Formulas for some confluent hypergeometric
functions in several variables∗
Jan Felipe van Diejen
Universidad de Talca, Chile
Representation Theory, Special Functions and Painlevé Equations
RIMS (Kyoto University), Japan, March 2015
∗
Joint work with Erdal Emsiz, Pontificia Universidad Católica de Chile
Themes of the talk:
Part A.:
Branching formula for multivariate Laguerre polynomials
Part B.:
Difference equation for class-one Whittaker functions
Part A.
Branching formula for multivariate Laguerre polynomials
I.A Multivariate Laguerre polynomials
II.A Recurrence relations
III.A Branching formula
IV.A Explicit formulas for Laguerre polynomials
I.A Multivariate Laguerre Polynomials
Macdonald 1988, Lassalle 1991, Baker and Forrester 1997, vD 1997, . . .
Orthogonality
— Weight function
(Selberg integral, Random Matrices, Calogero-Moser systems)
2
2
∆(x) = e −ω(x1 +···+xn ) |x1 · · · xn |2g1 −1
Y
|xj2 − xk2 |2g0
1≤j<k≤n
— Partitions
Λn := {λ = (λ1 , . . . , λn ) | λ1 ≥ · · · ≥ λn }
endowed with Lexicographical Ordering.
—Laguerre polynomials
Lλ (x) = x12λ1 · · · xn2λn + lower terms
such that:
–Lλ (x) invariant w.r.t. action of (signed) permutation group
–Lλ (x), λ ∈ Λn form orthogonal system w.r.t. ∆(x) (Gram-Schmidt).
Confluent hypergeometric differential equation
DLλ = 4ω(λ1 + · · · + λn )Lλ
2g1 − 1 ∂
∂ ∂2
−
+
2ωx
j
xj
∂xj
∂xj
∂xj2
1≤j≤n
X
1 ∂
∂ 1 ∂
∂ − 2g0
+
+
−
xj + xk ∂xj
∂xk
xj − xk ∂xj
∂xk
D=
X −
1≤j<k≤n
Relation to Calogero-Moser system
This confluent hypergeometric differential equation is equivalent to
the stationary Schrödinger equation with a harmonically confined
rational Calogero-Moser potential:
u(x1 , . . . , xn ) =
X (g1 − 1 )(g1 − 3 )
2 2
2
2
+
ω
x
j
xj2
1≤j≤n
X 1
1
+ g0 (g0 − 1)
+
(xj − xk )2
(xj + xk )2
1≤j6=k≤n
Constant term value
Lλ (0) = (−ω)−|λ|
Y
1≤j<k≤n
(1 + (k − j)g0 )λj −λk Y
((n − j)g0 + g1 )λj
((k − j)g0 )λj −λk
1≤j≤n
where (a)k = a(a + 1) . . . (a + k − 1) denotes the Pochammer symbol.
Orthogonality relations
(2π)n/2 δλ,µ
Lλ (x)Lµ (x)∆(x)dx = g n(n−1)+ng +2λ +···+2λ
n
1
1
ω 0
Rn
Y
×
Γ((n − j)g0 + g1 + λj )Γ(1 + (n − j)g0 + λj )
Z
1≤j≤n
×
Y
1≤j<k≤n
Γ((k − j + 1)g0 + λj − λk )Γ(1 + (k − j − 1)g0 + λj − λk )
Γ((k − j)g0 + λj − λk )Γ(1 + (k − j)g0 + λj − λk )
II.A Recurrence Relations
vD 1997
Elementary symmetric functions
X
Er (x) = (−ω)r
xj21 . . . xj2r
1≤j1 <···<jr ≤n
(r = 1, . . . , n)
Recurrence formula
Er (x)Lλ (x) =
X
µ,n
Cλ,r
(g0 , g1 , ω)Lµ (x)
µ∈Λn
µ∼r λ
µ ∼r λ iff µ = λ + eJ+ − eJ− with |J+ | + |J− | ≤ r
(where e1 , . . . , en is the standard unit basis, J+ , J− ⊂ {1, . . . , n} with J+ ∩ J− = ∅, and eJ =
P
j∈J
ej )
Recurrence coefficients
µ,n
Cλ,r
(g0 , g1 , ω) =
Lλ (0) n
n
(λ; g0 , g1 , ω)
V
(λ; g0 , g1 , ω)U(J
c
+ ∪J− ) ,r −|J+ |−|J− |
Lµ (0) J+ ,J−
with
n
Y
VJ (λ; g0 , g1 , ω) =
×
Y (j 0
j∈J+
j 0 ∈J−
×
1+
j∈J+
k6∈J+ ∪J−
n
g0
1+
− j)g0 + λj − λj 0
g0
Y
((n − j)g0 + λj )
j∈J−
g0
1+
Y
((n − j)g0 + g1 + λj )
j∈J+
(j 0
(k − j)g0 + λj − λk
− j)g0 + λj − λj 0 + 1
Y
)
1−
j∈J−
k6∈J+ ∪J−
g0
(k − j)g0 + λj − λk
p
UK ,p (λ; g0 , g1 , ω) = (−1) ×
X
Y
Y
((n − l)g0 + g1 + λl )
((n − l)g0 + λl )
L+ ,L− ⊂K l∈L+
L+∩L− =∅
|L+ |+|L− |=p
×
Y
Y
g0
1+
l∈L+ ,l 0 ∈L−
×
l∈L−
l∈L+
k∈K \(L+ ∪L− )
1+
g0
1+
(l 0 − l)g0 + λl − λl 0
g0
(k − l)g0 + λl − λk
)
(l 0 − l)g0 + λl − λl 0 + 1
Y
l∈L−
k∈K \(L+ ∪L− )
1−
g0
(k − l)g0 + λl − λk
Simplest case: r = 1 (generalized three-term recurrence)
Baker and Forrester 1997, vD 1997
− ωLλ (x)
X
xj2 =
1≤j≤n
X
X
Vj (λ)(Lλ+ej (x) − Lλ (x)) +
1≤j≤n
λ+ej ∈Λ
V−j (λ)(Lλ−ej (x) − Lλ (x))
1≤j≤n
λ−ej ∈Λ
Lλ (x) = Lλ (x)/Lλ (0)
Vj (λ) = ((n − j)g0 + g1 + λj )
Y 1+
1≤k≤n
k6=j
V−j (λ) = ((n − j)g0 + λj )
Y 1≤k≤n
k6=j
1−
g0
(k − j)g0 + λj − λk
g0
(k − j)g0 + λj − λk
λ2
0
λ1
III.A Branching Formula
vD-Emsiz 2014
Notation
—For λ, µ ∈ Λn the skew diagram λ \ µ is a horizontal strip iff
λ 1 ≥ µ 1 ≥ λ 2 ≥ µ2 ≥ · · · ≥ λ n ≥ µn
—We write µ λ iff there exists a ν ∈ Λn such that the skew
diagrams λ/ν and ν/µ are horizontal strips.
—mn − λ (λ1 ≤ m) is the partition such that
(mn − λ)j = m − λn+1−j
(j = 1, . . . , n).
—λ0 ∈ Λm (m ≥ λ1 ) denotes the conjugate partition of λ ∈ Λn
Branching formula
Lλ (x1 , . . . , xn , x) =
X
Lµ (x1 , . . . , xn )Lλ/µ (x)
µ∈Λn
µλ
Branching polynomials
Lλ/µ (x) =
X
k
(g0 , g1 , ω)x 2k
Bλ/µ
0≤k≤d
d = |{1 ≤ j ≤ n | λ0j − µ0j = 1}|.
Coefficients
(n+1)m −λ0 ,m
k
Bλ/µ
(g0 , g1 , ω) = ω k−m (−g0 )|λ|−|µ|−m Cnm −µ0 ,m−k (1/g0 , g1 /g0 , g0 ω)
(with k = 0, . . . , d and m = λ1 ).
Proof by degeneration from analogous branching formula for Koornwinder-Macdonald multivariate Askey-Wilson
polynomials, which is proved in turn by means of their recurrence relations and Mimachi’s reproducing kernel.
IV.A Explicit Formulas for Laguerre Polynomials
vD-Emsiz 2014
The multivariate Laguerre polynomials are given by
Lλ (x1 , . . . , xn ) =
X
Y
Lµ(i) /µ(i−1) (xi )
µ(i) ∈Λi ,i=1,...,n 1≤i≤n
µ(1) ···µ(n) =λ
where
(
branching polynomial
Lµ(i) /µ(i−1) (xi ) =
monic Laguerre polynomial
if i > 1
if i = 1
n=1
The monic Laguerre polynomial of degree m is given by
Lm (x) =
X
k
Bm/0
(g1 , ω)x 2k
0≤k≤m
with
m
,m
k
Bm/0
(g1 , ω) = ω k−m C00m ,m−k
(1/g0 , g1 /g0 , g0 ω)
Compare with standard 1 F1 representation:
Lm (x) = (−ω)−m (g1 )m 1 F1 (−m; g1 ; ωx 2 ).
Part B.
Difference equation for class-one Whittaker functions
I.B Toda system
II.B Whittaker function
III.B Difference equation
I.B Toda System
Kostant 1979, Olshanetsky-Perelomov 1983, Goodman-Wallach 1986, . . .
Quantum hamiltonian
HR := ∆ − 2
X
e −hα,xi
α∈S
Notation
∆ = Laplacian on Rn
h·, ·i = standard inner product on Rn
S = simple basis for a crystallographic root system R ⊂ Rn
Assumption R is irreducible and reduced.
Classical series
n
X
∂2
− 2 e −x1 +x2 + e −x2 +x3 + · · · + e −xn−1 +xn
HA =
∂xj2
j=1
HB =
n
X
j=1
HC =
∂2
− 2 e −x1 +x2 + e −x2 +x3 + · · · + e −xn−1 +xn +e −xn
2
∂xj
n
X
∂2
− 2 e −x1 +x2 + e −x2 +x3 + · · · + e −xn−1 +xn +e −2xn
2
∂xj
j=1
HD =
n
X
∂2
− 2 e −x1 +x2 + e −x2 +x3 + · · · + e −xn−1 +xn +e −xn−1 −xn
∂xj2
j=1
II.B Whittaker Function
Jacquet 1967, Kostant 1979, Hashizume 1982, Goodman-Wallach 1986, Baudoin-O’Connel 2011, . . .
Definition
The class-one Whittaker function Fξ (x) on R is a solution of the
Toda eigenvalue equation
HR Fξ (x) = hξ, ξiFξ (x)
that is smooth and of moderate exponential growth in x ∈ Rn and
holomorphic in the spectral variable ξ ∈ Cn .
Normalization conditions
Fw ξ (x) = Fξ (x)
(∀w ∈ W )
and
Y
lim e hw0 ξ,xi Fξ (x) =
x→+∞
ηα−hξ,α
∨i
Γ(hξ, α∨ i)
α∈R +
for Re(ξ) in the positive fundamental chamber.
Notation
W = Weyl group
w0 = longest element of q
W
2α
2
α∨ = hα,αi
ηα = hα,αi
III.C Difference Equation
vD-Emsiz 2014
For any dominant weight ω with hω, α∨ i ≤ 2 for all α ∈ R:
X
X
Uν,η (ξ)Vν (ξ)Fξ+ν (x) = e hω,xi Fξ (x)
ν∈P(ω) η∈Wν (wν−1 ω)
where
Y
Vν (ξ) =
α∈R
hν,α∨ i>0
Uν,η (ξ) =
ηα
hξ, α∨ i
Y
α∈Rν
hη,α∨ i>0
Y
α∈R
hν,α∨ i=2
ηα
hξ, α∨ i
ηα
1 + hξ, α∨ i
Y
α∈Rν
hη,α∨ i=2
−ηα
1 + hξ, α∨ i
Notation
Wν = the stabilizer {w ∈ W | w ν = ν} Rν = {α ∈ R | hν, α∨ i = 0}
wν = shortest element in W mapping ν to the dominant chamber
P(ω) = saturated set of weights on the convex hull of W ω
Proof
Based on q-difference equations for the Macdonald polynomials (Macdonald 1988, vD-Emsiz 2011) via
the following chain of degenerations:
Macdonald polynomials
vD-Emsiz 2014
=⇒
Heckman-Opdam hypergeometric functions
Shimeno 2008
=⇒
Whittaker functions
Simplest difference equation
When ω is minuscule (i.e. hω, α∨ i ≤ 1 for all α ∈ R) then:
X
ν∈W ω
Fξ+ν (x)
Y
α∈R
hν,α∨ i>0
(as P(ω) = W ω in this situation)
ηα
= e hω,xi Fξ (x)
hξ, α∨ i
Concrete example: difference equations for R of type A
X
Fξ+eJ (x)
J⊂{1,...,n}
|J|=r
Y
(ξj − ξk )−1 = e x1 +···+xr Fξ (x)
j∈J
k6=J
r = 1, . . . , n
cf. Ruijsenaars 1990 (classical mechanics), Babelon 2003 (via Mellin-Barnes type integrals), Kozlowski 2013 (via
quantum inverse scattering method)
Happy 60th Anniversary!
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