} = 1 (see [12, Corollary 4.6]). The term of highest degree |λ| of the polynomial P_{λ}^{ip}(x; τ) is the Jack polynomial P_{λ}(x; τ) (see [12, Corollary 4.7]):

lim_{r→∞} r^{−|λ|}P_{λ}^{ip}(rx; τ) = P_{λ}(x; τ).  (5.4)

Interpolation Jack polynomials are a limit case of interpolation Macdonald polynomials (see [25, Eq. (7.1)]):

lim_{q↑1}(q − 1)^{−|λ|}P_{λ}^{ip}(q^{x}; q, q^{τ}) = P_{λ}^{ip}(x; τ).  (5.5)

5.3. BC_{n}-type interpolation Macdonald polynomials.

Definition 5.1 (BC_{n}-type interpolation (or BC_{n}-type shifted) Macdonald polynomials). Let 0 < t < 1 and let a ∈ ℂ be generic. P_{λ}^{ip}(x; q, t, a) is the unique W_{n}-invariant λ-monic Laurent polynomial of degree |λ| such that

P_{λ}^{ip}(q^{μ}t^{δ}a; q, t, a) = 0 if μ does not contain λ.  (5.6)

(In particular, P_{λ}^{ip}(q^{μ}t^{δ}a; q, t, a) = 0 if |μ| ≤ |λ|, μ ≠ λ.)

These polynomials were first introduced by Okounkov [26] in a different notation and normalization. Okounkov [26, p. 185, Section 1] specifies the genericity of the parameter a (in his notation s) by the condition q^{i}t^{j}a^{k} ≠ 1 for i, j, k ∈ ℤ_{>0}. However, this may be too strong on the one hand and too weak on the other hand. Certainly the right-hand side of (5.10) (equivalently the normalization constant in [26, Definition 1.2]) should be nonzero. A requirement for this is that q^{i}t^{j}a^{2} ≠ 1 for i ∈ ℤ_{>0}, j ∈ ℤ_{≥0}.

Different approaches were given by Rains [29], and later by Noumi in unpublished slides of a lecture given in 2013 at a conference at Kyushu University. Our normalization follows Rains [29]. In terms of Rains’ P̄_{λ}^{∗(n)} and Okounkov’s P_{λ}^{∗}, we have (cf. [29, p. 76, Remark 1])

P_{λ}^{ip}(x; q, t, a) = P̄_{λ}^{∗(n)}(x; q, t, a) = (t^{n−1}a)^{|λ|}P_{λ}^{∗}(xt^{−δ}a^{−1}; q, t, a).  (5.7)

Note that Okounkov’s P_{λ}^{∗}(x; q, t, s) is W_{n}-symmetric in the variables x_{i}t^{n−i}s (i = 1, 2, …, n).

Just as for P_{λ}^{ip}(x; q, t), the top homogeneous term of P_{λ}^{ip}(x; q, t, a) equals the Macdonald polynomial P_{λ}(x; q, t) (see [26, Section 4]):

lim_{r→∞} r^{−|λ|}P_{λ}^{ip}(rx; q, t, a)) = P_{λ}(x; q, t).  (5.8)

There is also a limit from P_{λ}^{ip}(x; q, t, a) to P_{λ}^{ip}(x; q, t) (see [29, p. 75]):

lim_{a→∞} a^{−|λ|}P_{λ}^{ip}(ax; q, t, a) = P_{λ}^{ip}(x; q, t).  (5.9)

By combination of (5.7) with [26, Definitions 1.1 and 1.2], we get the following evaluation formula (with ⟨ . , . ⟩ denoting the standard inner product on ℝ^{n}):

P_{λ}^{ip}(q^{λ}t^{δ}a; q, t, a) = q^{−⟨λ,λ⟩}t^{−⟨λ,δ⟩} a^{−|λ|} ∏_{(i,j)∈λ}(1 − q^{λ_{i}−j+1}t^{λ′_{j}−i})(1 − a^{2}q^{λ_{i}+j−1}t^{λ′_{j}−i+2(n−λ′_{j})}).

(5.10)

By [29, Lemma 2.1] (see also [29, Corollary 3.7]) this can be rewritten as

P_{λ}^{ip}(q^{λ}t^{δ}a; q, t, a) = q^{−⟨λ,λ⟩}t^{−⟨λ,δ⟩} a^{−|λ|} ∏_{j=1}^{n} \frac{(qt^{n−j}; q)_{λ_{j}}(t^{2n−2j}a^{2}; q)_{2λ_{j}}}{(t^{n−j}a^{2}; q)_{λ_{j}}}

× ∏_{1≤i<j≤n} \frac{(t^{2n−i−j}a^{2}; q)_{λ_{i}+λ_{j}}}{(t^{2n−i−j+1}a^{2}; q)_{λ_{i}+λ_{j}}} \frac{(qt^{j−i−1}; q)_{λ_{i}−λ_{j}}}{(qt^{j−i}; q)_{λ_{i}−λ_{j}}} .  (5.11)

An elementary consequence of the definition of P_{λ}^{ip}(x; q, t, a) is a reduction formula (see [26, Proposition 2.1]):

P_{μ}^{ip}(x; q, t, a) = (−a)^{nμ_{n}}q^{− \frac{1}{2} nμ_{n}(μ_{n}−1)} ∏_{j=1}^{n} ((x_{j}a; q)_{μ_{n}}(x_{j}^{−1}a; q)_{μ_{n}})P_{μ−μ_{n}1^{n}}^{∗}(x; q, t, q^{μ_{n}}a).

(5.12)

6. Combinatorial formulas

6.1. Combinatorial formula for Macdonald polynomials. The combinatorial formula for Macdonald polynomials which we will use is a special case of [18, Ch. VI, Eq. (7.13′)]:

P_{λ}(x; q, t) = ∑_{T} ψ_{T} (q, t) ∏_{s∈λ} x_{T (s)},  (6.1)

where the sum is over all tableaux T of shape λ with entries in {1, …, n} and with ψ_{T} (q, t) defined in [18, Ch. VI] by formula (7.11′), by formula (ii) on p. 341 with C_{λ/μ} and S_{λ/μ} given after (6.22), and by formula (6.20). See [15, Section 1] for a summary of these results. Since the Macdonald polynomial is symmetric, we may as well sum over reverse tableaux instead of tableaux, with the definition of ψ_{T} (q, t) accordingly adapted. We will now give ψ_{T} (q, t) explicitly. See Subsections 3.1 and 3.2 for notation.

Recall that for a reverse tableaux T of shape λ with entries in {1, …, n} we write 0^{n} = λ^{(n)} ⊆ λ^{(n−1)} ⊆ ⋯ ⊆ λ^{(0)} = λ with T (s) = i for s in the horizontal strip λ^{(i−1)} − λ^{(i)}. Now

ψ_{T} (q, t) := ∏_{i=1}^{n} ψ_{λ^{(i−1)}/λ^{(i)}}(q, t),  ψ_{μ/ν}(q, t) = ∏_{s∈(R\C)_{μ/ν}} \frac{b_{ν}(s; q, t)}{b_{μ}(s; q, t)} ,  (6.2)

where

b_{μ}(s; q, t) := \frac{1 − q^{a_{μ}(s)}t^{l_{μ}(s)+1}}{1 − q^{a_{μ}(s)+1}t^{l_{μ}(s)}} .  (6.3)

By (3.4) and (4.1) we obtain that

P_{λ}(x; q, t) = ∑_{μ≤λ} u_{λ,μ}(q, t)m_{μ}(x) with u_{λ,μ}(q, t) = ∑_{T} ψ_{T} (q, t),  (6.4)

where the T -sum is over all reverse tableaux T of shape λ and weight μ, see [18, p. 378].

Let T_{λ} be the tableau of shape λ for which T (i, j) = n + 1 − i. This is the unique tableau of shape λ which has weight (λ_{n}, λ_{n−1}, …, λ_{1}). Since P_{λ}(x; q, t) is λ-monic, it follows from (6.1) that

ψ_{T_{λ}}(q, t) = 1.  (6.5)

6.2. Combinatorial formulas for (BC_{n}) interpolation Macdonald polynomials. For interpolation Macdonald polynomials P_{λ}^{ip}(x; q, t) and BC_{n}-type interpolation Macdonald polynomials P_{λ}^{ip}(x; q, t, a) there are combinatorial formulas similar to (6.1) and also involving ψ_{T} (q, t) given by (6.2):

P_{λ}^{ip}(x; q, t) = ∑_{T} ψ_{T} (q, t) ∏_{s∈λ} (x_{T (s)} − q^{a′_{λ}(s)}t^{n−T (s)−l′_{λ}(s)}),  (6.6)
P_{λ}^{ip}(x; q, t, a) = ∑_{T} ψ_{T} (q, t) ∏_{s∈λ} (x_{T (s)} − q^{a′_{λ}(s)}t^{n−T (s)−l′_{λ}(s)}a)
· (1 − (q^{a′_{λ}(s)}t^{n−T (s)−l′_{λ}(s)}a)^{−1}x_{T (s)}^{−1}).  (6.7)

See [25, Eq. (1.4)] for (6.6) and [26, Eq. (5.3)] for (6.7). The sums are over all reverse tableaux T of shape λ with entries in {1, …, n}. Note that the limits (5.3), (5.8) and (5.9) also follow by comparing (6.1), (6.6) and (6.7).

6.3. Combinatorial formulas for (interpolation) Jack polynomials. The combinatorial formula for Jack polynomials can be obtained as a limit case of the combinatorial formula (6.1) for Macdonald polynomials by using the limit (4.7) (see [18, p. 379]), but it can also be obtained independently, as was first done by Stanley [36, Theorem 6.3]:

P_{λ}(x; τ) = ∑_{T} ψ_{T} (τ) ∏_{s∈λ} x_{T (s)},  (6.8)

where the sum is over all reverse tableaux T of shape λ with entries in {1, …, n}. For the definition of ψ_{T} (τ) take 0^{n} = λ^{(n)} ⊆ λ^{(n−1)} ⊆ ⋯ ⊆ λ^{(0)} = λ as before and put

ψ_{T} (τ) := ∏_{i=1}^{n} ∏_{s∈(R\C)_{λ^{(i−1)}/λ^{(i)}}} \frac{b_{λ^{(i)}}(s; τ)}{b_{λ^{(i−1)}}(s; τ)}  (6.9)

with

b_{μ}(s; τ) := \frac{a_{μ}(s) + τ(l_{μ}(s) + 1)}{a_{μ}(s) + τl_{μ}(s) + 1} .

Note that

lim_{q↑1} b_{μ}(s; q, q^{τ}) = b_{μ}(s, τ) and lim_{q↑1} ψ_{T} (q, q^{τ}) = ψ_{T} (τ).

Hence, by (6.5) we have

ψ_{T_{λ}}(τ) = 1.  (6.10)

Similarly as for (6.4) we derive immediately that

P_{λ}(x; τ) = ∑_{μ≤λ} u_{λ,μ}(τ)m_{μ}(x) with u_{λ,μ}(τ) = ∑_{T} ψ_{T} (τ),

where the T -sum is over all reverse tableaux T of shape λ and weight μ.

The combinatorial formula for interpolation Jack polynomials (see [27, Eq. (2.4)]) was obtained in [25, Section 7] as a limit case of the combinatorial formula (6.6) for interpolation Macdonald polynomials by using the limit (5.5):

P_{λ}^{ip}(x; τ) = ∑_{T} ψ_{T} (τ) ∏_{s∈λ} (x_{T (s)} − a′_{λ}(s) − τ(n − T (s) − l′_{λ}(s))),  (6.11)

with the sum over all tableaux T of shape λ with entries in {1, …, n} and ψ_{T} (τ) given by (6.9). The limit (5.4) also follows by comparing (6.11) and (6.8). Furthermore, by comparing (6.7) and (6.8) we obtain the limit

lim_{q↑1} P_{λ}^{ip}(x; q, q^{τ}, q^{α}) = P_{λ}(x + x^{−1} − 2; τ),  (6.12)

and from (6.6) and (6.8) we obtain

lim_{q↑1} P_{λ}^{ip}(x; q, q^{τ}) = P_{λ}(x − 1^{n}; τ).  (6.13)

Remark 6.1. When we compare combinatorial formulas in the case of n and of n − 1 variables, we see that, in general, a combinatorial formula is equivalent to a branching formula, which expands a polynomial P_{λ} in x_{1}, …, x_{n} in terms of polynomials P_{μ} in x_{1}, …, x_{n−1} with the expansion coefficients depending on x_{n}. In particular, the branching formula for Macdonald polynomials is (see [15, Eqs. (1.9), (1.8)])

P_{λ}(x_{1}, …, x_{n−1}, x_{n}; q, t)) = ∑_{μ} P_{λ/μ}(x_{n}; q, t) P_{μ}(x_{1}, …, x_{n−1}; q, t),  (6.14)

where the sum runs over all partitions μ ⊆ λ of length < n such that λ−μ is a horizontal strip, and where, in notation (6.2),

P_{λ/μ}(z; q, t) = ψ_{λ/μ}(q, t) z^{|λ|−|μ|}.  (6.15)

The coefficients ψ_{λ/μ}(q, t) can be expressed in terms of Pieri coefficients for Macdonald polynomials by interchanging q and t and by passing to conjugate partitions λ′, μ′: ψ_{λ/μ}(q, t) = ψ′_{λ′/μ′}(t, q), see [18, Ch. VI, Eq. (6.24)].

Van Diejen and Emsiz [6] recently obtained a branching formula for Koornwinder polynomials. It has the same structure as (6.14), but the analogue of (6.15) becomes a sum of terms in the right-hand side. Each term is similar to the right-hand side of (6.15), with the monomial being replaced by a quadratic q-factorial, also depending on the parameter a_{1}. The analogues of the coefficients ψ_{λ/μ} can be expressed in terms of (earlier known) Pieri-type coefficients for Koornwinder polynomials. By taking highest degree parts in both sides of the new branching formula and by using (4.13), we are reduced to (6.15). However, the combinatorial formula (6.7) and its corresponding branching formula for BC_{n}-type interpolation Macdonald polynomials are quite different from the results in [6].

7. BC_{n}-type interpolation Jack polynomials

In view of the results surveyed until now, the following definition is quite natural.

Definition 7.1. Let τ > 0 and let α ∈ ℂ be generic. The BC_{n}-type interpolation Jack polynomial P_{λ}^{ip}(x; τ, α) is given as a limit of BC_{n}-type interpolation Macdonald polynomials,

P_{λ}^{ip}(x; τ, α) := lim_{q↑1}(1 − q)^{−2|λ|}P_{λ}^{ip}(q^{x}; q, q^{τ}, q^{α}).  (7.1)

Concerning the genericity of α ∈ ℂ we should have at least that the evaluation (7.5) is nonzero, i.e., i + jτ + 2α ≠ 0 for i ∈ ℤ_{>0} and j ∈ ℤ_{≥0}. That the limit (7.1) exists can be seen by substituting (6.7) in the right-hand side of (7.1). We obtain

P_{λ}^{ip}(x; τ, α) = ∑_{T} ψ_{T} (τ) ∏_{s∈λ} (x_{T (s)}^{2} − (a′_{λ}(s) + τ(n − T (s) − l′_{λ}(s)) + α)^{2}) (7.2)

with the sum over all reverse tableaux T of shape λ with entries in {1, …, n} and ψ_{T} (τ) given by (6.9). From (7.1), (7.2) and the properties of P_{λ}^{ip}(x; q, t) we see that P_{λ}^{ip}(x; τ, α) is a W_{n}-invariant polynomial of degree 2|λ| in x, where (ℤ_{2})^{n} now acts on the polynomial by sending some of the variables x_{i} to −x_{i} rather than to x_{i}^{−1}. By (6.10) it follows from (7.2) that P_{λ}^{ip}(x; τ, α) is (2λ)-monic.

It follows from (5.6) and (7.1) that

P_{λ}^{ip}(μ + τδ + α; τ, α) = 0 if μ does not contain λ.

By comparing (7.2), (6.11) and (6.8) we obtain the limits

lim_{r→∞} r^{−2|λ|}P_{λ}^{ip}(rx; τ, α) = P_{λ}(x^{2}; τ),  (7.3)
lim_{α→∞}(2α)^{−|λ|}P_{λ}^{ip}(x + α; τ, α) = P_{λ}^{ip}(x; τ).  (7.4)

From (5.10) and (5.11) together with (7.1), we obtain the evaluation formula

P_{λ}^{ip}(λ + τδ + α; τ, α)

= ∏_{(i,j)∈λ} (λ_{i} − j + 1 + τ(λ′_{j} − i))(2α + λ_{i} + j − 1 + τ(λ′_{j} − i + 2(n − λ′_{j}))

= ∏_{j=1}^{n} \frac{((n − j)τ + 1)_{λ_{j}}(2(n − j)τ + 2α)_{2λ_{j}}}{((n − j)τ + 2α)_{λ_{j}}}

× ∏_{1≤i<j≤n} \frac{((2n − i − j)τ + 2α)_{λ_{i}+λ_{j}}}{((2n − i − j + 1)τ + 2α)_{λ_{i}+λ_{j}}} \frac{((j − i − 1)τ + 1)_{λ_{i}−λ_{j}}}{((j − i)τ + 1)_{λ_{i}−λ_{j}}} . (7.5)

By (5.12) and (7.1) we get a reduction formula

P_{λ}^{ip}(x; τ, α) = (−1)^{nλ_{n}} ∏_{j=1}^{n} ((α + x_{j})_{λ_{n}}(α − x_{j})_{λ_{n}}) P_{λ−λ_{n}1^{n}}^{ip}(x; τ, λ_{n} + α).  (7.6)

8. Binomial formulas

8.1. Binomial formula for Koornwinder polynomials. Okounkov [26, Theorem 7.1] obtained the binomial formula for Koornwinder polynomials:

\frac{P_{λ}(x; q, t; a_{1}, a_{2}, a_{3}, a_{4})}{P_{λ}(t^{δ}a_{1}; q, t; a_{1}, a_{2}, a_{3}, a_{4})} = ∑_{μ⊆λ} \frac{P_{μ}^{ip}(q^{λ}t^{δ}a′_{1}; q, t, a′_{1})}{P_{μ}^{ip}(q^{μ}t^{δ}a′_{1}; q, t, a′_{1})} \frac{P_{μ}^{ip}(x; q, t, a_{1})}{P_{μ}(t^{δ}a_{1}; q, t; a_{1}, a_{2}, a_{3}, a_{4})} . (8.1)

As pointed out in [26], the duality (4.16) immediately follows from (8.1) in the self-dual case a_{1} = a′_{1}. In the general case (4.16) will follow from (8.1) together with the identity

\frac{P_{μ}(t^{δ}a_{1}; q, t; a_{1}, a_{2}, a_{3}, a_{4})}{P_{μ}^{ip}(q^{μ}t^{δ}a_{1}; q, t, a_{1})} = \frac{P_{μ}(t^{δ}a′_{1}; q, t; a′_{1}, a′_{2}, a′_{3}, a′_{4})}{P_{μ}^{ip}(q^{μ}t^{δ}a′_{1}; q, t, a′_{1})} ,

which is a consequence of the evaluation formulas (4.15) and (5.11).

Rains [29, Section 5] gives an alternative definition of Koornwinder polynomials involving triangularity and evaluation symmetry by which a version of (8.1) is an immediate consequence.

Because the inclusion partial ordering is compatible with the lexicographic ordering, we can use induction in λ with respect to the lexicographic ordering in order to show from (8.1) that

P_{λ}^{ip}(x; q, t, a_{1}) = ∑_{μ⊆λ} b_{λ,μ} P_{μ}(x; q, t; a_{1}, a_{2}, a_{3}, a_{4})

for certain coefficients b_{λ,μ} (more explicitly given in [29, Theorem 5.12]). Together with (4.10) this implies that

P_{λ}^{ip}(x; q, t, a_{1}) = ∑_{μ≤λ} c_{λ,μ} m̃_{μ}(x)  (8.2)

for certain coefficients c_{λ,μ}.

8.2. Binomial formula for Macdonald polynomials. Okounkov [24, Eq. (1.11)] gave the binomial formula for Macdonald polynomials:

\frac{P_{λ}(x; q, t)}{P_{λ}(t^{δ}; q, t)} = ∑_{μ⊆λ} \frac{P_{μ}^{ip}(q^{λ}t^{δ}; q, t)}{P_{μ}^{ip}(q^{μ}t^{δ}; q, t} \frac{P_{μ}^{ip}(x; q, t)}{P_{μ}(t^{δ}; q, t)} .  (8.3)

The binomial formula (8.3) immediately implies the duality formula (4.5) for Macdonald polynomials.

By comparing the binomial formulas (8.1) and (8.3) we can obtain a new limit formula.

Theorem 8.1. There is the limit

lim_{a_{1}→∞} a_{1}^{−|λ|} P_{λ}(a_{1}x; q, t; a_{1}, a_{2}, a_{3}, a_{4}) = P_{λ}(x; q, t).  (8.4)

Proof. Rewrite (8.1) as

\frac{a_{1}^{−|λ|} P_{λ}(a_{1}x; q, t; a_{1}, a_{2}, a_{3}, a_{4})}{a_{1}^{−|λ|} P_{λ}(t^{δ}a_{1}; q, t; a_{1}, a_{2}, a_{3}, a_{4})}

= ∑_{μ⊆λ} \frac{a_{1}^{−|μ|}P_{μ}^{ip}(q^{λ}t^{δ}a′_{1}; q, t, a′_{1})}{a_{1}^{−|μ|}P_{μ}^{ip}(q^{μ}t^{δ}a′_{1}; q, t, a′_{1})} \frac{a_{1}^{−|μ|} P_{μ}^{ip}(a_{1}x; q, t, a_{1})}{a_{1}^{−|μ|} P_{μ}(t^{δ}a_{1}; q, t; a_{1}, a_{2}, a_{3}, a_{4})} .

Now let a_{1} → ∞. The result follows by (5.9) and (8.3) if we can show that (8.4) holds for x = t^{δ}. This, in its turn, follows by comparing (4.15) and (4.4). □

As pointed out to me by Ole Warnaar, the limit (8.4) also follows from the limit (4.13) (yielding Macdonald polynomials as highest degree part of Koornwinder polynomials) together with the known fact (although not in the literature) that the coefficients u_{λ,μ} (|μ| < |λ|) in (4.10) are bounded as a_{1} → ∞.

8.3. Binomial formula for Jack polynomials. The binomial formula for Jack polynomials was given by Okounkov and Olshanski [27, p. 72]:

\frac{P_{λ}(1 + x, τ)}{P_{λ}(1; τ)} = ∑_{μ⊆λ} \frac{P_{μ}^{ip}(λ + τδ; τ)}{P_{μ}^{ip}(μ + τδ; τ)} \frac{P_{μ}(x; τ)}{P_{μ}(1; τ)} .  (8.5)

It is a limit case of (8.3) because of the limits (4.7), (4.9) (twice), (5.5) (twice) and (6.13). If we compare (8.5) with Macdonald [21, Eqs. (6.15), (6.24)], Lassalle [16, § 3] or Yan [38, Eq. (10)], we see that P_{μ}^{ip}(λ + τδ; τ)/P_{μ}^{ip}(μ + τδ; τ) equals the generalized binomial coefficient \binom{λ}{μ} defined in these references.

9. Binomial formula for BC_{n}-type Jacobi polynomials

Let α′ be given by (4.22).

Theorem 9.1. For BC_{n}-type Jacobi polynomials we have the binomial formula

\frac{P_{λ}(x; τ; α, β)}{P_{λ}(0; τ; α, β)} = ∑_{μ⊆λ} \frac{P_{μ}^{ip}(λ + τδ + α′; τ, α′)}{P_{μ}^{ip}(μ + τδ + α′; τ, α′)} \frac{P_{μ}(x; τ)}{P_{μ}(0; τ; α, β)} .  (9.1)

Proof. From (8.1) we obtain

\frac{P_{λ}(x; q, q^{τ}; q^{α+1}, −q^{β+1}, 1, −1)}{P_{λ}(q^{τδ+α+1}; q, q^{τ}; q^{α+1}, −q^{β+1}, 1, −1)}

= ∑_{μ⊆λ} \frac{P_{μ}^{ip}(q^{λ+τδ+α′}; q, q^{τ}, q^{α′})}{P_{μ}^{ip}(q^{μ+τδ+α′}; q, q^{τ}, q^{α′})} \frac{P_{μ}^{ip}(x; q, q^{τ}, q^{α+1})}{P_{μ}(q^{τδ+α+1}; q, q^{τ}; q^{α+1}, −q^{β+1}, 1, −1)} .

Now let q ↑ 1 and apply (4.19), (4.23) (twice), (7.1) (twice) and (6.12). □

By comparing the binomial formulas (9.1) and (8.5) we arrive at the limit

lim_{α→∞} P_{λ}(x; τ; α, β) = P_{λ}(x − 1^{n}; τ),  (9.2)

where 1^{n} is the n-vector with all coordinates equal to 1. The limit (9.2) was given slightly more generally in [31, Theorem 4.2] and goes back to an unpublished result by Beerends and the author. For the proof of (9.2) let α → ∞ in the right-hand side of (9.1) and use (7.4). Then we will obtain the right-hand side of (8.5) if we can prove that (9.2) is valid for x = 0. But in that case (9.2) follows by comparing (4.21) and (4.8).

Remark 9.2. In (9.1) we have an expansion

P_{λ}(x; τ; α, β) = ∑_{μ⊆λ} b_{λ,μ} P_{μ}(x; τ)  (9.3)

with

b_{λ,μ} = \frac{P_{λ}(0; τ; α, β)}{P_{μ}(0; τ; α, β)} \frac{P_{μ}^{ip}(λ + τδ + α′; τ, α′)}{P_{μ}^{ip}(μ + τδ + α′; τ, α′)} ,  (9.4)

where, by (7.2),

P_{μ}^{ip}(λ+τδ+α′; τ, α′) = ∑_{T} ψ_{T} (τ) ∏_{s∈μ} ((λ_{T (s)}+τδ_{T (s)}+α′)^{2}−(τδ_{T (s)}+α′+a′_{λ}(s)−τl′_{λ}(s))^{2}).

(9.5)

On the other hand, Macdonald [21, p. 58] (also in Beerends and Opdam [2, Eq. (5.12)]) gives (9.3) with

b_{λ,μ} = (−1)^{|λ|−|μ|} \frac{P_{λ}(1; τ)}{P_{μ}(1; τ)} c_{λ/μ}(2τ(n−1)+α′+1) ∏_{j=1}^{n}(μ_{j}+τ(n+j−2)+α+1)_{λ_{j}−μ_{j}} . (9.6)

Here

c_{λ/μ}(C) = ∑_{T} f_{T} (C)  (9.7)

for a certain function f_{T}, and the sum is over all standard tableaux T of shape λ/μ. Macdonald derives (9.7) by solving a recurrence relation [21, Eq. (9.16)] for c_{λ/μ}(C) (C and λ fixed). Thus both in Macdonald’s formula (9.6) and in formula (9.4) the expansion coefficients b_{λ,μ} are essentially given combinatorially by tableau sums ((9.7) and (9.5), respectively). However the tableau sums are quite different: over standard tableaux of shape λ/μ in Macdonald’s case, and over reverse tableaux T of shape μ with entries in {1, …, n} in the case of (9.5). I do not see how to match these tableau sums with each other.

10. Explicit expressions for n = 2

The combinatorial formulas (6.1), (6.6), (6.7) for Macdonald polynomials and their (BC_{n}-type) interpolation versions all have the form

∑_{T} ψ_{T} (q, t) ∏_{s∈λ} f(x_{T (s)}).  (10.1)

Here ψ_{T} (q, t) is given by (6.2), while f is an elementary (Laurent) polynomial which may also depend on s, T (s), t and a. The sum is over all reverse tableaux T of shape λ with entries in {1, …, n}.

Now let n = 2 and λ := (m, 0). Then the possible reverse tableaux T of shape (m, 0) with entries in {1, 2} are the tableaux T_{k} (k = 0, 1, …, m) given by

T_{k}(1, j) = 2 if j = 1, …, k and = 1 if j = k + 1, …, m.  (10.2)

So we have to compute ψ_{T_{k}}(q, t), which is given by (6.2) as a double product involving (R\C)_{λ^{(0)}/λ^{(1)}} = {(1, 1), …, (1, k)} and (R\C)_{λ^{(1)}/λ^{(2)}} = ∅. Hence

ψ_{T_{k}}(q, t) = ∏_{s∈(R\C)_{λ^{(0)}/λ^{(1)}}} \frac{b_{(k)}(s; q, t)}{b_{(m)}(s; q, t)} ,

where b_{μ}(s; q, t) is defined by (6.3). This yields for j = 1, …, k that

b_{(k)}((1, j); q, t) = \frac{1 − q^{k−j}t}{1 − q^{k−j+1}} , b_{(m)}((1, j); q, t) = \frac{1 − q^{m−j}t}{1 − q^{m−j+1}} .

Hence

ψ_{T_{k}}(q, t) = \frac{(q^{k−1}t; q^{−1})_{k} (q^{m}; q^{−1})_{k}}{(q^{k}; q^{−1})_{k} (q^{m−1}t; q^{−1})_{k}} = \frac{(t, q^{−m}; q)_{k}}{(q, q^{1−m}t^{−1}; q)_{k}} (qt^{−1})^{k}.  (10.3)

10.1. Explicit expression for BC_{2}-type interpolation Macdonald polynomials. Consider in (10.1) ∏_{j=1}^{m} f(x_{T_{k}(1,j)}) for the case of (6.1). By taking into account (10.2) this product becomes

∏_{j=1}^{k} ((x_{2} − q^{j−1}a)(1 − (q^{j−1}a)^{−1}x_{2}^{−1})) ∏_{j=k+1}^{m} ((x_{1} − q^{j−1}ta)(1 − (q^{j−1}ta)^{−1}x_{1}^{−1}))

= (−1)^{m}q^{− \frac{1}{2} m(m−1)}(ta)^{−m}(tax_{1}, tax_{1}^{−1}; q)_{m} \frac{(ax_{2}, ax_{2}^{−1}; q)_{k}}{(tax_{1}, tax_{1}^{−1}; q)_{k}} t^{k}. (10.4)

Then take (10.1), a single sum over T = T_{k} (k = 0, …, m), with (10.3) and (10.4) substituted. The result is

P_{m,0}^{ip}(x_{1}, x_{2}; q, t, a) = \frac{(tax_{1}, tax_{1}^{−1}; q)_{m}}{q^{\frac{1}{2} m(m−1)}(−ta)^{m}} _{4}ϕ_{3}({q^{−m}, t, ax_{2}, ax_{2}^{−1} \atop q^{1−m}t^{−1}, tax_{1}, tax_{1}^{−1}} ; q, q) .  (10.5)

Next we want to obtain an explicit expression for P_{m_{1},m_{2}}^{ip} by using the reduction formula (5.12). In the present case, this takes the form

P_{m_{1},m_{2}}^{ip}(x_{1}, x_{2}; q, t, a)

= a^{−2m_{2}}q^{−m_{2}(m_{2}−1)}(ax_{1}, ax_{1}^{−1}, ax_{2}, ax_{2}^{−1}; q)_{m_{2}} P_{m_{1}−m_{2},0}^{ip}(x_{1}, x_{2}; q, t, q^{m_{2}}a).

In combination with (10.5) this gives

P_{m_{1},m_{2}}^{ip}(x_{1}, x_{2}; q, t, a) = \frac{q^{− \frac{1}{2} m_{1}(m_{1}−1)− \frac{1}{2} m_{2}(m_{2}−1)}}{(−t)^{m_{1}−m_{2}}a^{m_{1}+m_{2}}} (ax_{1}, ax_{1}^{−1}, ax_{2}, ax_{2}^{−1}; q)_{m_{2}}

× (q^{m_{2}}tax_{1}, q^{m_{2}}tax_{1}^{−1}; q)_{m_{1}−m_{2}} _{4}ϕ_{3}({q^{−m_{1}+m_{2}}, t, q^{m_{2}}ax_{2}, q^{m_{2}}ax_{2}^{−1} \atop q^{1−m_{1}+m_{2}}t^{−1}, q^{m_{2}}tax_{1}, q^{m_{2}}tax_{1}^{−1}} ; q, q) . (10.6)

In order to make the symmetry in x_{1}, x_{2} in (10.6) obvious, we may expand the second line of this formula as

∑_{k=0}^{m_{1}−m_{2}} \frac{(q^{−m_{1}+m_{2}}, t; q)_{k} q^{k}}{(q^{1−m_{1}+m_{2}}t^{−1}, q; q)_{k}} (q^{m_{2}+k}tax_{1}, q^{m_{2}+k}tax_{1}^{−1}; q)_{m_{1}−m_{2}−k} (q^{m_{2}}ax_{2}, q^{m_{2}}ax_{2}^{−1}; q)_{k}.

Now substitute there a version of the q-Pfaff-Saalschütz formula [8, Eq. (II.12)]:

\frac{(q^{m_{2}+k}tax_{1}, q^{m_{2}+k}tax_{1}^{−1}; q)_{m_{1}−m_{2}−k}}{(q^{k}t, q^{k+2m_{2}}ta^{2}; q)_{m_{1}−m_{2}−k}} = _{3}ϕ_{2}({q^{−m_{1}+m_{2}+k}, q^{m_{2}}ax_{1}, q^{m_{2}}ax_{1}^{−1} \atop q^{1−m_{1}+m_{2}}t^{−1}, q^{2m_{2}+k}ta^{2}} ; q, q)

and expand the _{3}ϕ_{2}. Then (10.6) takes the form

P_{m_{1},m_{2}}^{ip}(x_{1}, x_{2}; q, t, a) = \frac{q^{− \frac{1}{2} m_{1}(m_{1}−1)− \frac{1}{2} m_{2}(m_{2}−1)}}{(−t)^{m_{1}−m_{2}}a^{m_{1}+m_{2}}} (t, q^{2m_{2}}ta^{2}; q)_{m_{1}−m_{2}}

× (ax_{1}, ax_{1}^{−1}, ax_{2}, ax_{2}^{−1}; q)_{m_{2}} ∑_{\begin{matrix}j,k≥0 \\ j+k≤m_{1}+m_{2} \end{matrix}} \frac{(q^{−m_{1}+m_{2}}; q)_{j}}{(q^{1−m_{1}+m_{2}}t^{−1}, q; q)_{j}} (q^{m_{2}}ax_{1}, q^{m_{2}}ax_{1}^{−1}; q)_{j}

× \frac{(q^{−m_{1}+m_{2}}; q)_{k}}{(q^{1−m_{1}+m_{2}}t^{−1}, q; q)_{k}} (q^{m_{2}}ax_{1}, q^{m_{2}}ax_{1}^{−1}; q)_{k} \frac{q^{j+k}}{(q^{2m_{2}}ta^{2}; q)_{j+k}} . (10.7)

10.2. (Interpolation) Macdonald polynomials for n = 2. For explicit formulas of P_{m_{1},m_{2}}^{ip}(x_{1}, x_{2}; q, t) and P_{m_{1},m_{2}}(x_{1}, x_{2}; q, t) we may either give a derivation analogous to the one for (10.6), now starting from (6.6) or (6.1), or apply, much quicker, the limits (5.9) or (5.8) to (10.6). For P_{m_{1},m_{2}}^{ip}(x_{1}, x_{2}; q, t) we obtain

P_{m_{1},m_{2}}^{ip}(x_{1}, x_{2}; q, t) = x_{1}^{m_{1}}x_{2}^{m_{2}} (x_{1}^{−1}, x_{2}^{−1}; q)_{m_{2}}(q^{m_{2}}tx_{1}^{−1}; q)_{m_{1}−m_{2}}

× _{3}ϕ_{2}({q^{−m_{1}+m_{2}}, t, q^{m_{2}}x_{2}^{−1} \atop q^{1−m_{1}+m_{2}}t^{−1}, q^{m_{2}}tx_{1}^{−1}} ; q, \frac{qx_{2}}{tx_{1}} ) .

By series inversion, this yields

P_{m_{1},m_{2}}^{ip}(x_{1}, x_{2}; q, t) = x_{1}^{m_{2}}x_{2}^{m_{1}} (x_{1}^{−1}; q)_{m_{2}} (x_{2}^{−1}; q)_{m_{1}} _{3}ϕ_{2}({q^{−m_{1}+m_{2}}, t, q^{−m_{1}+1}t^{−1}x_{1} \atop q^{1−m_{1}+m_{2}}t^{−1}, q^{−m_{1}+1}x_{2}} ; q, q) .

(10.8)

Different explicit expansions for interpolation Macdonald polynomials in the case where n = 2 are given by Morse [22, Theorem 1.1], [23, Theorem 1].

For P_{m_{1},m_{2}}(x_{1}, x_{2}; q, t) we obtain

P_{m_{1},m_{2}}(x_{1}, x_{2}; q, t) = x_{1}^{m_{1}}x_{2}^{m_{2}} _{2}ϕ_{1}({q^{−m_{1}+m_{2}}, t \atop q^{1−m_{1}+m_{2}}t^{−1}} ; q, \frac{qx_{2}}{tx_{1}} )  (10.9)

= \frac{(q; q)_{m_{1}−m_{2}}}{(t; q)_{m_{1}−m_{2}}} ∑_{j=0}^{m_{1}−m_{2}} \frac{(t; q)_{j}(t; q)_{m_{1}−m_{2}−j}}{(q; q)_{j}(q; q)_{m_{1}−m_{2}−j}} x_{1}^{m_{1}−j}x_{2}^{m_{2}+j}  (10.10)

= \frac{(q; q)_{m_{1}−m_{2}}}{(t; q)_{m_{1}−m_{2}}} (x_{1}x_{2})^{\frac{1}{2} (m_{1}+m_{2})} C_{m_{1}−m_{2}} (\frac{x_{1} + x_{2}}{2(x_{1}x_{2})^{\frac{1}{2}}} ; t | q) . (10.11)

Here we used the q-ultraspherical polynomial (see [1, Eq. (4.4)]):

C_{m}(cos θ; t | q) := ∑_{j=0}^{m} \frac{(t; q)_{j}(t; q)_{m−j}}{(q; q)_{j}(q; q)_{m−j}} e^{i(m−2j)θ}.  (10.12)

Formula (10.11) was earlier given by Morse [23, Theorem 2].

10.3. Jack polynomials and (BC_{n}) interpolation Jack polynomials for n = 2. We can get explicit formulas for the n = 2 cases of BC_{n}-type interpolation Jack polynomials, interpolation Jack polynomials and Jack polynomials by applying the limit formulas (7.1), (5.5), (4.7) to (10.6), (10.8), (10.9), respectively. We obtain

P_{m_{1},m_{2}}^{ip}(x_{1}, x_{2}; τ, α) = (−1)^{m_{1}+m_{2}} (α + x_{1}, α − x_{1}, α + x_{2}, α − x_{2})_{m_{2}}

× (m_{2} + τ + α + x_{1}, m_{2} + τ + α − x_{1})_{m_{1}−m_{2}}

× _{4}F_{3}({−m_{1} + m_{2}, τ, m_{2} + α + x_{2}, m_{2} + α − x_{2} \atop 1 − m_{1} + m_{2} − τ, m_{2} + τ + α + x_{1}, m_{2} + τ + α − x_{1}} ; 1) , (10.13)

P_{m_{1},m_{2}}^{ip}(x_{1}, x_{2}; τ)

= (−1)^{m_{1}+m_{2}}(−x_{1})_{m_{2}}(−x_{2})_{m_{1}} _{3}F_{2}({−m_{1} + m_{2}, t, −m_{1} + 1 − τ + x_{1} \atop 1 − m_{1} + m_{2} − τ, −m_{1} + 1 + x_{2}} ; 1) . (10.14)

P_{m_{1},m_{2}}(x_{1}, x_{2}; τ) = x_{1}^{m_{1}}x_{2}^{m_{2}} _{2}F_{1}({−m_{1} + m_{2}, τ \atop 1 − m_{1} + m_{2} − τ}; \frac{x_{2}}{x_{1}} )

= \frac{(m_{1} − m_{2})!}{(τ)_{m_{1}−m_{2}}} ∑_{j=0}^{m_{1}−m_{2}} \frac{(τ)_{j}(τ)_{m_{1}−m_{2}−j}}{j! (m_{1} − m_{2} − j)!} x_{1}^{m_{1}−j}x_{2}^{m_{2}+j}

= \frac{(m_{1} − m_{2})!}{(τ)_{m_{1}−m_{2}}} (x_{1}x_{2})^{\frac{1}{2} (m_{1}+m_{2})} C_{m_{1}−m_{2}}^{τ} (\frac{x_{1} + x_{2}}{2(x_{1}x_{2})^{\frac{1}{2}}} ) .  (10.15)

Here we used the ultraspherical polynomial (see [7, Section 10.9]):

C_{m}^{τ}(cos θ) := ∑_{j=0}^{m} \frac{(τ)_{j}(τ)_{m−j}}{j! (m − j)!} e^{i(m−2j)θ}.  (10.16)

11. Binomial formulas for n = 2

In this section we will only discuss the n = 2 case of the binomial formulas for BCtype polynomials (Koornwinder and Jacobi), because they lead to explicit expressions of these polynomials. We will not discuss here the n = 2 cases of the binomial formulas for the Macdonald and Jack polynomials, although they have interesting aspects from the point of view of special functions.

11.1. Binomial formula for Koornwinder polynomials for n = 2. The binomial formula (8.1) takes for n = 2 the form

\frac{P_{m_{1},m_{2}}(x_{1}, x_{2}; q, t; a_{1}, a_{2}, a_{3}, a_{4})}{P_{m_{1},m_{2}}(ta_{1}, a_{1}; q, t; a_{1}, a_{2}, a_{3}, a_{4})} = ∑_{k_{2}=0}^{m_{2}} ∑_{k_{1}=k_{2}}^{m_{1}} \frac{P_{k_{1},k_{2}}^{ip}(q^{m_{1}}ta′_{1}, q^{m_{2}}a′_{1}; q, t, a′_{1})}{P_{k_{1},k_{2}}^{ip}(q^{k_{1}}ta′_{1}, q^{k_{2}}a′_{1}; q, t, a′_{1})}

× \frac{P_{k_{1},k_{2}}^{ip}(x_{1}, x_{2}; q, t, a_{1})}{P_{k_{1},k_{2}}(ta_{1}, a_{1}; q, t; a_{1}, a_{2}, a_{3}, a_{4})} . (11.1)

This gives an explicit expression for Koornwinder polynomials for n = 2, since we have explicit expressions for everything on the right-hand side. For the first quotient on the right-hand side of (11.1) we get by (10.6):

\frac{P_{k_{1},k_{2}}^{ip}(q^{m_{1}}ta′_{1}, q^{m_{2}}a′_{1}; q, t, a′_{1})}{P_{k_{1},k_{2}}^{ip}(q^{k_{1}}ta′_{1}, q^{k_{2}}a′_{1}; q, t, a′_{1})} = \frac{(q^{m_{1}}ta_{1}^{′2}, q^{−m_{1}}t^{−1}, q^{m_{2}}a_{1}^{′2}, q^{−m_{2}}; q)_{k_{2}}}{(q^{k_{1}}ta_{1}^{′2}, q^{−k_{1}}t^{−1}, q^{k_{2}}a_{1}^{′2}, q^{−k_{2}}; q)_{k_{2}}}

× \frac{(q^{m_{1}+k_{2}}t^{2}a_{1}^{′2}, q^{m_{2}−k_{1}}; q)_{k_{1}−k_{2}}}{(q^{k_{1}+k_{2}}t^{2}a_{1}^{′2}, q^{k_{2}−k_{1}}; q)_{k_{1}−k_{2}}} _{4}ϕ_{3}({q^{−k_{1}+k_{2}}, t, q^{m_{2}+k_{2}}a_{1}^{′2}, q^{m_{2}−k_{2}} \atop q^{1−k_{1}+k_{2}}t^{−1}, q^{m_{2}+k_{2}}t^{2}a_{1}^{′2}, q^{−m_{1}+k_{2}}} ; q, q) . (11.2)

As for the second quotient on the right-hand side the numerator is obtained from (10.6), i.e.,

P_{k_{1},k_{2}}^{ip}(x_{1}, x_{2}; q, t, a_{1}) = \frac{q^{− \frac{1}{2} k_{1}(k_{1}−1)− \frac{1}{2} k_{2}(k_{2}−1)}}{(−t)^{k_{1}−k_{2}}a_{1}^{k_{1}+k_{2}}} (a_{1}x_{1}, a_{1}x_{1}^{−1}, a_{1}x_{2}, a_{2}x_{2}^{−1}; q)_{k_{2}}

× (q^{k_{2}}ta_{1}x_{1}, q^{k_{2}}ta_{1}x_{1}^{−1}; q)_{k_{1}−k_{2}} _{4}ϕ_{3}({q^{−k_{1}+k_{2}}, t, q^{k_{2}}a_{1}x_{2}, q^{k_{2}}a_{1}x_{2}^{−1} \atop q^{1−k_{1}+k_{2}}t^{−1}, q^{k_{2}}ta_{1}x_{1}, q^{k_{2}}ta_{1}x_{1}^{−1}} ; q, q) , (11.3)

and the denominator is obtained from (4.15), i.e.,

P_{k_{1},k_{2}}(ta_{1}, a_{1}; q, t; a_{1}, a_{2}, a_{3}, a_{4}) = t^{−k_{1}}a_{1}^{−k_{1}+k_{2}} \frac{(ta_{1}^{′2}; q)_{k_{1}} (a_{1}^{′2}; q)_{k_{2}}}{(t^{2}a_{1}^{′2}; q)_{2k_{1}} (a_{1}^{′2}; q)_{2k_{2}}}

× (ta_{1}a_{2}, ta_{1}a_{3}, ta_{1}a_{4}; q)_{k_{1}} (a_{1}a_{2}, a_{1}a_{3}, a_{1}a_{4}; q)_{k_{2}} \frac{(t^{2}a_{1}^{′2}; q)_{k_{1}+k_{2}}}{(ta_{1}^{′2}; q)_{k_{1}+k_{2}}} \frac{(t^{2}; q)_{k_{1}−k_{2}}}{(t; q)_{k_{1}−k_{2}}} . (11.4)

11.2. Binomial formula for BC_{2}-type Jacobi polynomials. We can specialize the binomial formula (9.1) for BC_{n}-type Jacobi polynomials to n = 2 and get everything on the right-hand side explicitly:

\frac{P_{m_{1},m_{2}}(x_{1}, x_{2}; τ; α, β)}{P_{m_{1},m_{2}}(0, 0; τ; α, β)}

= ∑_{k_{2}=0}^{m_{2}} ∑_{k_{1}=k_{2}}^{m_{1}} \frac{P_{k_{1},k_{2}}^{ip}(m_{1} + τ + α′, m_{2} + α′; τ, α′)}{P_{k_{1},k_{2}}^{ip}(k_{1} + τ + α′, k_{2} + α′; τ, α′)} \frac{P_{k_{1},k_{2}}(x_{1}, x_{2}; τ)}{P_{k_{1},k_{2}}(0, 0; τ; α, β)} . (11.5)

The first quotient on the right-hand side can be explicitly evaluated as a limit case of (11.2) by using (7.1):

\frac{P_{k_{1},k_{2}}^{ip}(m_{1} + τ + α′, m_{2} + α′; τ, α′)}{P_{k_{1},k_{2}}^{ip}(k_{1} + τ + α′, k_{2} + α′; τ, α′)} = \frac{(m_{1} + τ + 2α′, −m_{1} − τ, m_{2} + 2α′, −m_{2})_{k_{2}}}{(k_{1} + τ + 2α′, −k_{1} − τ, k_{2} + 2α′, −k_{2})_{k_{2}}}

× \frac{(m_{2} + k_{2} + 2τ + 2α′, m_{2} − k_{2})_{k_{1}−k_{2}}}{(k_{1} + k_{2} + 2τ + 2α′, k_{2} − k_{1})_{k_{1}−k_{2}}}

× _{4}F_{3}({−k_{1} + k_{2}, τ, m_{2} + k_{2} + 2α′, m_{2} − k_{2} \atop 1 − k_{1} + k_{2} − τ, m_{2} + k_{2} + 2τ + 2α′, −m_{1} + k_{2}} ; 1) . (11.6)

For the numerator of the second quotient we have (10.15):

P_{k_{1},k_{2}}(x_{1}, x_{2}; τ) = \frac{(q; q)_{k_{1}−k_{2}}}{(t; q)_{k_{1}−k_{2}}} (x_{1}x_{2})^{\frac{1}{2} (k_{1}+k_{2})} C_{k_{1}−k_{2}}^{τ} (\frac{x_{1} + x_{2}}{2(x_{1}x_{2})^{\frac{1}{2}}} ) .  (11.7)

The denominator of the second quotient can be obtained by specializing (7.5):

P_{k_{1},k_{2}}(0, 0; τ; α, β) = (−1)^{k_{1}+k_{2}} \frac{(τ + 2α′, τ + α + 1)_{k_{1}} (2α′, α + 1)_{k_{2}}}{(2τ + 2α′)_{2k_{1}} (2α′)_{2k_{2}}}

× \frac{(2τ + 2α′)_{k_{1}+k_{2}}}{(τ + 2α′)_{k_{1}+k_{2}}} \frac{(2τ)_{k_{1}−k_{2}}}{(τ)_{k_{1}−k_{2}}} . (11.8)

The explicit formula for BC_{2}-type Jacobi polynomials which is given by combination of (11.5), (11.6), (11.7) and (11.8) was obtained in a very different way by Koornwinder and Sprinkhuizen [14, Corollary 6.6].

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T. H. Koornwinder, Korteweg-de Vries Institute, University of Amsterdam,, P.O. Box 94248, 1090 GE Amsterdam, The Netherlands; E-mail: T.H.Koornwinder@uva.nl