86
J. Harnad
where
Δ(x) =
1≤i
n−j
i
) 1≤i,j,≤n
(34)
is the Vandermonde determinant.
Eulerian Wronskian Representation
It follows from the recursion relations
β(D + k − 1)φ k = φ k−1 , k ∈ Z,
(35)
that
τ
(G,β)
X
= γ n
n
i=1
x
n−1
i
det
D i−1 φ n (x j )
1≤i,j,≤n
Δ(x)
,
(36)
where
γ n :=
β
1
2 n(n−1)
n
i=1 ρ −i
.
(37)
3.4 Matrix Integral Representation of τ (G,β) ([X]) [2, 7]
Wronskian Representation: Rational Case
For rational weight generating functions G = G c,d , and any n ∈ N + , let
φ
(c,d,β)
n
(e
y ) =
C n
A
(c,d,β)
n
(s)e
ys ds,
A
(c,d,β)
n
(s) :=
C
(c,d,β)
n
Γ (1 − n − s)
L
l=1 Γ
s +
1
βc l
(−κ c,d ) s
2πi
M
m=1 Γ
s −
1
βd m
.
Define the diagonal matrix Y = diag(y 1 , . . . y n )
X = e
Y , Y = ln(X), x i = e
y i , i = 1, . . . , n.
(38)
J. Harnad
where
Δ(x) =
1≤i
n−j
i
) 1≤i,j,≤n
(34)
is the Vandermonde determinant.
Eulerian Wronskian Representation
It follows from the recursion relations
β(D + k − 1)φ k = φ k−1 , k ∈ Z,
(35)
that
τ
(G,β)
X
= γ n
n
i=1
x
n−1
i
det
D i−1 φ n (x j )
1≤i,j,≤n
Δ(x)
,
(36)
where
γ n :=
β
1
2 n(n−1)
n
i=1 ρ −i
.
(37)
3.4 Matrix Integral Representation of τ (G,β) ([X]) [2, 7]
Wronskian Representation: Rational Case
For rational weight generating functions G = G c,d , and any n ∈ N + , let
φ
(c,d,β)
n
(e
y ) =
C n
A
(c,d,β)
n
(s)e
ys ds,
A
(c,d,β)
n
(s) :=
C
(c,d,β)
n
Γ (1 − n − s)
L
l=1 Γ
s +
1
βc l
(−κ c,d ) s
2πi
M
m=1 Γ
s −
1
βd m
.
Define the diagonal matrix Y = diag(y 1 , . . . y n )
X = e
Y , Y = ln(X), x i = e
y i , i = 1, . . . , n.
(38)