An Analytic Formula for the A_2 Jack Polynomials

In this letter I shall review my joint results with Vadim Kuznetsov and Evgeny Sklyanin [Indag. Math. 14 (2003), 451-482, math.CA/0306242] on separation of variables for the $A_n$ Jack polynomials. This approach originated from the work [RIMS Kokyuroku 919 (1995), 27-34, solv-int/9508002] where the integral representations for the $A_2$ Jack polynomials was derived. Using special polynomial bases I shall obtain a more explicit expression for the $A_2$ Jack polynomials in terms of generalised hypergeometric functions.


Introduction
In middle 1990's Vadim Kuznetsov and Evgeny Sklyanin started to work on applications of the Separation of Variables (SoV) method to symmetric polynomials. When I read their papers on SoV for Jack and Macdonald polynomials, I was quite charmed by a beauty of this approach. It connected together few different areas of mathematics and mathematical physics including symmetric polynomials, special functions and integrable models.
Later on we started to work together on development of SoV for symmetric functions. During my visits to Leeds around 2001-2003 we realised that the separating operator for the A n Jack polynomials can be constructed using Baxter's Q-operator following a general approach [8] developed by Vadim and Evgeny Sklyanin. Vadim was a very enthusiastic and active person. His creative energy always encouraged me to work harder on joint projects. His area of expertise ranged from algebraic geometry to condensed matter physics. He was an excellent mathematician and a very strong physicist.
During my visits to Leeds University I was charmed by the English countryside. I will never forget long walks in the forest with Vadim's family and quiet dinners at his home.
The sad news about his death was a real shock for me. His untimely death is a big loss for the scientific community. However, he will be long remembered by many people for his outstanding scientific achievements.

Quantum Calogero-Sutherland model
The quantum Calogero-Sutherland model [1,21] describes a system of n particle on a circle with coordinates 0 ≤ q i ≤ π, i = 1, . . . , n. The Hamiltonian and momentum are The ground state of the model is given by q ≡ (q 1 , . . . , q n ) (2.2) with the ground state energy E 0 = 1 6 g 2 (n 3 − n). Further we shall assume that g > 0 that simplifies description of the eigenvectors.
The Calogero-Sutherland model is completely integrable [14] and there is a commutative ring of differential operators H i , i = 1, . . . , n [16], which contains the Hamiltonian and momentum (2.1).
Conjugating the Hamiltonian H with vacuum (2.2) we obtain Then we obtain that Jack polynomials P (1/g) λ (x) are eigenfunctions of the operator H g and the eigenvalues are given by It follows that symmetric Jack polynomials are orthogonal with respect to the scalar product where x stands for the complex conjugate of x and due to conditions g > 0 and (2.4), where q i are real, we have P . Jack polynomials are homogeneous of the degree |λ| In fact, (2.10) corresponds to a centre-of-mass separation in the Calogero-Sutherland model. Under simultaneous shift of all parts of the partition λ by an integer the Jack polynomials undergo a simple multiplicative transformation, which can be written as P (1/g) λ 1 ,...,λn = (x 1 . . . x n ) λn P (1/g) λ 1 −λn,...,λ n−1 −λn,0 (x 1 , . . . , x n ). (2.11) We shall use properties (2.10), (2.11) in our construction of the A 2 Jack polynomials.

Separation of variables for Jack polynomials
In this section we shall briefly review basic facts about separation of variables [17,18] for Jack polynomials [7,6]. Our main reference is [6]. We shall start with definition of separated polynomials. For each partition λ defines a function f λ (x) [7,9] as f λ (y) = y λn (1 − y) 1−ng n F n−1 a 1 , . . . , a n ; b 1 , . . . , b n−1 y , where the hypergeometric function n F n−1 [2,19] is defined as n F n−1 a 1 , . . . , a n ; the Pocchammer symbol (a) n = Γ(a+n) Γ(a) and parameters a i , b i are related to the partition λ = {λ 1 , . . . , λ n } as follows The important property of this function is described by the following Theorem 1. f λ (y) is a polynomial in y of the cumulative degree λ 1 and also has another representation f λ (y) = b λ y λn This theorem proved in [6] allows to evaluate f λ (1) = b λ and we shall use this fact to calculate a correct normalisation of the A 2 Jack polynomials.
In fact, we can directly expand (3.1) into the product of two series in y and multiply them to produce an expansion of the form with coefficients ξ k (λ; g) being expressed in terms of n+1 F n hypergeometric functions (see [9] for the case of Macdonald polynomials). However, the coefficients of y k will be, in general, infinite series depending on λ and g. The Theorem 1 shows that, in fact, all those coefficients are rational functions and the expansion really truncates at y λ 1 . It is not clear how to produce an explicit form of those coefficients in terms of hypergeometric functions of one variable with finite number of terms. It can be easily done for the case n = 2 and later we will show how to do that for the case n = 3. Now we shall formulate two main theorems of [6].
There exists an integral operator S n , which maps symmetric polynomials in n variables into symmetric polynomials, and this operator factorises Jack polynomials Action of this operator S n on elementary symmetric polynomials e k was described in [6]. This symmetric separation of variables corresponds to the so called dynamical normalization of the Baker-Akhiezer function (see, for example, [10]).
However, using the property (2.10) we can trivially separate one variable x n . After that we need to separate the n − 1 remaining variables. It results in the so called standard separation Theorem 3. There exist an integral operator S s n which maps symmetric polynomials in n variables into polynomials symmetric only in n − 1 variables, In two next sections we shall describe the action of S 2 and S s 3 explicitly.

The A 1 case
In this section we shall consider the simplest n = 2 case in standard and dynamical normalisations.
In the standard normalisation the separating operator S s 2 is trivial, c λ = b λ , λ = (λ 1 , λ 2 ) and we simply obtain where the separating polynomial and we used a well known formula for 2 F 1 [2]. The last formula in (4.2) gives an explicit expansion of f λ (x) in series of x with rational coefficients. Although formulas (4.1), (4.2) give explicit representation for the A 1 Jack polynomials, a symmetry w.r.t. x 1 and x 2 is not obvious. To obtain such symmetric representation it is instructive to construct a separation which corresponds to the dynamical normalisation of the Baker-Akhiezer function.
Introduce the following basis in the space of symmetric polynomials S(x 1 , x 2 ) in two variables and define the operator S 2 acting on S(x 1 , x 2 ) as Such operator S 2 can be easily represented as an integral operator with a simple kernel (see [9] for Macdonald polynomials) but we do not need here its explicit form. What is more important is that it maps the A 1 Jack polynomials into the product of separated polynomials The action of the inverse operator S −1 2 on the basis p mn is easily constructed from (4.4) and to obtain a symmetric representation of P (1/g) λ (x 1 , x 2 ) we have to expand the rhs of (4.5) in p mn . To do that we use Watson's formula [22] and the Appell's function F 4 is defined as After little algebra we obtain from (4.2)-(4.7) Finally we can rewrite (4.8) in the basis of elementary symmetric polynomials e 1 = x 1 + x 2 , e 2 = x 1 x 2 . Expanding p mn in e 1 , e 2 and evaluating remaining sums we come to Another symmetric representation for P (1/g) λ (x 1 , x 2 ) which first appeared in [5] is given in terms of Gegenbauer polynomials C g n (x). Expanding e 1 in series of x 1 and x 2 we obtain In fact, this formula can be directly obtained from representation (4.1), (4.2), but we preferred to use the symmetric separation (4.5). As was shown in [10], the separating operator S n−1 for the dynamical normalisation has the same structure as the separating operator S s n in the standard normalisation. In the next section we shall use this fact to construct a representation for the A 2 Jack polynomials.
5 The A 2 case and the expansion of products of separated polynomials Now we shall consider n = 3 case in the standard normalisation. Using (2.10) we obtain where we introduced a short notation p λ (x 1 , x 2 ) for the A 2 Jack polynomials with x 3 = 1. Define a reduced separating operatorŜ 3 which acts only on the basis p mn (see (4.3)) in the space of symmetric polynomials in two variables S(x 1 , x 2 ) aŝ The explicit construction ofŜ 3 as an integral operator was given in [7,6]. The operatorŜ 3 separates variables in p λ (x 1 , x 2 ) where f λ (x) is given by (3.1) with n = 3. To construct p λ (x 1 , x 2 ) we have to invert the action ofŜ 3 . It follows from (5.2) that in order to apply to (5.3)Ŝ −1 3 we have to expand the rhs of (5.3) in p mn and it is easy to see that coefficients c m,n (λ; g) depend on the partition λ only via λ 13 and λ 23 . In fact, formula (5.4) is the main nontrivial result of this paper. Setting x 2 = 1 in (5.4) we obtain as a particular case the expansion for f λ (x 1 ) of the form (3.6) with coefficients ξ k (λ, g) = c m, 0 (λ, g)/b λ . (5.5) Now we shall calculate coefficients c m,n (λ, g). For convenience we shall omit dependence of c m,n (λ; g) on λ and g.
Let us remind that the hypergeometric function 3 F 2 satisfies 3-th order differential equation of the form [2,19] Using (3.1) we can easily obtain a third order differential equation for the functions f λ (x i ), i = 1, 2. Now we can rewrite these equations in terms of new variables Therefore, for the rhs of (5.4) we obtain two differential equations in u and v which can be rewritten as recurrence relations for the coefficients c m,n . All calculations are straightforward and we shall omit the details. The first recurrence relation for c mn looks amazingly simple and basically allows to find c m,n in a closed form c m+1,n (m + 1)(1 − 2g + m − r 1 )(1 − g + m − r 2 ) + c m,n+1 (n + 1)(3g + n − 1)(3g + n) where we denoted r 1 = λ 1,3 , r 2 = λ 2,3 .
The relation (5.8) immediately suggests to make a substitution The second recurrence relation for c m,n leads to the following relation for a m,n m(2g + r 1 − m)(g + r 2 − m)a m−1,n − n(3g + n − 2)(3g + n − 1)a m,n−1 However, due to (5.11) we need the latter relation only for the calculation of a m,0 which corresponds to the case n = 0 in (5.12). In fact, it is much easier to obtain the 3-term recurrence relation for a m,0 not from (5.12), but from the 3-term recursion for the coefficients c m,0 itself. This latter recursion follows from (3.6), (5.5) and the third order differential equation for f λ (x). Now let us introduce a generating function and rewrite the recurrence relation for a m,0 as a third order differential equation for h(x). Then this equation can be easily reduced to the standard hypergeometric equation (5.6) for the function 3 F 2 . This equation has three solutions with exponents ρ = 2g, g − r 2 , −r 1 (5.14) at x = 0 which can be written as and for any choice of ρ a hypergeometric function 4 F 3 reduces to 3 F 2 . Now we note that for ρ = 2g and ρ = g − r 2 the hypergeometric function in the rhs of (5.15) truncates and we obtain from (5.13), (5.15) the following equivalent expressions for the coefficients a m,0 and To calculate the correct normalisation coefficients α 1 , α 2 for (5.16), (5.17) we take the limit x 1 → 0, x 2 → 1 in (5.4). Then only the term c 0,0 contributes to the rhs of (5.4) and we use (3.5), (5.9) to calculate a 0,0 .
From the other side the hypergeometric series in (5.16), (5.17) becomes Saalschützian 3 F 2 (i.e. 1 + a 1 + a 2 + a 3 = b 1 + b 2 ) and can be evaluated using the so called Saalschütz's theorem [2,19] 3 F 2 a, b, −n; Then we obtain Now we can substitute (5.16)-(5.17) into (5.11). Expanding hypergeometric functions into series we can interchange the order of summations and calculate the sum over l in (5.11). Substituting the result into (5.9) we finally come to the following explicit expressions for c m,n As we mentioned above the coefficients of the expansion of the separating polynomial f λ (x) in x are given by c m,0 , i.e. expressed in terms of hypergeometric functions 4 F 3 . It is quite surprising that the coefficients c m,n of the expansion (5.4) are still given in terms of 4 F 3 by formulas (5.21), (5.22) and do not involve higher hypergeometric functions. This happens probably because the basis p mn is very special.

A representation for the A 2 Jack polynomials
Now we are ready to give explicit formulas for the A 2 Jack polynomials. Applying the opera-torŜ −1 3 to (5.3) and using (5.1), (5.2), (5.4) we obtain two representations Formulas (6.1), (6.2) simplify when λ 2 = λ 3 and λ 1 = λ 2 accordingly. First consider the case of one-row partitions (λ, 0, 0). It is convenient to set x 3 = 1. Then we obtain from (6.1) Expanding the last term we can transform this formula to another basis Finally restoring dependence on x 3 we shall rewrite (6.4) in the basis of elementary symmetric functions In fact, (6.5) corresponds to the first fundamental weight ω 1 of A 2 and is a special case of the Proposition 2.2 from [20] for arbitrary number of variables for all partitions µ such that Now from (6.2) we can similarly calculate P for all partitions µ = (1 m 1 2 m 2 3 m 3 ), |µ| = 2r. It seems that (6.8) has the following generalisation to n variables for all partitions µ = (1 m 1 · · · n mn ), |µ| = (n − 1)λ. We are not aware of any appearances of (6.9) in the literature. Note that formulas for the A 2 Jack polynomials were already obtained earlier in [15]. They were generalised in [11] for partitions of length 3 with any number of variables and further in [12] for the general A n case.
We tried hard to obtain directly (6.1), (6.2) from (6.10), (6.11) with no success. The corresponding transformations can be reduced to multiple sum identities involving a large number of Pochhammer symbols. It is absolutely unclear how to prove them.
Finally, it would be interesting to rewrite formulas (6.1), (6.2) in the basis e 1 , e 2 , e 3 like (6.5) for a general partition λ = (λ 1 , λ 2 , λ 3 ). However, it involves sums of truncated hypergeometric functions 4 F 3 and we did not succeed at this stage to obtain any compact expression for general partitions of length 3.

Conclusions
In this letter we applied the separation of variables method to construct a formula for the A 2 Jack polynomials based on the results of [7]. We explicitly evaluated the coefficients of the expansion of the A 2 Jack polynomials in a special basis.
It is a difficult problem to show that our formulas (6.1), (6.2) can be reduced to results of [15,11]. In fact, their structure is completely different. Nevertheless, our approach can be useful to find a simpler representation for Jack and Macdonald polynomials.