Table Of ContentSymmetry, Integrability and Geometry: Methods and Applications SIGMA 3(2007), 014, 11 pages
⋆
An Analytic Formula for the A Jack Polynomials
2
Vladimir V. MANGAZEEV
Department of Theoretical Physics, Research School of Physical Sciences and Engineering,
The Australian National University, Canberra, Australia
E-mail: [email protected]
Received November 01, 2006, in final form January 05, 2007; Published online January 24, 2007
7
Original article is available at http://www.emis.de/journals/SIGMA/2007/014/
0
0
2 Abstract. In this letter I shall review my joint results with Vadim Kuznetsov and Evgeny
n Sklyanin[Indag. Math. 14(2003),451–482]onseparationofvariables(SoV)fortheAnJack
a polynomials. Thisapproachoriginatedfromthe work[RIMS Kokyuroku919(1995),27–34]
J
where the integral representations for the A2 Jack polynomials was derived. Using special
4
polynomial bases I shall obtain a more explicit expression for the A2 Jack polynomials in
2
terms of generalised hypergeometric functions.
]
A Key words: Jack polynomials; integral operators; hypergeometric functions
C
2000 Mathematics Subject Classification: 05E05;33C20; 82B23
.
h
t Dedicated to the memory of Vadim Kuznetsov
a
m
1 Introduction
[
1
In middle 1990’s Vadim Kuznetsov and Evgeny Sklyanin started to work on applications of the
v
Separation of Variables (SoV) method to symmetric polynomials. When I read their papers on
7
7 SoV for Jack and Macdonald polynomials, I was quite charmed by a beauty of this approach.
6
It connected together few different areas of mathematics and mathematical physics including
1
0 symmetric polynomials, special functions and integrable models.
7 Later onwestarted toworktogether ondevelopmentof SoVforsymmetricfunctions. During
0
my visits to Leeds around 2001–2003 we realised that the separating operator for the A Jack
/ n
h polynomials can be constructed using Baxter’s Q-operator following a general approach [8]
t
a developed by Vadim and Evgeny Sklyanin.
m
Vadim was a very enthusiastic and active person. His creative energy always encouraged
:
v me to work harder on joint projects. His area of expertise ranged from algebraic geometry to
i condensed matter physics. He was an excellent mathematician and a very strong physicist.
X
During my visits to Leeds University I was charmed by the English countryside. I will never
r
a forget long walks in the forest with Vadim’s family and quiet dinners at his home.
Thesad newsabouthisdeath was arealshock forme. His untimely death isabigloss forthe
scientific community. However, he will be long remembered by many people for his outstanding
scientific achievements.
2 Quantum Calogero–Sutherland model
The quantum Calogero–Sutherland model [1, 21] describes a system of n particle on a circle
with coordinates 0 ≤ q ≤ π, i= 1,...,n. The Hamiltonian and momentum are
i
1 n ∂2 g(g−1) n ∂
H = − + , P = −i . (2.1)
2 ∂q2 sin2(q −q ) ∂q
i=1 i i<j i j j=1 j
X X X
⋆ThispaperisacontributiontotheVadimKuznetsovMemorialIssue“IntegrableSystemsandRelatedTopics”.
The full collection is available at http://www.emis.de/journals/SIGMA/kuznetsov.html
2 V.V. Mangazeev
The ground state of the model is given by
g
Ω(q) = sin(q −q ) , q ≡ (q ,...,q ) (2.2)
i j 1 n
(cid:20)i<j (cid:21)
Y
with the ground state energy E = 1g2(n3 − n). Further we shall assume that g > 0 that
0 6
simplifies description of the eigenvectors.
The Calogero–Sutherland model is completely integrable [14] and there is a commutative
ring of differential operators H , i= 1,...,n [16], which contains the Hamiltonian and momen-
i
tum (2.1).
The eigenfunctions are labelled by partitions λ ≡ (λ ,...,λ ), λ ≥ λ ≥ 0, i= 1,...,n−1
1 n i i+1
and can be written as
Ψ (q) = Ω(q)P(1/g)(x), (2.3)
λ λ
where P(1/g)(x), x ≡ (x ,...,x ) are symmetric Jack polynomials in n variables [3, 13, 20]
λ 1 n
x = e2iqi, i= 1,...,n. (2.4)
i
Further for a partition λ = (λ ,...,λ ), we define its length l(λ) = n, its weight |λ| =
1 n
λ +···+λ (see [13]) and use a short notation λ = λ −λ .
1 n ij i j
Conjugating the Hamiltonian H with vacuum (2.2) we obtain
1
Ω−1(q)HΩ(q) = H +E , (2.5)
g 0
2
n 2
∂ x +x ∂ ∂
i j
H = x +g x −x . (2.6)
g i i j
∂x x −x ∂x ∂x
i=1(cid:18) i(cid:19) i<j i j (cid:18) i j(cid:19)
X X
Then we obtain that Jack polynomials P(1/g)(x) are eigenfunctions of the operator H
λ g
H P(1/g)(x) =E P(1/g)(x) (2.7)
g λ g λ
and the eigenvalues are given by
n
E = λ [λ +g(n+1−2i)]. (2.8)
g i i
i=1
X
It follows that symmetric Jack polynomials are orthogonal with respect to the scalar product
g
1 dx dx
1 n
(P ,P )= ··· (1−x /x )
λ µ (2πi)n x x i j
1 n
|x1I|=1 |xnI|=1 Yi6=j
×P(1/g)(x)P(1/g)(x) = 0 ifλ 6= µ, (2.9)
λ µ
where x stands for the complex conjugate of x and due to conditions g > 0 and (2.4), where
q are real, we have P(1/g)(x) = P(1/g)(x−1).
i λ λ
Jack polynomials are homogeneous of the degree |λ|
P(1/g)(x ,...,x )= x|λ|P(1/g)(x /x ,...,x /x ,1). (2.10)
λ 1 n n λ 1 n n−1 n
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) = (x ...x )λnP(1/g) (x ,...,x ). (2.11)
λ1,...,λn 1 n λ1−λn,...,λn−1−λn,0 1 n
We shall use properties (2.10), (2.11) in our construction of the A Jack polynomials.
2
An Analytic Formula for the A Jack Polynomials 3
2
3 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 func-
tion f (x) [7, 9] as
λ
a ,...,a ;
f (y) = yλn(1−y)1−ng F 1 n y , (3.1)
λ n n−1 b ,...,b
1 n−1
(cid:18) (cid:12) (cid:19)
(cid:12)
where the hypergeometric function F [2, 19] is(cid:12)defined as
n n−1
a ,...,a ; ∞ (a ) ···(a ) ym
F 1 n y = 1 m n m , |y| < 1, (3.2)
n n−1 b ,...,b (b ) ···(b ) m!
(cid:18) 1 n−1 (cid:12) (cid:19) mX=0 1 m n−1 m
(cid:12)
(cid:12)
Γ(a+n)
the Pocchammer symbol (a) = and parameters a , b are related to the partition λ =
n Γ(a) i i
{λ ,...,λ } as follows
1 n
a = λ −λ +1−(n−i+1)g, b = a +g. (3.3)
i n i j j
The important property of this function is described by the following
Theorem 1. f (y) is a polynomial in y of the cumulative degree λ and also has another
λ 1
representation
λ1,2 λn−1,n n−1
(−λ ) (ig)
f (y) = b yλn ··· (1−y)ki i,i+1 ki k1+···+ki , (3.4)
λ λ
k ! ((i+1)g)
kX1=0 knX−1=0Yi=1 i k1+···+ki
where
n−1
((n−i+1)g)
b = λi,n. (3.5)
λ
((n−i)g)
i=1 λi,n
Y
Thistheorem provedin[6]allows toevaluatef (1) = b andweshallusethisfacttocalculate
λ λ
a correct normalisation of the A Jack polynomials.
2
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
λ1
f (y) = ykξ (λ;g) (3.6)
λ k
kX=λn
with coefficients ξ (λ;g) being expressed in terms of F hypergeometric functions (see [9]
k n+1 n
for the case of Macdonald polynomials). However, the coefficients of yk 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 yλ1. 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].
4 V.V. Mangazeev
Theorem 2. There exists an integral operator S , which maps symmetric polynomials in n va-
n
riables into symmetric polynomials,
S 1 = 1 (3.7)
n
and this operator factorises Jack polynomials
n
S [P(1/g)](x ,...,x ) = c b−n f (x ), (3.8)
n λ 1 n λ λ λ i
i=1
Y
where
(g(j −i+1))
c = P(1/g)(1,...,1) = λi,j. (3.9)
λ λ (g(j −i))
1≤i<j≤n λi,j
Y
Action of this operator S on elementary symmetric polynomials e was described in [6].
n k
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 . After that we
n
need to separate the n−1 remaining variables. It results in the so called standard separation
Theorem 3. There exist an integral operator Ss which maps symmetric polynomials in n va-
n
riables into polynomials symmetric only in n−1 variables,
Ss1 = 1 (3.10)
n
and
n−1
Ss[P(1/g)](x ,...,x ) = c b1−n|x |λ f (x ). (3.11)
n λ 1 n λ λ n λ i
i=1
Y
In two next sections we shall describe the action of S and Ss explicitly.
2 3
4 The A case
1
In this section we shall consider the simplest n = 2 case in standard and dynamical normalisa-
tions.
In the standard normalisation the separating operator Ss is trivial, c = b , λ = (λ ,λ ) and
2 λ λ 1 2
we simply obtain
P(1/g)(x ,x ) = xλ1+λ2f (x /x ), (4.1)
λ 1 2 2 λ 1 2
where the separating polynomial
f (x) = xλ2(1−x)1−2g F (−λ +1−2g,1−g;−λ +1−g;x)
λ 2 1 12 12
= xλ2 F (−λ ,g;−λ +1−g;x) (4.2)
2 1 12 12
and we used a well known formula for F [2]. The last formula in (4.2) gives an explicit
2 1
expansion of f (x) in series of x with rational coefficients.
λ
Althoughformulas(4.1),(4.2)giveexplicitrepresentationfortheA Jackpolynomials,asym-
1
metryw.r.t.x andx isnotobvious. Toobtainsuchsymmetricrepresentationitisinstructiveto
1 2
construct aseparation which correspondsto thedynamical normalisation of theBaker–Akhiezer
function.
An Analytic Formula for the A Jack Polynomials 5
2
Introduce the following basis in the space of symmetric polynomials S(x ,x ) in two variab-
1 2
les x , x
1 2
p ≡ p (x ,x ) = (x x )m[(1−x )(1−x )]n, m,n ≥ 0 (4.3)
mn mn 1 2 1 2 1 2
and define the operator S acting on S(x ,x ) as
2 1 2
(g)
n
S [p ] = p . (4.4)
2 mn mn
(2g)
n
Such operator S can be easily represented as an integral operator with a simple kernel (see [9]
2
for Macdonald polynomials) but we do not need here its explicit form. What is more important
is that it maps the A Jack polynomials into the product of separated polynomials
1
(g)
S [P(1/g)](x ,x ) = λ12 f (x )f (x ). (4.5)
2 λ 1 2 (2g) λ 1 λ 2
λ12
The action of the inverse operator S−1 on the basis p is easily constructed from (4.4) and to
2 mn
(1/g)
obtain a symmetric representation of P (x ,x ) we have to expand the rhs of (4.5) in p .
λ 1 2 mn
To do that we use Watson’s formula [22]
F (−n,b;c;x ) F (−n,b;c;x )
2 1 1 2 1 2
(c−b)
n Z
= F (−n,b;c,1−n+b−c;x x ,(1−x )(1−x )), n ∈ (4.6)
4 1 2 1 2 +
(c)
n
and the Appell’s function F is defined as
4
∞ (a) (b) xmyn
m+n m+n
F (a,b;c,d;x,y) = . (4.7)
4
(c) (d) m!n!
m n
m,n=0
X
After little algebra we obtain from (4.2)–(4.7)
(−λ ) (g) p (x ,x )
P(1/g)(x ,x ) = (x x )λ2 12 m+n m+n mn 1 2 . (4.8)
λ 1 2 1 2 (−λ +1−g) (g) m!n!
12 m n
0≤mX+n≤λ12
Finally we can rewrite (4.8) in the basis of elementary symmetric polynomials e = x + x ,
1 1 2
e = x x . Expanding p in e , e and evaluating remaining sums we come to
2 1 2 mn 1 2
(λ )! eiej
P(1/g)(x ,x ) = (e )λ2(−1)λ12 12 (g) 1 2(−1)i+j. (4.9)
λ 1 2 2 (g) i+j i!j!
λ12 i+2Xj=λ12
(1/g)
Another symmetric representation for P (x ,x ) which first appeared in [5] is given in terms
λ 1 2
g
of Gegenbauer polynomials C (x). Expanding e in series of x and x we obtain
n 1 1 2
Pλ(1/g)(x1,x2) = (x1x2)12|λ|((gλ)12)!Cλg12 12 (x1/x2)1/2 +(x2/x1)1/2 . (4.10)
λ12 (cid:18) (cid:19)
(cid:2) (cid:3)
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 Ss in the
n
standard normalisation. In the next section we shall use this fact to construct a representation
for the A Jack polynomials.
2
6 V.V. Mangazeev
5 The A case and the expansion of products
2
of separated polynomials
Now we shall consider n = 3 case in the standard normalisation. Using (2.10) we obtain
(1/g) |λ| (1/g) |λ|
P (x ,x ,x ) = x P (x /x ,x /x ,1) ≡ x p (x /x ,x /x ), (5.1)
λ 1 2 3 3 λ 1 3 2 3 3 λ 1 3 2 3
where we introduced a short notation p (x ,x ) for the A Jack polynomials with x = 1.
λ 1 2 2 3
Define a reduced separating operator Sˆ which acts only on the basis p (see (4.3)) in the
3 mn
space of symmetric polynomials in two variables S(x ,x ) as
1 2
(2g)
Sˆ [p ] = np . (5.2)
3 mn mn
(3g)
n
The explicit construction of Sˆ as an integral operator was given in [7, 6]. The operator Sˆ
3 3
separates variables in p (x ,x )
λ 1 2
Sˆ [p ](x ,x ) = c b−2f (x )f (x ), (5.3)
3 λ 1 2 λ λ λ 1 λ 2
where f (x) is given by (3.1) with n = 3.
λ
To construct p (x ,x ) we have to invert the action of Sˆ . It follows from (5.2) that in order
λ 1 2 3
to apply to (5.3) Sˆ−1 we have to expand the rhs of (5.3) in p
3 mn
f (x )f (x )= (x x )λ3 c (λ;g)p (x ,x ) (5.4)
λ 1 λ 2 1 2 m,n mn 1 2
0≤mX+n≤λ1,3
and it is easy to see that coefficients c (λ;g) depend on the partition λ only via λ and λ .
m,n 13 23
In fact, formula (5.4) is the main nontrivial result of this paper. Setting x = 1 in (5.4) we
2
obtain as a particular case the expansion for f (x ) of the form (3.6) with coefficients
λ 1
ξ (λ,g) = c (λ,g)/b . (5.5)
k m,0 λ
Now we shall calculate coefficients c (λ,g). For convenience we shall omit dependence of
m,n
c (λ;g) on λ and g.
m,n
Let us remind that the hypergeometric function F satisfies 3-th order differential equation
3 2
of the form [2, 19]
2 3
d
δ (δ+b −1)−x (δ+a ) F (a ,a ,a ;b ,b ;x) = 0, δ ≡ x . (5.6)
i i 3 2 1 2 3 1 2
dx
" #
i=1 i=1
Y Y
Using (3.1) we can easily obtain a third order differential equation for the functions f (x ),
λ i
i = 1,2. Now we can rewrite these equations in terms of new variables
u = x x , v = (1−x )(1−x ), p = umvn. (5.7)
1 2 1 2 m,n
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 . All calculations are straightforward
m,n
and we shall omit the details. The first recurrence relation for c looks amazingly simple and
mn
basically allows to find c in a closed form
m,n
c (m+1)(1−2g+m−r )(1−g+m−r )
m+1,n 1 2
+c (n+1)(3g+n−1)(3g+n)
m,n+1
+c (2g+m+n)(r −m−n)(g+m+n−r ) = 0, (5.8)
mn 1 2
where we denoted r = λ , r =λ .
1 1,3 2 2,3
An Analytic Formula for the A Jack Polynomials 7
2
The relation (5.8) immediately suggests to make a substitution
(2g) (−r ) (g−r )
m+n 1 m+n 2 m+n
c = a . (5.9)
mn mn
m!n!(3g−1) (3g) (1−2g−r ) (1−g−r )
n n 1 m 2 m
Then (5.8) reduces to
a +a = a (5.10)
m+1,n m,n+1 mn
which can be solved in terms of a
m,0
n
n
a = (−1)l a . (5.11)
m,n m+l,0
l
l=0 (cid:18) (cid:19)
X
The second recurrence relation for c leads to the following relation for a
m,n m,n
m(2g+r −m)(g+r −m)a −n(3g+n−2)(3g +n−1)a
1 2 m−1,n m,n−1
+(m+n−r )(2g+m+n)(g−r +m+n)a
1 2 m+1,n
+[n(3g+n−1)(3g +n−2)−3m(m+1)n−n(r −1)(r −1)
1 2
+2(m−r )[g(1+r )+(r −m)m]+2(3g+r +r )mn
2 1 1 1 2
−g(g−1)(r +3r −5m)−g(2g −3+r +2r )n]a = 0. (5.12)
1 2 1 2 mn
However, due to (5.11) we need the latter relation only for the calculation of a which corre-
m,0
sponds to the case n = 0 in (5.12). In fact, it is much easier to obtain the 3-term recurrence
relation for a not from (5.12), but from the 3-term recursion for the coefficients c itself.
m,0 m,0
This latter recursion follows from (3.6), (5.5) and the third order differential equation for f (x).
λ
Now let us introduce a generating function
h(x) = x1−ρ a xm (5.13)
m,0
m≥0
X
and rewrite the recurrence relation for a as a third order differential equation for h(x).
m,0
Then this equation can be easily reduced to the standard hypergeometric equation (5.6) for the
function F . This equation has three solutions with exponents
3 2
ρ = 2g,g−r ,−r (5.14)
2 1
at x= 0 which can be written as
g−r −ρ,3g−ρ,2g−r −ρ,1;
h(x) = x1−ρ(1−x)3g−2 F 1 2 x (5.15)
4 3 2g+1−ρ,1−ρ−r ,1−ρ+g−r
1 2
(cid:18) (cid:12) (cid:19)
(cid:12)
and for any choice of ρ a hypergeometric function F reduces to F . (cid:12)
4 3 3 2
Now we note that for ρ = 2g and ρ = g − r the hypergeometric function in the rhs
2
of (5.15) truncates and we obtain from (5.13), (5.15) the following equivalent expressions for the
coefficients a
m,0
(1−g) −r , g, −g−r ,1−2g−m;
a = α m F 2 1 1 (5.16)
m,0 1 (2g) 4 3 1−g−r ,1−2g−r ,g−m
m (cid:18) 2 1 (cid:12) (cid:19)
(cid:12)
and (cid:12)
(1−2g−r ) r −r , g, 2g+r ,1−g+r −m;
a = α 2 m F 2 1 2 2 1 . (5.17)
m,0 2 (g−r ) 4 3 1−g+r −r ,1+g+r ,2g+r −m
2 m (cid:18) 2 1 2 2 (cid:12) (cid:19)
(cid:12)
(cid:12)
8 V.V. Mangazeev
To calculate the correct normalisation coefficients α , α for (5.16), (5.17) we take the limit
1 2
x → 0, x → 1 in (5.4). Then only the term c contributes to the rhs of (5.4) and we use
1 2 0,0
(3.5), (5.9) to calculate a .
0,0
From the other side the hypergeometric series in (5.16), (5.17) becomes Saalschu¨tzian F
3 2
(i.e. 1+a +a +a = b +b ) and can be evaluated using the so called Saalschu¨tz’s theorem
1 2 3 1 2
[2, 19]
a,b,−n; (c−a) (c−b)
n n
F = . (5.18)
3 2 c,1+a+b−c−n (c) (c−a−b)
(cid:18) (cid:19) n n
Then we obtain
(r −r )! (3g) (2g)
α = 1 2 r1 r2 , (5.19)
1
r ! (2g) (1−g)
1 r1−r2 r2
r ! (1+g) (g) (3g) (2g)
α = 2 r1 r1−r2 r1 r2. (5.20)
2
r !(1+g) (2g) (2g) (g)
1 r2 r1−r2 r1 r2
Now wecansubstitute(5.16)–(5.17)into(5.11). Expandinghypergeometricfunctionsintoseries
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
(r −r )! (3g) (2g) (−r ) (g−r ) (1−g)
c = 1 2 r1 r2 1 m+n 2 m+n m
m,n
r ! (2g) (1−g) m!n!(1−2g−r ) (1−g−r ) (3g)
1 r1−r2 r2 1 m 2 m n
−r , g, −g−r ,1−2g−m−n;
× F 2 1 1 (5.21)
4 3 1−g−r ,1−2g−r ,g−m
2 1
(cid:18) (cid:12) (cid:19)
(cid:12)
or (cid:12)
r ! (1+g) (g) (3g) (2g) (1−2g−r ) (−r ) (2g)
c = 2 r1 r1−r2 r1 r2 2 m 1 m+n m+n
m,n
r !(1+g) (2g) (2g) (g) (1−2g−r ) (1−g−r ) (3g)
1 r2 r1−r2 r1 r2 1 m 2 m n
1 r −r , g, 2g+r ,1−g+r −m−n;
× F 2 1 2 2 1 . (5.22)
m!n! 4 3 1−g+r −r ,1+g+r ,2g+r −m
2 1 2 2
(cid:18) (cid:12) (cid:19)
(cid:12)
As we mentioned above the coefficients of the expansion of the s(cid:12)eparating polynomial f (x)
λ
in x are given by c , i.e. expressed in terms of hypergeometric functions F . It is quite
m,0 4 3
surprising that the coefficients c of the expansion (5.4) are still given in terms of F by for-
m,n 4 3
mulas (5.21), (5.22)anddonotinvolve higherhypergeometric functions. Thishappensprobably
because the basis p is very special.
mn
6 A representation for the A Jack polynomials
2
Now we are ready to give explicit formulas for the A Jack polynomials. Applying the opera-
2
tor Sˆ−1 to (5.3) and using (5.1), (5.2), (5.4) we obtain two representations
3
(g) (2g) (λ )!
P(1/g) (x ,x ,x )= λ23 λ13 12 (x x )λ3xλ1+λ2−λ3
(λ1,λ2,λ3) 1 2 3 (g) (1−g) (λ )! 1 2 3
λ12 λ23 13
1 (−λ ) (g−λ ) (1−g)
13 m+n 23 m+n m
× (6.1)
m!n!(1−2g−λ ) (1−g−λ ) (2g)
13 m 23 m n
0≤mX+n≤λ13
m n
−λ , g, −g−λ ,1−2g−m−n; x x x x
× F 23 13 1 1 2 1− 1 1− 2
4 3(cid:18) 1−g−λ23,1−2g−λ13,g−m (cid:12) (cid:19)(cid:20) x23 (cid:21) (cid:20)(cid:18) x3(cid:19)(cid:18) x3(cid:19)(cid:21)
(cid:12)
(cid:12)
An Analytic Formula for the A Jack Polynomials 9
2
and
(g) (λ )!
P(1/g) (x ,x ,x )= λ13+1 23 (x x )λ3xλ1+λ2−λ3
(λ1,λ2,λ3) 1 2 3 (g) (λ )! 1 2 3
λ23+1 13
1 −λ , g, 2g+λ ,1−g+λ −m−n;
× F 12 23 23 1 (6.2)
m!n! 4 3 1+g+λ ,1−g−λ ,2g+λ −m
23 12 23
0≤mX+n≤λ13 (cid:18) (cid:12) (cid:19)
m (cid:12) n
(−λ13)m+n(1−2g−λ23)m(2g)m+n x1x2 x1 (cid:12)x2
× 1− 1− .
(1−2g−λ ) (1−g−λ ) (2g) x2 x x
13 m 23 m n (cid:20) 3 (cid:21) (cid:20)(cid:18) 3(cid:19)(cid:18) 3(cid:19)(cid:21)
Formulas (6.1), (6.2) simplify when λ = λ and λ = λ accordingly.
2 3 1 2
First consider the case of one-row partitions (λ,0,0). It is convenient to set x = 1. Then
3
we obtain from (6.1)
(2g) (−λ) (g) (x x )m[(1−x )(1−x )]n
(1/g) λ m+n m+n 1 2 1 2
P (x ,x ,1) = . (6.3)
λ,0,0 1 2 (g) (1−2g−λ) (2g) m!n!
λ m n
0≤m+n≤λ
X
Expanding the last term we can transform this formula to another basis
(−1)i+j (−λ) (g) (g)
P(1/g)(x ,x ,1) = i+2j i+j λ−i−2j(x +x )i(x x )j. (6.4)
λ,0,0 1 2 i!j! (g) 1 2 1 2
λ
0≤i+2j≤λ
X
Finallyrestoringdependenceonx weshallrewrite(6.4)inthebasisofelementarysymmetric
3
functions e = x +x +x , e = x x +x x +x x and e = x x x .
1 1 2 3 2 1 2 1 3 2 3 3 1 2 3
λ! (g)
P(1/g)(x ,x ,x ) = (−1)λ (−1)i+j+k i+j+k eiejek. (6.5)
λ,0,0 1 2 3 (g) i!j!k! 1 2 3
λ
i+2j+3k=λ
X
In fact, (6.5) corresponds to the first fundamental weight ω of A and is a special case of
1 2
the Proposition 2.2 from [20] for arbitrary number of variables
r! (g)
P(1/g)(x) = e l(µ) (−1)r−l(µ) (6.6)
(r) (g)r |µ|=r µ mi!
X
i≥1
Y
for all partitions µ such that
µ = (1m12m2···), |µ|= r, l(µ) = m , e = em1em2··· . (6.7)
i µ 1 2
X
(1/g)
Now from (6.2) we can similarly calculate P (x ,x ,x ) which corresponds to the second
λ,λ,0 1 2 3
fundamental weight ω of A
2 2
λ! (g) (−1)m3
P(1/g) (x ,x ,x ) = e λ−m3 (6.8)
(λ,λ,0) 1 2 3 (g) µ 3 2
λ
|µ|=2λ λ− m ! m !
X i i
(cid:18) i=1 (cid:19) i=1
P Q
for all partitions µ = (1m12m23m3), |µ|= 2r.
It seems that (6.8) has the following generalisation to n variables
λ! (g) (−1)mn
P(1/g) (x ,...,x ) = e λ−mn (6.9)
(λn−1,0) 1 n (g) µ n n−1
λ
|µ|=(n−1)λ λ− m ! m !
X i i
(cid:18) i=1 (cid:19) i=1
P Q
10 V.V. Mangazeev
for all partitions µ = (1m1···nmn), |µ|= (n−1)λ. We arenot aware of any appearances of (6.9)
in the literature.
Note that formulas for the A Jack polynomials were already obtained earlier in [15]. They
2
weregeneralised in[11]forpartitions oflength 3withany numberofvariables andfurtherin[12]
for the general A case.
n
The approach used in [15, 11] is based on the inversion of the Pieri formulas. For the A case
2
formulas of [15] can be written as
min(λ1−λ2,λ2)
P = βi P P , (6.10)
(λ1,λ2,0) λ1,λ2 (λ1−λ2−i,0,0) (λ2−i,λ2−i,0)
i=0
X
λ2
P = γi P P , (6.11)
(λ1,λ2,0) λ1,λ2 (λ1+i,0,0) (λ2−i,0,0)
i=0
X
where βi and γi are explicitly given factorised coefficients. Note also that (6.11) is
λ1,λ2 λ1,λ2
a specialisation of the result [4] for partitions of length 2 and any number of variables.
It is interesting to compare (6.10), (6.11) with our formulas (6.1), (6.2). Using the expres-
sions (6.5) and (6.8) we can rewrite (6.10), (6.11) in terms of quintuple sum in the basis of
elementary symmetric functions. However, our formulas (6.1), (6.2) contain only triple sums in
a different basis.
We tried hard to obtain directly (6.1), (6.2) from (6.10), (6.11) with no success. The corre-
sponding 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 beinteresting to rewrite formulas (6.1), (6.2) in the basis e , e , e like (6.5)
1 2 3
for a general partition λ = (λ ,λ ,λ ). However, it involves sums of truncated hypergeometric
1 2 3
functions F and we did not succeed at this stage to obtain any compact expression for general
4 3
partitions of length 3.
7 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 Jack polynomials in a special basis.
2
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.
Acknowledgments
I greatly benefited in my understanding of the quantum SoV from very stimulating discussions
with Vadim Kuznetsov and Evgeny Sklyanin. A warm atmosphere at Leeds University, critical
and creative talks with Vadim are things which will be greatly missed.
References
[1] Calogero F., Solution of a three-body problem in one dimension, J. Math. Phys. 10 (1969), 2191–2196.
[2] Erd´elyi A.(Editor), Highertranscendental functions, Vol.I, McGraw-Hill Book Company,1953.
[3] JackH.,Aclassofsymmetricfunctionswithaparameter, Proc. Royal Soc. Edinburgh (A)69(1970), 1–18.
[4] Jing N.H., J´ozefiak T., A formula for two-row Macdonald functions, Duke Math. J. 67 (1992), 377–385.
