Quantum Integrable Systems and Special Functions.I

A.M. Perelomov^{1}

Max-Planck-Institut für Mathematik, Vivatsgasse 7, Bonn, Germany

Abstract

The class of quantum integrable systems associated with root systems was introduced in [OP 1977] as a generalization of the Calogero–Sutherland systems [Ca 1971], [Su 1972]. For the potential v(q) = κ(κ − 1) sin^{−2} q, the wave functions of such systems are related to polynomials in l variables (l is a rank of root system) and they are a generalization of Gegenbauer polynomials and Jack polynomials [Ja 1970]. In [Pe 1998a], it was proved that the series for the product of two such polynomials is a κ-deformation of the Clebsch–Gordan series. This yields recurrence relations for these polynomials, in particular, for generalized zonal polynomials on symmetric spaces. The present paper follows papers [Pe 1998a], [PRZ 1998]. In last of them, the recurrence relations were used to compute the explicit expressions for A_{2} type polynomials, i.e., for the wave functions of the three-body Calogero–Sutherland system. As it was shown by Ragoucy, Zaugg and the author of this paper (see Appendix 2), the similar results are also valid in A_{2} case for the more general two-parameter deformation ((q, t)- deformation) introduced by Macdonald [Ma 1988].

1 Introduction

The class of quantum integrable systems associated with root systems was introduced in [OP 1977] (see also [OP 1978] and [OP 1983]) as a generalization of the Calogero–Sutherland systems [Ca 1971], [Su 1972]. Such systems depend on one real parameter κ (for root systems of the type A_{n}, D_{n} and E_{6}, E_{7}, E_{8}), on two parameters (for the type B_{n}, C_{n}, F_{4} and G_{2}) and on three parameters for the type BC_{n}. These parameters are related to the coupling constants of the quantum system.

For the potential v(q) = κ(κ−1) sin^{−2} q and special values of parameter κ, the wave functions correspond to the characters of the compact simple Lie groups (κ = 1) [We 1925/26] or to zonal spherical functions on symmetric spaces (κ = 1/2, 2, 4) [Ha 1958], [He 1978]. At arbitrary values of κ, they provide an interpolation between these objects.

This class has many remarkable properties. Here we mention only one: the wave functions of such systems are a natural generalization of special functions (hypergeometric functions) to the case of several variables. The history of this problem and some results may be found in [OP 1983]. In [Pe 1998a], it was shown that the product of two wave functions is a finite linear combination of analogous functions, namely, of functions that appear in the corresponding Clebsch–Gordan series. In other words, this deformation (κ-deformation) does not change the Clebsch–Gordan series. For rank 1, we obtain the well-known cases of the Legendre, Gegenbauer and Jacobi polynomials, and the limiting cases of the Laguerre and Hermite polynomials (see for example [Vi 1968]). Some other cases were also considered in [He 1955], [Ja 1970], [Ko 1974], [Ja 1975], [Vr 1976], [Se 1977], [KS 1978], [Ma 1982], [Pr 1984], [Vr 1984], [HO 1987], [He 1987], [Op 1988a], [Op 1988b], [Ma 1988], [La 1989], [Op 1989], [St 1989], [KS 1995] and [CP 1997]. In [PRZ 1998], κ-deformed Clebsch–Gordan series was used in order to obtain the explicit expressions for the generalized Gegenbauer polynomials^{2} of type A_{2} what gives the explicit solution of the three-body Calogero–Sutherland model. For special values of κ = 1/2, 2, 4, these formulae give the explicit expressions for zonal polynomials of type A_{2}.

1 On leave of absence from Institute for Theoretical and Experimental Physics, 117259 Moscow, Russia.

Current e-mail address: perelomo@posta.unizar.es.

In Appendix 2, are presented analogous results obtained by E. Ragoucy, Ph. Zaugg and the author for two-parameter family of polynomials of type A_{2} introduced by Ruijsenaars [Ru 1987] and Macdonald [Ma 1988].

2 General description

The systems under consideration are described by the Hamiltonian (for more details, see [OP 1983])

H = \frac{1}{2} p^{2} + U(q),  p^{2} = (p, p) = ∑_{j=1}^{l} p_{j}^{2},  (2.1)

where p = (p_{1}, ..., p_{l}), p_{j} = − i ∂/∂q_{j}, is a momentum operator, and q = (q_{1}, ..., q_{l}) is a coordinate vector in the l-dimensional vector space V ∼ ℝ^{l} with standard scalar product (· , ·). The potential U(q) is constructed by means of a certain system of vectors R^{+} = {α} in V (the so-called root systems):

U = ∑_{α∈R^{+}} g_{α}^{2} v(q_{α}),  q_{α} = (α, q),  g_{α}^{2} = κ_{α}(κ_{α} − 1),  g_{α} = g_{β}, if (α, α) = (β, β).

(2.2)

Such systems are completely integrable for potentials of five types (see [OP 1983] for A_{l}; [HO 1987], [He 1987], [Op 1988a], [Op 1988b] and [Op 1989] for a general case). They are a generalization of the Calogero–Sutherland systems [Ca 1971], [Su 1972] for which {α} = {e_{i} − e_{j}}, {e_{j}} being a standard basis in V.

In this paper, we consider in details only the case of A_{2} with potential v(q) = sin^{−2} q.

3 Root systems

We give here only basic definitions. For more details, see [Hu 1972], [Ma 1988] and [OV 1990].

2 In some papers, the name Jack polynomials [Ja 1970] is used. However, the Jack polynomials are a very special case of polynomials under consideration. So, we prefer to use the name generalized Gegenbauer polynomials for the general case and Jack polynomials for the special case.

Let V be a l-dimensional real vector space with a standard scalar product (· , ·), (α, β) = ∑ α_{j}β_{j}, and let s_{α} be a reflection in the hyperplane through origin orthogonal to the vector α,

s_{α} q = q − (q, α^{∨}) α,  α^{∨} = \frac{2}{(α, α)} α .  (3.1)

Consider a finite set of non-zero vectors R = {α} generating V and satisfying the following conditions:

1. For any α ∈ R, the reflection s_{α} conserves R : s_{α}R = R.

2. For all α, β ∈ R, we have (α^{∨}, β) ∈ ℤ.

The set {s_{α}} generates the finite group W (R) which is called the Weyl group of R. The root system R is called reduced one if only ±α are vectors being collinear to α in R.

Let us choose the hyperplane which does not contain any root. This hyperplane divide the root system on two subsets, R = R^{+} ⋃ R^{−}, where R^{+} is called the set of positive roots. In R^{+}, there is the basis {α_{1}, …, α_{l}} such that α = ∑_{j} n_{j}α_{j}, n_{j} ≥ 0, for any α ∈ R^{+}. It is called the set of simple roots. The root system R is called irreducible if it cannot decompose onto two non-empty subsets R_{1} and R_{2} being orthogonal to each other.

Let {α_{1}, …, α_{l}} be the set of simple roots, let R^{+} be the set of positive roots, and {λ_{j}} be a dual basis or the basis of fundamental weights: (λ_{j}, α_{k}) = δ_{jk}.

Let Q be the root lattice, and Q^{+} be the cone of positive roots

Q = {β: β = ∑_{j=1}^{l} m_{j}α_{j}, m_{j} ∈ ℤ };  Q^{+} = {γ: γ = ∑_{j=1}^{l} n_{j}α_{j}, n_{j} ∈ ℕ} .  (3.2)

Let P be a weight lattice, and P^{+} be a cone of dominant weights:

P = {λ: λ = ∑_{j=1}^{l} m_{j}λ_{j}, m_{j} ∈ ℤ};  P^{+} = {μ: μ = ∑_{j=1}^{l} n_{j}λ_{j}, n_{j} ∈ ℕ} .  (3.3)

Following [Vr 1976] and [Ma 1988], we define a partial order on P as follows: λ ≥ μ if and only if λ − μ ∈ Q^{+}, or (λ, λ_{j}) ≥ (μ, λ_{j}) for all j = 1, …, l. The set of linear combinations over ℝ of functions f_{λ}(q) = exp {2i(λ, q)}, λ ∈ P, q ∈ V, may be considered as the group algebra A over ℝ of the free Abelian group P. For any λ ∈ P, we denote the corresponding element of A as e^{λ} ∼ f_{λ}(q). So, e^{λ} e^{μ} = e^{λ+μ}, (e^{λ})^{−1} = e^{−λ}, and the identity element of A as e^{0} = 1. Then e^{λ}, λ ∈ P, form an ℝ-basis of A.

The Weyl group W (R) acts on P and hence also on A : s(e^{λ}) = e^{sλ} for s ∈ W and λ ∈ P. Let A^{W} denotes the subalgebra of W -invariant elements of A. It is evident that each W -orbit in P contains only one point in P^{+}. Consequently, the monomial symmetric functions

m_{λ} = ∑_{μ∈W ·λ} e^{μ},  λ ∈ P^{+}  (3.4)

form the ℝ-basis of A^{W}.

4 The Clebsch–Gordan series

Let us recall the main results of [Pe 1998a] and specialize them to the A_{2} case with potential v(q) = sin^{−2} q.

The Schrödinger equation for this quantum system has the form

H Ψ^{κ} = E(κ) Ψ^{κ};  H = − Δ_{2} + U(q_{1}, q_{2}, q_{3}),  Δ_{2} = ∑_{j=1}^{3} \frac{∂^{2}}{∂q_{j}^{2}}  (4.1)

with potential

U(q_{1}, q_{2}, q_{3}) = κ(κ − 1) (sin^{−2}(q_{1} − q_{2}) + sin^{−2}(q_{2} − q_{3}) + sin^{−2}(q_{3} − q_{1})) .  (4.2)

The ground state wave function and its energy are

Ψ_{0}^{κ}(q) = (∏_{j<k}^{3} sin(q_{j} − q_{k}))^{κ} ,  E_{0}(κ) = 8 κ^{2}.  (4.3)

Substituting Ψ_{λ}^{κ} = Φ_{λ}^{κ} Ψ_{0}^{κ} in (4.1), we obtain

−Δ^{κ} Φ_{λ}^{κ} = ε_{λ}(κ) Φ_{λ}^{κ},  Δ^{κ} = Δ_{2} + Δ_{1}^{κ},  ε_{λ}(κ) = E_{λ}(κ) − E_{0}(κ).  (4.4)

Here the operator Δ_{1}^{κ} takes the form

Δ_{1}^{κ} = κ ∑_{j<k}^{3} cot(q_{j} − q_{k}) (\frac{∂}{∂q_{j}} − \frac{∂}{∂q_{k}} ) .  (4.5)

It is easy to see that the set of symmetric polynomials in variables exp(2iq_{j}) is invariant under the action of Δ^{κ}. Such polynomial m_{λ} is labelled by the SU(3) highest weight λ = mλ_{1} + nλ_{2}, with m, n being non-negative integers, and λ_{1,2} being two fundamental weights. In general,

Φ_{λ}^{κ} = ∑_{μ≤λ} C_{λ}^{μ}(κ) m_{μ}, μ, λ ∈ P^{+},  m_{μ} = ∑_{ν∈W ·μ} e^{2i(q,ν)},  (4.6)

where P^{+} denotes the cone of dominant weights, W is the Weyl group, and C_{λ}^{μ}(κ) are some constants.

As it was shown in [Pe 1998a], the product of two wave functions is a finite sum of wave functions (a sort of the κ-deformed Clebsch–Gordan series):

Φ_{μ}^{κ} Φ_{λ}^{κ} = ∑_{ν∈D_{μ}(λ)} C_{μλ}^{ν}(κ) Φ_{ν}^{κ}.  (4.7)

In this equation, D_{μ}(λ) = (D_{μ} + λ) ∩ P^{+}, where D_{μ} is a weight diagram of the representation with the highest weight μ.

Since Φ_{μ}^{κ} are symmetric functions of exp(2iq_{j}), it is convenient to use a new set of variables:

\begin{matrix}z_{1} = e^{2iq_{1}} + e^{2iq_{2}} + e^{2iq_{3}}, \\ z_{2} = e^{2i(q_{1}+q_{2})} + e^{2i(q_{2}+q_{3})} + e^{2i(q_{3}+q_{1})}, \\ z_{3} = e^{2i(q_{1}+q_{2}+q_{3})}. \end{matrix}  (4.8)

In the centre-of-mass frame (∑_{i} q_{i} = 0), the wave functions depend only on two variables choosen as z_{1} and z_{2} (in this case, z_{3} = 1). In these variables, up to a normalisation factor, we have

Δ^{κ} = (z_{1}^{2} − 3z_{2}) ∂_{1}^{2} + (z_{2}^{2} − 3z_{1}) ∂_{2}^{2} + (z_{1}z_{2} − 9) ∂_{1}∂_{2} + (3κ + 1) (z_{1}∂_{1} + z_{2}∂_{2}),  (4.9)

where ∂_{i} = ∂/∂z_{i}. Corresponding eigenvalues are

ε_{m,n}(κ) = m^{2} + n^{2} + mn + 3 κ(m + n).  (4.10)

We shall use the normalization for polynomials Φ_{λ}^{κ} such that the coefficient at the highest monomial is equal to one. Denoting them by P_{m,n}^{κ}, we have

P_{m,n}^{κ}(z_{1}, z_{2}) = ∑_{p,q} C_{m,n}^{p,q}(κ) z_{1}^{p} z_{2}^{q} = z_{1}^{m} z_{2}^{n} + lower terms,  (4.11)

with p + q ≥ m + n and p − q ≡ m − n (mod 3). As it is easy to see, the first polynomials are

P_{0,0}^{κ} = 1,  P_{1,0}^{κ} = z_{1},  P_{0,1}^{κ} = z_{2}.  (4.12)

Simple consequences of (4.7) for P_{λ}^{κ} = P_{1,0}^{κ} or P_{0,1}^{κ} are [Pe 1998]

z_{1} P_{m,n}^{κ} = P_{m+1,n}^{κ} + a_{m,n}(κ) P_{m,n−1}^{κ} + c_{m}(κ) P_{m−1,n+1}^{κ},

(4.13)

z_{2} P_{m,n}^{κ} = P_{m,n+1}^{κ} + ã_{m,n}(κ) P_{m−1,n}^{κ} + c_{n}(κ) P_{m+1,n−1}^{κ},

where

a_{m,n}(κ) = ã_{n,m}(κ) = c_{n}(κ) c_{m+n+κ}(κ),

(4.14)

c_{m}(κ) = \frac{e(m)}{e(κ + m)},  e(m) = \frac{m}{m − 1 + κ}.

Below we shall construct such polynomials using these recurrence relations.

5 A_{1} case

Formulae of this section are well-known, see for example [Vi 1968]. However, the presentation of results being not standard is very useful for more complicated cases, for example, for A_{2}.

Recall that the unitary irreducible representation d of the Lie algebra A_{1} ∼ su(2) is characterized by one integer non-negative number l, d = d_{l}, and the Clebsch–Gordan series has the well-known form:

d_{k} ⊗ d_{l} = ⊕_{m=0}^{m_{1}} d_{|k−l|+2m},  m_{1} = m_{max} = \frac{1}{2} (k + l − |k − l|).  (5.1)

We begin with the simplest case

d_{1} ⊗ d_{l} = d_{l−1} ⊕ d_{l+1}, l ≥ 1,
d_{1} ⊗ d_{0} = d_{1}.

Corresponding formulae for characters are

\begin{matrix}χ_{1}(θ) · χ_{l}(θ) = χ_{l−1}(θ) + χ_{l+1}(θ), l ≥ 1;  χ_{l}(θ) = e^{ilθ} + e^{i(l−2)θ} + ⋯ + e^{−ilθ}, \\ χ_{l}(θ) = P_{l}^{1}(z) = z^{l} + ⋯,  z = 2 cos θ. \end{matrix}  (5.2)

Hence for polynomials P_{l}^{1}(z) we have the formula

z P_{l}^{1}(z) = P_{l+1}^{1}(z) + P_{l−1}^{1}(z),  P_{0}^{1} = 1, P_{1}^{1} = z.  (5.3)

This formula may be considered as a basic recurrence relation for these polynomials. From this, it follows the equation for the generating function

F^{1}(z, u) = ∑_{l=0}^{∞} P_{l}^{1}(z) u^{l},

(5.4)

z F^{1}(z, u) = u^{−1} (F^{1}(z, u) − 1) + u F^{1}(z).

Hence

F^{1}(z, u) = (1 − zu + u^{2})^{−1} .  (5.5)

Let us define now the κ-deformed generating function and corresponding system of polynomials

F^{κ}(z, u) = (1 − zu + u^{2})^{−κ} = ∑_{l=0}^{∞} P̃_{l}^{κ}(z) u^{l}.  (5.6)

This is a basic formula. We shall demonstrate below how to get all other formulae from it. First we have

P̃_{l}^{κ}(z) = \frac{(κ)_{l}}{l!} P_{l}^{κ}(z), P_{l}^{κ}(z) = z^{l} − \frac{l(l − 1)}{l − 1 + κ} z^{l−2} + ⋯,  (5.7)

where

(x)_{n} = x(x + 1) ⋯ (x + n − 1).  (5.8)

Then

\begin{matrix}F_{z}^{κ} = κu F^{κ+1}, & F_{zz}^{κ} = κ(κ + 1)u^{2} F^{κ+2}, \\ F_{u}^{κ} = κ(z − 2u) F^{κ+1}, & F_{uu}^{κ} = κ(κ + 1) (z − 2u)^{2} F^{κ+2} − 2κ F^{κ+1}, \end{matrix}  (5.9)

where F_{z} = ∂F/∂z, F_{u} = ∂F/∂u, and so on.

From this, it is easy to get the equation

(D_{z}^{2} − 4 ∂_{z}^{2} + 2κ D_{z}) F^{κ}(z, u) = (D_{u}^{2} + 2κ D_{u}) F^{κ}(z, u),  (5.10)

where D_{z} = z ∂_{z}, D_{u} = u ∂_{u}. Content-Length: 452 X-Lines: 11 Status: O Substituting the expansion (5.6) for F^{κ}, we obtain the differential equation for P̃_{l}^{κ},

((z^{2} − 4) ∂_{z}^{2} + (2κ + 1)z ∂_{z}) P̃_{l}^{κ} = l(l + 2κ) P̃_{l}^{κ}.  (5.11)

Let us rewrite now (5.9) in the form

\begin{matrix}(1 − zu + u^{2}) F_{u}^{κ} = κ(z − 2u) F^{κ}, \\ F_{u}^{κ} − z (D_{u} + κ) F^{κ} + u (D_{u} + 2κ) F^{κ} = 0. \end{matrix}  (5.12)

From this, it follows the recurrence relation

(n + 1) P̃_{n+1}^{κ} − z(n + κ) P̃_{n}^{κ} + (n − 1 + 2κ) P̃_{n−1}^{κ} = 0.  (5.13)

At κ = 1, it is equivalent to the basic recurrence relation (5.3), and for polynomilas P_{n}^{κ}(z) we have

z P_{n}^{κ}(z) = P_{n+1}^{κ}(z) + a_{n}(κ) P_{n−1}^{κ}(z),  a_{n}(κ) = \frac{n(n − 1 + 2κ)}{(n − 1 + κ)(n + κ)},  P_{0}^{κ} = 1, P_{1}^{κ} = z.

(5.14)

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So, we may interprete (5.13), (5.14) as a special deformation of the Clebsch–Gordan series for the Lie algbera A_{1} ∼ su(2).

From (5.9), also it follows

F_{z}^{κ} − u(D_{z} + κ) F^{κ} + u^{2} F_{z}^{κ} = 0.  (5.15)

Therefore, we have

∂_{z} P̃_{n+1}^{κ} − (z∂_{z} + κ) P̃_{n}^{κ} + ∂_{z} P̃_{n−1}^{κ} = 0.  (5.16)

Combining the expressions for F_{u}^{κ} and F_{z}^{κ}, we obtain

uF_{u}^{κ} = (z − 2u) F_{z}^{κ}.  (5.17)

Hence

n P̃_{n}^{κ} − z ∂_{z}P̃_{n}^{κ} + 2 ∂_{z}P̃_{n−1}^{κ} = 0.  (5.18)

Using (5.9), we get

(1 − \frac{1}{2} zu) F_{z}^{κ} = \frac{1}{2} u(u F_{u}^{κ} + 2κ F^{κ}).  (5.19)

From this, it follows that

∂_{z} P̃_{n+1}^{κ} − \frac{1}{2} z ∂_{z}P̃_{n}^{κ} = \frac{1}{2} (n + 2κ) P̃_{n}^{κ}.  (5.20)

We also have

(4 − z^{2}) ∂_{z}F^{κ} = 4 κu F^{κ} − z(z − 2u)u F_{u}^{κ} = u(4κ + 2 D_{u}) F^{κ} − zD_{u} F^{κ}.  (5.21)

Hence

(4 − z^{2}) ∂_{z}P̃_{n}^{κ} = 2 (2κ + n − 1) P̃_{n−1}^{κ} − zn P̃_{n}^{κ},  (5.22)

or

((4 − z^{2}) \frac{d}{dz} + nz) P̃_{n}^{κ} = 2 (2κ + n − 1) P̃_{n−1}^{κ}.  (5.23)

In similar way, we obtain

((4 − z^{2}) \frac{d}{dz} − (n + 1)z) P̃_{n}^{κ} = − (n + 2 − 2κ) P̃_{n+1}^{κ},  (5.24)

and

\frac{1}{2} (2n + 1) (4 − z^{2}) ∂_{z}P̃_{n}^{κ} = (n + 1)(n + 2κ − 1) P̃_{n−1}^{κ} − n(n + 2 − 2κ) P̃_{n+1}^{κ}.  (5.25)

Content-Length: 1716 X-Lines: 47 Status: O Finally let us give the important Rodrigues formula:

P̃_{n}^{κ}(z) = \frac{1}{2^{n} n!} \frac{1}{(z^{2} − 4)^{κ−(1/2)}} \frac{d^{n}}{dz^{n}} (z^{2} − 4)^{n+κ−(1/2)} .  (5.26)

Note also that

\frac{d}{dz} P̃_{n}^{κ}(z) = κ P̃_{n−1}^{κ+1}(z)  (5.27)

and that from the identity

F^{κ} = F^{κ_{1}} · F^{κ_{2}}, κ = κ_{1} + κ_{2}

it follows the formula

P̃_{n}^{κ}(z) = ∑_{n_{1}+n_{2}=n} P̃_{n_{1}}^{κ_{1}}(z) P̃_{n_{2}}^{κ_{2}}(z).  (5.28)

6 A_{2} case

Now we proceed to the case of A_{2} ∼ su(3). In this case, the representation d of A_{2} is characterized by two non-negative numbers d = d_{mn}. We have two fundamental representations

d_{10},  d_{01};  (dim d_{10} = dim d_{01} = 3),

and also representations

d_{n0},  d_{0n};  (dim d_{n0} = dim d_{0n} = \frac{1}{2} (n + 1)(n + 2)) ,

d_{n1} (dim d_{n1} = (n + 1)(n + 3)), d_{nn} (dim d_{nn} = (n + 1)^{3}).

We start with the Clebsch–Gordan series

\begin{matrix}d_{10} ⊗ d_{n+1,0} = d_{n+2,0} ⊕ d_{n,1}, \\ d_{01} ⊗ d_{n0} = d_{n,1} ⊕ d_{n−1,0}. \end{matrix}  (6.1)

Excluding d_{n,1}, we obtain

d_{n−1,0} ⊖ (d_{01} ⊗ d_{n0}) ⊕ (d_{10} ⊗ d_{n+1,0}) ⊖ d_{n+2,0} = 0,

or

χ_{n−1,0} − z_{2} χ_{n,0} + z_{1} χ_{n+1,0} − χ_{n+2,0} = 0,  (6.2)

where are introduced the notations:

\begin{matrix}z_{1} = χ_{10} = e^{iθ_{1}} + e^{iθ_{2}} + e^{iθ_{3}}, \\ z_{2} = χ_{01} = e^{−iθ_{1}} + e^{−iθ_{2}} + e^{−iθ_{3}}. \end{matrix}  (6.3)

From this, we obtain the expression for the generating function

F_{0}^{1}(z_{1}, z_{2}; u) = ∑_{n=0}^{∞} χ_{n0}(z_{1}, z_{2}) u^{n},
F_{0}^{1}(z_{1}, z_{2}; u) = (1 − z_{1}u + z_{2} u^{2} − u^{3})^{−1} .  (6.4)

Content-Length: 3413 X-Lines: 81 Status: O Let us define now the κ-deformed functions P̃_{n,0}^{κ}(z_{1}, z_{2}) by the formula

F^{κ}(z_{1}, z_{2}; u) = (1 − z_{1} u + z_{2} u^{2} − u^{3})^{−κ} = ∑_{n=0}^{∞} P̃_{n,0}^{κ}(z_{1}, z_{2}) u^{n}.  (6.5)

Analogously to A_{1} case, we may obtain all other formulae from this one. Differentiating F^{κ} on u, z_{1} and z_{2}, we get

F_{u}^{κ} = κ (z_{1} − 2z_{2} u + 3u^{2}) F^{κ+1},

\begin{matrix}F_{z_{1},z_{1}}^{κ} = κ(κ + 1)u^{2} F^{κ+2}, & F_{z_{2},z_{2}}^{κ} = κ(κ + 1)u^{4} F^{κ+2}, \\ F_{z_{1},z_{2}}^{κ} = − κ(κ + 1)u^{3} F^{κ+2}, & u F_{u}^{κ} = κ (z_{1}u − 2 z_{2}u^{2} + 3 u^{3}) F^{κ+1}, \end{matrix}

and

D_{u}^{2} F^{κ} = {κ (z_{1}u − 4 z_{2}u^{2} + 9u^{3}) (1 − z_{1}u + z_{2}u^{2} − u^{3})
+ κ(κ + 1) (z_{1}u − 2z_{2}u^{2} + 3u^{3})^{2}} F^{κ+2}, D_{u} = u∂_{u},

or

\begin{matrix}(1 − z_{1}u + z_{2} u^{2} − u^{3}) F_{u}^{κ} = κ (z_{1} − 2 z_{2}u + 3 u^{2}) F^{κ}, \\ (n + 3) P̃_{n+3}^{κ} − z_{1}(n + 2) P̃_{n+2}^{κ} + z_{2}(n + 1) P̃_{n+1}^{κ} − n P̃_{n}^{κ} \\ = κz_{1} P̃_{n+2}^{κ} − 2 κz_{2} P̃_{n+1}^{κ} + 3 κ P̃_{n}^{κ}. \end{matrix}  (6.6)

So, we obtain the important recurrence formula

(n + 3) P̃_{n+3,0}^{κ} = (n + 2 + κ)z_{1} P̃_{n+2,0}^{κ} − (n + 1 + 2κ)z_{2} P̃_{n+1,0}^{κ} + (n + 3κ) P̃_{n,0}.  (6.7)

Now let us differentiate F^{κ} on z_{1}. We have

F_{z_{1}}^{κ} = κu F^{κ+1}.  (6.8)

Hence

∂_{z_{1}}P_{n,0}^{κ} = n P_{n−1,0}^{κ+1}.  (6.9)

Analogously,

F_{z_{2}}^{κ} = − κu^{2} F^{κ+1}.  (6.10)

Therefore

∂_{z_{2}}P_{n,0}^{κ} = − \frac{n(n − 1)}{κ + n − 1} P_{n−2,0}^{κ+1}.

Finally we have the basic differential equation for F^{κ}(z_{1}, z_{2}; u):

((D_{z_{1}}^{2} + D_{z_{2}}^{2} + D_{z_{1}}D_{z_{2}}) − 3 z_{2}∂_{z_{1}}^{2} − 3 z_{1}∂_{z_{2}}^{2} − 9 ∂_{z_{1}}∂_{z_{2}} + 3 κ(D_{z_{1}} + D_{z_{2}})) F^{κ}
= (D_{u}^{2} + 3 κD_{u}) F^{κ}(z_{1}, z_{2}; u);
D_{z_{1}} = z_{1} ∂_{z_{1}},  D_{z_{2}} = z_{2} ∂z_{2},  D_{u} = u ∂_{u}.

Let us note that the normalization of polynomials P̃_{n,0}^{κ}(z_{1}, z_{2}) follows from the expression (6.5) for the generating function. Namely,

P̃_{n0}^{κ}(z_{1}, z_{2}) = \frac{(κ)_{n}}{n!} z_{1}^{n} + ⋯ = \frac{(κ)_{n}}{n!} P_{n,0}^{κ}(z_{1}, z_{2}),  (6.11)

where

(κ)_{n} = (κ)(κ + 1) ⋯ (κ + n − 1).

The main property of this normalization is that P̃_{n,0}^{κ}(z_{1}, z_{2}) has a polynomial dependence on the parameter κ.

Now we shall consider other Clebsch–Gordan series for κ = 1:

d_{1,0} ⊗ d_{n+1,0} = d_{n+2,0} ⊕ d_{n,1}.

According to [Pe 1998], the analogous formula is valid for an arbitrary value of κ, i.e.,

a_{n}z_{1} P̃_{n+1,0}^{κ} = b_{n} P̃_{n+2,0}^{κ} + c_{n} P̃_{n,1}^{κ},  (6.12)

where coefficients a_{n}, b_{n} and c_{n} do not depend on z_{1} and z_{2} but may depend on κ. Comparing of coefficients at z_{1}^{n+2} yields

a_{n} = κ + n + 1, b_{n} = n + 2.

From this relation, we may determine the function P̃_{n,1} up to the normalizing constant c_{n}:

c_{n}(κ) P̃_{n,1}^{κ} = (κ + n + 1)z_{1} P̃_{n+1,0}^{κ} − (n + 2) P̃_{n+2,0}^{κ}.  (6.13)

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Let us calculate now the generating function for both left-hand and right-hand sides of this equation,

G^{κ} = ∑_{n=0}^{∞} c_{n}(κ) P̃_{n,1}^{κ}(z_{1}, z_{2}),

(6.14)

G^{κ} = \frac{κz_{1}}{u} (F_{0}^{κ} − 1) + \frac{z_{1}}{u} F_{1}^{κ} − \frac{1}{u^{2}} (F_{1}^{κ} − κz_{1}u),

where

F_{0}^{κ} = ∑_{n=0}^{∞} P̃_{n,0}^{κ} u^{n}; F_{1}^{κ} = ∑_{n=0}^{∞} n P̃_{n,0}^{κ} u^{n} = D_{u} F_{0}^{κ},

and

G^{κ} = \frac{1}{u^{2}} (κz_{1}u − (1 − z_{1}u)D_{u}) F_{0}^{κ}.

Finally,

G^{κ} = κ (2z_{2} − (z_{1}z_{2} + 3) u + 2 z_{1}u^{2}) F_{0}^{κ+1}.  (6.15)

From this, it follows three-term recurrence relation

P̃_{n,1}^{κ} = κ (2 z_{2} P̃_{n,0}^{κ+1} − (z_{1}z_{2} + 3) P̃_{n−1,0}^{κ+1} + 2 z_{1} P̃_{n−2,0}^{κ+1}) .

Content-Length: 3189 X-Lines: 71 Status: O Let us give also the explicit expression for P̃_{n,0}^{κ}(z_{1}, z_{2}) and P̃_{n,1}^{κ}(z_{1}, z_{2}) in other variables

z_{1} = e^{iθ_{1}} + e^{iθ_{2}} + e^{iθ_{3}} = x_{1} + x_{2} + x_{3},
z_{2} = e^{−iθ_{1}} + e^{−iθ_{2}} + e^{−iθ_{3}} = x_{1}x_{2} + x_{2}x_{3} + x_{3}x_{1}.

As it was shown in [Pr 1984],

P̃_{n,0}^{κ} = ∑_{m_{1},m_{2},m_{3}} C_{m_{1},m_{2},m_{3}}^{n,0}(κ) x_{1}^{m_{1}}x_{2}^{m_{2}}x_{3}^{m_{3}},  m_{1} + m_{2} + m_{3} = n,  x_{1} x_{2} x_{3} = 1, (6.16)

where quantities C_{m_{1},m_{2},m_{3}}^{n,0}(κ) are κ-deformed three-nomial coefficients

C_{m_{1},m_{2},m_{3}}^{n,0}(κ) = \frac{n!}{m_{1}! m_{2}! m_{3}!} \frac{(κ)_{m_{1}}(κ)_{m_{2}}(κ)_{m_{3}}}{(κ)_{n}} .  (6.17)

The function P̃_{n,1}^{κ} may be defined by the formula

P̃_{n−2,1}^{κ}(z_{1}, z_{2}) = z_{1}P̃_{n−1,0}^{κ} − P̃_{n,0},  (6.18)
P̃_{n−2,1}^{κ}(z_{1}, z_{2}) = ∑_{m_{1},m_{2},m_{3}} C_{m_{1},m_{2},m_{3}}^{n,1}(κ) x_{1}^{m_{1}}x_{2}^{m_{2}}x_{3}^{m_{3}}.

We obtain

C_{m_{1},m_{2},m_{3}}^{n,1}(κ) = C_{m_{1}−1,m_{2},m_{3}}^{n−1,0}(κ) + C_{m_{1},m_{2}−1,m_{3}}^{n−1,0}(κ) + C_{m_{1},m_{2},m_{3}−1}^{n−1,0} − C_{m_{1},m_{2},m_{3}}^{n,0}(κ). (6.19)
Substituting here the explicit expression for C_{m_{1},m_{2},m_{3}}^{n,0}(κ), we get
C_{m_{1},m_{2},m_{3}}^{n,1}(κ) = C_{m_{1},m_{2},m_{3}}^{n,0}(κ) S_{m_{1},m_{2},m_{3}}^{n,1},  (6.20)

where

S_{m_{1},m_{2},m_{3}}^{n,1} = \frac{κ − 1 + n}{n} (\frac{m_{1}}{κ − 1 + m_{1}} + \frac{m_{2}}{κ − 1 + m_{2}} + \frac{m_{3}}{κ − 1 + m_{3}} − \frac{n}{κ − 1 + n}) . (6.21)

At κ = 1, the quantity S_{m_{1},m_{2},m_{3}}^{n,1}(κ) is equal to 2. As κ → ∞, it has an order κ^{−1} and is a symmetric function of m_{1}, m_{2}, m_{3}. So, it should have a form

S_{m_{1},m_{2},m_{3}}^{n,1} = \frac{α(κ − 1)^{2} + β(κ − 1) + γ}{(κ − 1 + m_{1})(κ − 1 + m_{2})(κ − 1 + m_{3})} .  (6.22)

Now let us follow [PRZ 1998] and construct the general polynomials in terms of the simplest polynomials (Jack polynomials) P_{m,0}^{κ} and P_{0,n}^{κ}. We get

P_{m,0}^{κ} P_{0,n}^{κ} = ∑_{i=0}^{min(m,n)} γ_{m,n}^{i} P_{m−i,n−i}^{κ}.  (6.23)

This is a consequence of equation (4.7), with the notable difference that the sum on the righthand side is over a restricted domain (actually, it parallels exactly the SU(3) Clebsch–Gordan decomposition).

To prove this, let us assume that (6.23) is valid up to (m, n). Then using (4.13) and (4.14), we get

P_{m,0}^{κ} P_{0,n+1}^{κ} = ∑_{i=0}^{min(m,n+1)} γ_{m,n+1}^{i} P_{m−i,n+1−i}^{κ} + c_{n} δ_{m,n+1}^{i} P_{m+1−i,n−1−i}^{κ},  (6.24)

where we defined

γ_{m,n+1}^{i} = γ_{m,n}^{i} + ã_{m−i+1,n−i+1} γ_{m,n}^{i−1} − c_{n} c_{m−i+1} γ_{m,n−1}^{i−1},  (6.25)
δ_{m,n+1}^{i} = c_{n}^{−1} c_{n−i} γ_{m,n}^{i} − γ_{m,n−1}^{i} − a_{m−i+1,n−i} γ_{m,n−1}^{i−1}
+ c_{n−1} c_{κ+n−1} γ_{m,n−2}^{i−1}.  (6.26)

From the polynomial normalization, we already know that γ_{m,n}^{0} = 1. After a straigthforward computation, the solution to (6.25) proved to be equal

γ_{m,n}^{i} = \frac{e(2κ + m + n + 1 − i)_{−i}}{e(1)_{i} e(m + 1)_{−i} e(n + 1)_{−i}} ,  (6.27)

where

e(m) = \frac{m}{m − 1 + κ} .

It implies that δ_{m,n+1}^{i} = 0 in (6.26). Let us give also the more explicit expression^{3}

γ_{m,n}^{i} = \frac{(κ)^{i}(m)_{i} (n)_{i} (3κ + m + n − 1 − i)_{i}}{i! (κ + m − 1)_{i} (κ + n − 1)_{i} (2κ + m + n − i)_{i}} ,  (6.28)

where

\begin{matrix}(x)^{i} = x(x + 1) ⋯ (x + i − 1), \\ (x)_{i} = x(x − 1) ⋯ (x − i + 1). \end{matrix}  (6.29)

Content-Length: 730 X-Lines: 19 Status: O The constructive aspect of this formula is in its inverted form.

Theorem 1. The generalized Gegenbauer polynomials P_{m,n}^{κ} of type A_{2} are given by the formula

P_{m,n}^{κ} = ∑_{i=0}^{min(m,n)} β_{m,n}^{i} P_{m−i,0}^{κ} P_{0,n−i}^{κ} ,  (6.30)

where the constants β_{m,n}^{i} are

β_{m,n}^{i} = \frac{(−1)^{i}}{i!} \frac{3κ + m + n − 2i}{3κ + m + n − i} \frac{(m)_{i} (n)_{i} (κ)_{i} (3κ + m + n − 1)_{i}}{(κ + m − 1)_{i}(κ + n − 1)_{i}(2κ + m + n − 1)_{i}} .  (6.31)

Note that β_{m,n}^{i} are obtained by using of the relation

β_{m,n}^{i} = − ∑_{j=0}^{i−1} β_{m,n}^{j} γ_{m−j,n−j}^{i−j} .  (6.32)

From this theorem, we see that the construction of a general polynomial P_{m,n}^{κ} is similar to the construction of SU(3) representations from tensor products of two fundamental representations. Content-Length: 1447 X-Lines: 39 Status: O

Likewise, we can consider other types of decompositions, such as

P_{m,0}^{κ} P_{n,0}^{κ} = ∑_{i=0}^{min(m,n)} γ̃_{m,n}^{i} P_{m+n−2i,i}^{κ}.  (6.33)

3 Note that this expression for γ_{m,n}^{i} may be obtained from the general Macdonald formula [Ma 1995], however, the way of proof given in [PRZ 1998] is more convenient here.

The proof is analogous to (6.23) (see footnote 3). The coefficients γ̃_{m,n}^{i} are given by the formula

γ̃_{m,n}^{i} = \frac{(κ)^{i} (m)_{i} (n)_{i} (2κ + m + n − 1 − i)_{i}}{i! (κ + m − 1)_{i} (κ + n − 1)_{i} (κ + m + n − i)_{i}} .  (6.34)

Theorem 2. There is another formula for polynomials P_{m,n}^{κ} at m ≥ n:

γ̃_{m+n,n}^{n} P_{m,n}^{κ} = ∑_{i=0}^{n} β̃_{m,n}^{i} P_{m+n+i,0}^{κ} P_{n−i,0}^{κ},  (6.35)

where

β̃_{m,n}^{i} = \frac{(−1)^{i}}{i!} (κ)_{i} \frac{(m + 2i)}{m} \frac{(κ + m + n)^{i}}{(m + n + 1)^{i}} \frac{(m)^{i}}{(κ + m + 1)^{i}} \frac{(n)_{i}}{(κ + n − 1)_{i}} .  (6.36)

This theorem follows directly from equation (6.33). The coefficients β̃_{m,n}^{i} are found by using of the relation

β̃_{m,n}^{i} = − (γ̃_{m+n+i,n−i}^{n−i})^{−1} ∑_{j=0}^{i−1} β̃_{m,n}^{j} γ̃_{m+n+j,n−j}^{n−i}.  (6.37)

As a by-product, let us specialize equation (6.30) to the case κ = 1, where P_{m,n}^{κ} are nothing but the SU(3) characters. We get

P_{m,n}^{1} = P_{m,0}^{1} P_{0,n}^{1} − P_{m−1,0}^{1} P_{0,n−1}^{1}.  (6.38)

From this, we easily deduce the generating function for SU(3) characters (see e.g. [PS 1978])

G^{1}(u, v) = ∑_{m,n=0}^{∞} u^{m} v^{n} P_{m,n}^{1} = \frac{1 − uv}{(1 − z_{1} u + z_{2} u^{2} − u^{3}) (1 − z_{2} v + z_{1} v^{2} − v^{3})}.  (6.39)

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Closing this section, let us underline that for κ=1/2, 1, 2 and 4, the obtained formulae yield the explicit expression of zonal polynomials for certain symmetric spaces listed below.

Table. List of symmetric spaces with A_{2} as a restricted root system

Type  X^{−}  X^{+}  κ = m/2
of symmetric (non-compact)  (compact) (m is a multiplicity

of roots)

A I  SL(3, ℝ)/𝕊𝕆(⊯) SU(3)/SO(3)  1/2
A 0  SL(3, ℂ)/𝕊𝕌(⊯)  SU(3)  1
A II  SU_{6}^{∗}/Sp(3)  SU(6)/Sp(3)  2
E IV  E_{6}(−26)/F_{4}  E_{6}(−78)/F_{4}  4

7 The integral representation for the case of N = 3

The integral representation for the case of N = 2 coincides with the integral representation for the Gegenbauer polynomials and is well known (see for example [Vi 1968]).

For the special values of κ (κ = 1/2, 1 , 2 , 4), wave functions are related to zonal spherical functions. The integral representation of these functions was obtained by Harish–Chandra by integrating on the some Lie group K [Ha 1958], [He 1978].

Following [CP 1997] let us consider now the A_{2} case (N = 3) for κ = 1/2, K = SO(3, ℝ). The element k ∈ K = SO(3, ℝ) may be represented by three unit vectors orthogonal to each other

n, l, m;  n^{2} = l^{2} = m^{2} = 1,  (n, l) = (l, m) = (m, n) = 0.

Here the integral representation for zonal spherical polynomials has the form

Φ_{pq}(x) = ∫_{K} [Ξ_{1}(x_{j}; n)]^{p} [Ξ_{2}(x_{j}; n, l)]^{q} dμ(n, l),  x_{j} = e^{iq_{j}},  (7.1)

where

Ξ_{1}(x_{j}; n) = n_{1}^{2} x_{1} + n_{2}^{2} x_{2} + n_{3}^{2} x_{3}, Ξ_{2}(x_{j}; n, l) = ∑_{j<k}(n_{j} l_{k} − n_{k} l_{j})^{2} x_{j} x_{k},

and the integration is taken on the orthogonal group K = SO(3, ℝ), which is equivalent to the space of two unit orthogonal vectors n and l.

Noting that m_{k} = ϵ_{kij} n_{i} l_{j}, we also have x_{1}x_{2} = x_{3}^{−1}, . . .. Hence Φ_{pq} is given by (7.1) where

Ξ_{2}(x_{j}; n, l) = m_{1}^{2} x_{1}^{−1} + m_{2} x_{2}^{−1} + m_{3} x_{3}^{−1}.  (7.2)

For vectors n and m, the standard parametrization through the Euler angles φ, θ and ψ may be used:

n = (cos φ sin θ, sin φ sin θ, cos θ), m = cos ψ · a + sin ψ · b,
a = (− sin φ, cos φ, 0), b = (− cos φ cos θ, − sin φ cos θ, sin θ)  (7.3)

with dμ(k) = dμ(n, m) = A sin θ dθ dφ dψ. Hence in the preceding expression we have a threedimensional integral which may be calculated by using of the generating functions.

8 The generating function for the case of N = 3

Following [CP 1997] let us define the generating function by the formula

F (x_{1}, x_{2}, x_{3}; t_{1}, t_{2}) = ∑ Φ_{l_{1},l_{2}}(x_{1}, x_{2}, x_{3}) t_{1}^{l_{1}} t_{2}^{l_{2}}.  (8.1)

Then we obtain the integral representation

F (x_{1}, x_{2}, x_{3}; t_{1}, t_{2}) = ∫ [∏_{j=1}^{2}(1 − Ξ_{j}(x; n, ψ)t_{j})]^{−1} dμ(n) dμ(ψ).  (8.2)

Let a and b be two unit orthogonal vectors in two-dimensional plane orthogonal to the vector n. Then on this plane an arbitrary unit vector m has the form cos ψ · a + sin ψ · b. Integrating first on dμ(ψ) we get:

F (x_{1}, x_{2}, x_{3}; t_{1}, t_{2}) = ∫ B^{−1}(n) C^{−1/2}(n) dμ(n), ∫ dμ(n) = 1,  (8.3)

where

B = 1 − (n_{1}^{2}x_{1} + n_{2}^{2}x_{2} + n_{3}^{2}x_{3}) t_{1},  C = (1 − x_{2}^{−1}t_{2}) (1 − x_{3}^{−1}t_{2}) n_{1}^{2} + · · · .  (8.4)

The crucial step for the further integration is the use of the formula

B^{−1} C^{−1/2} = ∫_{0}^{1} dξ [B(1 − ξ^{2}) + Cξ^{2}]^{−3/2} .  (8.5)

Using this, we obtain

F (x_{1}, x_{2}, x_{3}; t_{1}, t_{2}) = ∫_{0}^{1} dξ ∫ [E(x_{1}, x_{2}, x_{3}; t_{1}, t_{2}, n, ξ)]^{−3/2} dμ(n),  (8.6)

where

E(x_{1}, x_{2}, x_{3}; t_{1}, t_{2}, n, ξ) = ∑_{j} e_{j}(x_{1}, x_{2}, x_{3}; t_{1}, t_{2}, ξ) n_{j}^{2}.  (8.7)

We can now integrate on dμ(n). Finally we obtain the one-dimensional integral representation for the generating function

F (x_{1}, x_{2}, x_{3}; t_{1}, t_{2}) = ∫_{0}^{1} dξ [H(x_{1}, x_{2}, x_{3}; t_{1}, t_{2}, ξ)]^{−1/2},  (8.8)

where H = h_{1} h_{2} h_{3}, and functions h_{j}(ξ; t_{1}, t_{2}) are given by formulae

h_{j}(ξ; t_{1}, t_{2}) = 1 − d_{j}(t_{1}, t_{2}) (1 − ξ^{2}) ,  d_{j}(t_{1}, t_{2}) = (x_{j}t_{1} + x_{j}^{−1}t_{2} − t_{1}t_{2}) .  (8.9)
From this it follows that if z_{1} = x_{1} + x_{2} + x_{3} and z_{2} = x_{1}x_{2} + x_{2}x_{3} + x_{3}x_{1}, then
H = a_{0}^{3} − a_{0}^{2} [z_{1}τ_{1} + z_{2}τ_{2}] + a_{0} [z_{2}τ_{1}^{2} + z_{1}τ_{2}^{2} + (z_{1}z_{2} − 3)τ_{1}τ_{2}]
− [τ_{1}^{3} + τ_{2}^{3} + τ_{1}τ_{2} [(z_{2}^{2} − 2z_{1}) τ_{1} + (z_{1}^{2} − 2z_{2}) τ_{2}]] ,  (8.10)

where a_{0} = 1 + (1 − ξ^{2}) t_{1}t_{2}, τ_{1} = (1 − ξ^{2}) t_{1}, τ_{2} = (1 − ξ^{2}) t_{2}. Note that from (8.10) it follows that integral (8.8) is elliptic one and may be expressed in terms of standard elliptic integrals. Content-Length: 543 X-Lines: 19 Status: O

Expanding F (x_{1}, x_{2}, x_{3}; t_{1}, t_{2}) in power series of the variable t_{2} we obtain

F (x_{1}, x_{2}, x_{3}; t_{1}, t_{2}) = ∑_{q=0}^{∞} F_{q}(x_{1}, x_{2}, x_{3}; t_{1}) t_{2}^{q}.  (8.11)

We have

F_{0}(x_{1}, x_{2}, x_{3}; t) = ∫_{0}^{1} dξ [H_{0}]^{−1/2},  (8.12)

and

F_{1} = \frac{1}{2} ∫_{0}^{1} dξ H_{1} [H_{0}]^{−3/2},  (8.13)

where

H_{0} = 1 − z_{1} τ_{1} + z_{2} τ_{1}^{2} − τ_{1}^{3},
H_{1} = (1 − ξ^{2}) z_{2} − [3ξ^{2} + z_{1}z_{2} (1 − ξ^{2})] τ_{1} + [2z_{1} ξ^{2} + (1 − ξ^{2}) z_{2}^{2}] τ_{1}^{2} − z_{2} τ_{1}^{3}.

From integral representation (8.8), many useful formulae may be obtained. Here we give only one of them, namely, as z_{1}, z_{2} → ∞,

Φ_{m,n}(z_{1}, z_{2}) ≈ A_{m,n} z_{1}^{p} z_{2}^{q},  A_{m,n} = \frac{(1/2)_{m} (1/2)_{n}}{(1)_{m} (1)_{n}} \frac{(1)_{m+n}}{(3/2)_{m+n}} .  (8.14)

One can also be shown [Pe 1998b] that for arbitrary κ the analogous integral representation is valid

F^{κ}(z_{1}, z_{2}; t_{1}, t_{2}) = ∑_{m,n=1}^{∞} A_{m,n}(κ) P_{m,n}^{κ}(z_{1}, z_{2}) t_{1}^{m} t_{2}^{n} = ∫_{0}^{1}[H]^{−κ}dμ^{κ}(ξ),  (8.15)

where H is given by (8.10), P_{m,n}^{κ}(z_{1}, z_{2}) ∼ z_{1}^{m}z_{2}^{n} as z_{1}, z_{2} → ∞, and

dμ^{κ}(ξ) = [ξ(1 − ξ^{2})]^{2κ−1}, A_{m,n}(κ) = \frac{1}{2} \frac{Γ(κ)Γ(2κ)}{Γ(3κ)} \frac{(κ)_{m} (κ)_{n}}{(1)_{m} (1)_{n}} \frac{(2κ)_{m+n}}{(3κ)_{m+n}} .  (8.16)

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Note that for the case under consideration another integral representation for N = 3 was obtained in [KS 1995] in terms of _{3}F_{2} hypergeometric series.

Appendix 1. List of polynomials P_{m,n}^{κ} with m + n ≤ 4

Following [PRZ 1998], we list here the polynomials P_{m,n}^{κ} with m + n ≤ 4:

P_{2,0}^{κ} = z_{1}^{2} − \frac{2}{κ + 1} z_{2},
P_{1,1}^{κ} = z_{1}z_{2} − \frac{3}{2κ + 1},
P_{3,0}^{κ} = z_{1}^{3} − \frac{6}{κ + 2} z_{1}z_{2} + \frac{6}{(κ + 1)(κ + 2)},
P_{2,1}^{κ} = z_{1}^{2}z_{2} − \frac{2}{κ + 1} z_{2}^{2} − \frac{3κ + 1}{(κ + 1)^{2}} z_{1},
P_{4,0}^{κ} = z_{1}^{4} − \frac{12}{κ + 3} z_{1}^{2}z_{2} + \frac{12}{(κ + 2)(κ + 3)} z_{2}^{2} + \frac{24}{(κ + 2)(κ + 3)} z_{1},
P_{3,1}^{κ} = z_{1}^{3}z_{2} − \frac{6}{κ + 2} z_{1}z_{2}^{2} − \frac{3(3κ + 2)}{(κ + 2)(2κ + 3)} z_{1}^{2} + \frac{30}{(κ + 2)(2κ + 3)} z_{2},
P_{2,2}^{κ} = z_{1}^{2}z_{2}^{2} − \frac{2}{κ + 1} (z_{1}^{3} + z_{2}^{3}) − \frac{12(κ − 1)}{(κ + 1)(2κ + 3)} z_{1}z_{2} + \frac{9(κ − 1)}{(κ + 1)^{2}(2κ + 3)} .

Appendix 2. Some formulae for Macdonald Polynomials for A_{2} case (by A.M. Perelomov, E. Ragoucy and Ph. Zaugg)

Here we give only some necessary for us information. For other results and details, see [Ma 1995].

The Macdonald polynomials of type A_{2} may be defined as polynomial eigenfunctions of the Macdonald difference equation

M^{1} P_{m,n}^{(q,t)}(x_{1}, x_{2}, x_{3}) = λ P_{m,n}^{(q,t)}(x_{1}, x_{2}, x_{3}),  P_{m,n} = z_{1}^{m}z_{2}^{n} + lower terms,  (8.1)
z_{1} = x_{1} + x_{2} + x_{3},  z_{2} = x_{1}x_{2} + x_{2}x_{3} + x_{3}x_{1},  z_{3} = x_{1}x_{2}x_{3},

where

M^{1} = ∏_{j≠k} \frac{(tx_{j} − x_{k})}{(x_{j} − x_{k})} T_{j},  T_{1} f(x_{1}, x_{2}, x_{3}) = f(qx_{1}, x_{2}, x_{3}), ….  (8.2)

The Macdonald polynomials satisfy the following recurrence relation

z_{1} P_{m,n}^{(q,t)}(z_{1}, z_{2}, z_{3}) = P_{m+1,n}^{(q,t)} + a_{m,n}(q, t) P_{m,n−1}^{(q,t)} + b_{m,n}(q, t) P_{m−1,n+1}^{(q,t)},  (8.3)

where

a_{mn} = c_{n} c̃_{m+n},  b_{mn} = c_{m},
c_{m}(q, t) = \frac{(1 − q^{m})(1 − t^{2}q^{m−1})}{(1 − tq^{m})(1 − tq^{m−1})},  (8.4)

c̃_{m+n}(q, t) = (\frac{1 − tq^{m+n}}{1 − t^{2}q^{m+n}} ) (\frac{1 − t^{3}q^{m+n−1}}{1 − t^{2}q^{m+n−1}} ) .

Note that

c_{m}(q, 1) = c_{m}(q, q) = 1,  c̃_{m+n}(q, 1) = c̃_{m+n}(q, q) = 1.  (8.5)

Now let us consider the case t = q^{k}, k being an integer. In this case, the explicit expression for the generating function of polynomials P_{n,0}^{(k)} ≡ P_{n,0}^{(q,t)} has the form

G^{(k)}(u) = ∏_{j=0}^{k−1} F^{1}(q^{j}u),
G^{(k)}(u) = ∑ C_{n}^{k} P_{n,0}^{(k)} u^{n},  (8.6)
C_{n}^{k} = \frac{[k]_{n}}{[1]_{n}} ,  [x]_{n} = [x][x + 1] ⋯ [x + n − 1], [x] = \frac{1 − q^{x}}{1 − x} .

From this we get a three-term recurrence relation being analogous to (6.7)

[n + 1] P̃_{n+1}^{k} = [k + n] z_{1} P̃_{n}^{k} − [2k + n − 1] z_{2} P̃_{n−1}^{k} + [3k + n − 2] z_{3} P̃_{n−2}^{k},  (8.7)

which follows from the recurrence relation

(1 − z_{1}u + z_{2}u^{2} − z_{3}u^{3}) G^{(k)}(u) = (1 − z_{1}uq^{k} + z_{2}u^{2}q^{2k} − z_{3}u^{3}q^{3k}) G(qu).  (8.8)

From here, we obtain

P̃_{0}^{κ} = 1, P̃_{1}^{κ} = [κ] z_{1}, P̃_{2}^{κ} = \frac{[κ + 1][κ]}{[1][2]} z_{1}^{2} − \frac{[2κ]}{[2]} z_{2}, … P̃_{m,0} = \frac{[1]_{m}}{[κ]_{m}} P_{m,0}.

Content-Length: 2165 X-Lines: 53 Status: O Let us give also the formula for the generating function for the case of arbitrary t:

G^{(q,t)}(u) = ∏_{j=0}^{∞} \frac{(1 − q^{j}tu)}{(1 − q^{j}u)} = ∑_{j=0}^{∞} c_{j}(q, t) u^{j}.  (8.9)

Using the results from the book [Ma 1995], it is not difficult to get the formulae

P_{m,0}^{(q,t)} P_{0,n}^{(q,t)} = ∑_{i=0}^{min (m,n)} γ_{m,n}^{i} P_{m−i,n−i}^{(q,t)} ,  (8.10)

where

γ_{m,n}^{i} = \frac{(t; q)_{i} (q^{m}; q^{−1})_{i} (q^{n}; q^{−1})_{i} (q^{m+n−1−i}t^{3}; q^{−1})_{i}}{(q; q)_{i} (q^{m−1}t; q^{−1})_{i} (q^{n−1}t; q^{−1})_{i} (q^{m+n−i}t^{2}; q^{−1})_{i}};  (8.11)

and

P_{m,0}^{(q,t)} P_{n,0}^{(q,t)} = ∑_{i=0}^{min (m,n)} γ̃_{m,n}^{i} P_{m+n−2i,i}^{(q,t)} ,  (8.12)

where

γ̃_{m,n}^{i} = \frac{(t; q)_{i} (q^{m}; q^{−1})_{i} (q^{n}; q^{−1})_{i} (q^{m+n−i−1}t^{2}; q^{−1})_{i}}{(q; q)_{i}(q^{m−1}t; q^{−1})_{i}(q^{n−1}t; q^{−1})_{i}(q^{m+n−i}t; q^{−1})_{i}} .  (8.13)

Inverting formulae (8.10) and (8.11) as in the [PRZ 1998], we come to Theorem 1a. The Macdonald polynomilas P_{m,n}^{(q,t)} of type A_{2} are given by the formula

P_{m,n}^{(q,t)} = ∑_{i=0}^{min(m,n)} β_{m,n}^{i} P_{m−i,0}^{(q,t)} P_{0,n−i}^{(q,t)},  (8.14)

where constants have the form

β_{m,n}^{i} = (−1)^{i} q^{i(i−1)/2} \frac{(t; q^{−1})_{i}}{(q; q)_{i}} \frac{1 − q^{m+n−2i} t^{3}}{1 − q^{m+n−i} t^{3}}

× \frac{(q^{m}; q^{−1})_{i} (q^{n}; q^{−1})_{i} (q^{m+n−1}t^{3}; q^{−1})_{i}}{(q^{m−1}t; q^{−1})_{i} (q^{n−1}t; q^{−1})_{i} (q^{m+n−1}t^{2}; q^{−1})_{i}} .

In similar way, we get Theorem 2a. The Macdonald polynomials P_{m,n}^{(q,t)} of type A_{2} are given by the formula

γ̃_{m+n,n}^{n} P_{m,n}^{(q,t)} = ∑_{i=0}^{n} β̃_{m,n}^{i} P_{m+n+i,0}^{(q,t)} P_{n−i,0}^{(q,t)},  m ≥ n  (8.15)

where constants γ̃_{m,n}^{i} are given by formula (8.13), and

β̃_{m,n}^{i} = (−1)^{i} q^{i(i−1)/2} \frac{(t; q^{−1})_{i}}{(q; q)_{i}} \frac{(1 − q^{m+2i})}{(1 − q^{m})} \frac{(q^{m}; q)_{i} (q^{n}; q^{−1})_{i} (q^{m+n}t; q)_{i}}{(q^{m+n+1}; q)_{i} (q^{m+1}t; q)_{i} (q^{n−1}t; q^{−1})_{i}} .  (8.16)

From Theorems 1a and 2a, many interesting identities may be obtained. Here we give one of them:

S_{n,l}^{k} = ∑_{i=0}^{l}(−1)^{i} q^{i(i−1)/2} \frac{[3k + n − 2i]}{[3k + n − i]} (∏_{j=1}^{i} \frac{[k − j + 1]}{[j]} \frac{[3k + n − j]}{[2k + n − j]})

× ( ∏_{j=0}^{l−i−1} \frac{[k + j]}{[j + 1]} \frac{[3k + n − l − i − j − 1]}{[2k + n − l − i − j]}) = 0,  (8.17)

where [n] = (1 − q^{n})/(1 − q). Content-Length: 5667 X-Lines: 115 Status: O The simplest version of this formula is

∑_{i=0}^{l}(−1)^{i} q^{i(i−1)/2} \frac{[k + l − i − 1]!}{[i]! [l − i]! [k − i]!} = 0.  (8.18)

We give below the list of polynomilas P_{m,n}^{(q,t)} at m + n ≤ 4
P_{0,0}^{(q,t)} = 1, P_{1,0}^{(q,t)} = z_{1}, P_{0,1}^{(q,t)} = z_{2},
P_{2,0}^{(q,t)} = z_{1}^{2} − \frac{(1 − q)(1 + t)}{(1 − qt)} z_{2},
P_{1,1}^{(q,t)} = z_{1}z_{2} − \frac{(1 − q)(1 + t + t^{2})}{(1 − qt^{2})} z_{3},
P_{3,0}^{(q,t)} = z_{1}^{3} − \frac{(1 − q)(2 + q + t + 2qt)}{1 − q^{2}t} z_{1}z_{2} + \frac{(1 − q)^{2}(1 + q)(1 + t + t^{2})}{(1 − qt)(1 − q^{2}t)} z_{3},
P_{2,1}^{(q,t)} = z_{1}^{2}z_{2} − \frac{(1 − q)(1 + t)}{1 − qt} z_{2}^{2} − \frac{(1 − q^{2})(1 − qt^{3})}{(1 − qt)^{2}(1 + qt)} z_{1}z_{3},
P_{4,0}^{(q,t)} = z_{1}^{4} − \frac{(1 − q)(3 + 2q + q^{2} + t + 2qt + 3q^{2}t)}{1 − q^{3}t} z_{1}^{2}z_{2}

+ \frac{(1 − q)^{2}(1 + q + q^{2})(1 + t)(1 + qt)}{(1 − q^{2}t)(1 − q^{3}t)} z_{2}^{2}

+ \frac{(1 − q)^{2}(1 + q)(2 + q + q^{2} + t + 2qt + q^{2}t + t^{2} + qt^{2} + 2q^{2}t^{2})}{(1 − q^{2}t)(1 − q^{3}t)} z_{1}z_{3},

P_{3,1}^{(q,t)} = z_{1}^{3}z_{2} − \frac{(1 − q)(2 + q + t + 2qt)}{1 − q^{2}t} z_{1}z_{2}^{2} − \frac{(1 − q^{3})(1 − q^{2}t^{3})}{(1 − q^{2}t)(1 − q^{3}t^{2})} z_{1}^{2}z_{3}

+ \frac{(1 − q)^{2}}{(1 − q^{2}t)(1 − q^{3}t^{2})}

× (2 + 2q + q^{2} + 2t + 4qt + 3q^{2}t + q^{3}t + t^{2}+ 3qt^{2} + 4q^{2}t^{2} + 2q^{3}t^{2} + qt^{3} + 2q^{2}t^{3} + 2q^{3}t^{3}) z_{2}z_{3};

P_{2,2}^{(q,t)} = z_{1}^{2}z_{2}^{2} − \frac{(1 − q)(1 + t)}{1 − qt} (z_{2}^{3} + z_{1}^{3}z_{3})

− \frac{(1 − q)(3q + q^{2} − 3t + 2q^{2}t + q^{3}t − t^{2} − 2qt^{2} + 3q^{3}t^{2} − qt^{3} − 3q^{2}t^{3})}{(1 − qt)(1 − q^{3}t^{2})} z_{1}z_{2}z_{3}

+ \frac{(1 − q)^{2}(1 + q)(1 + t + t^{2})(q − t + q^{2}t − qt^{2} + q^{3}t^{2} − q^{2}t^{3})}{(1 − qt)(1 − q^{2}t^{2})(1 − q^{3}t^{2})} z_{3}^{2}.

Acknowledgment

The paper was complete during my stay at Max-Planck-Institut für Mathematik Bonn. I am grateful the members of the Institute for hospitality.

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