146
H. Matsuzoe and A. Takatsu
6.4.2 Expression of the Refined Riemann Metrics
We compute the exact value of
g
(q,a)
μμ (ξ ) =
4
(3 − q) 2
1
j=0
b
2
j
(Z q σ ) 2(1−q) σ 2
R
x − μ
σ
2 p q (x; ξ)
2q−1
− q (x, ξ)
j dx
=
4
(3 − q) 2
1
j=0
b
2
j
(Z q σ ) 2(1−q) σ 2 (q, 2, 1, j; ξ),
g
(q,a)
σ σ (ξ ) =
1
j=0
b
2
j
(Z q σ ) 2(1−q) σ 2
R
1 −
x − μ
σ
2
2
p q (x; ξ)
2q−1
− q (x, ξ)
j dx
=
1
j=0
b
2
j
(Z q σ ) 2(1−q) σ 2
2
k=0
2
k
(−1)
k
(q, 2, k, j; ξ),
(6.8)
for ξ ∈ R × q,a , where we set
(q, n, k, j; ξ) :=
R
x − μ
σ
2k p q (x; ξ)
(n−1)(q−1)+q
− q (x, ξ)
j
dx.
Lemma 6 For n ∈ N, k ∈ {0, 1, . . . , n} and ξ = (μ, σ ) ∈ R × q,a , then
(q, n, k, 0; ξ)
=
⎧
⎪ ⎨
⎪ ⎩
σ
(Z q σ ) (n−1)(q−1)+q
3 − q
q − 1
k+
1
2
B
3 − q
2(q − 1)
+ n − k,
1
2
+ k
if q > 1,
(2k − 1)!!
if q = 1,
where by convention (2 · 0 − 1)!! := 1.
Proof We apply the change of variables with
y =
1
2
x − μ
σ
2
if q = 1, and y =
q − 1
3 − q
x − μ
σ
2
otherwise.
For q = 1, we observe that