6 Gauge Freedom of Entropies on q-Gaussian Measures
137
on [1, ∞) converges if and only if 2/(1 − q) < −1, that is 1 < q < 3, so does the
improper integral of the function x → Rexp q (−x
2
) on R. We observe that
R
Rexp q (−x
2
)dx = 2
∞
0
1 − (1 − q)x
2
1
1−q dx
=
1
√
q − 1
∞
0
(1 + r )
1
1−q r
−
1
2 dr
=
1
√
q − 1
B
3 − q
2(q − 1)
,
1
2
,
where we used that
B(t − s, s) =
∞
0
r
s−1
(1 + r ) t dr
for t > s > 0.
In the case q < 1, the support of the function x → Rexp q (−x
2
) on R is
−
1
√
1 − q
,
1
√
1 − q
implying that
R
Rexp q (−x
2
)dx = 2
1
√
(1−q)
0
1 − (1 − q)x
2
1
1−q dx
=
1
√
1 − q
1
0
[1 − r ]
1
1−q r
−
1
2 dx
=
1
√
1 − q
B
2 − q
1 − q
,
1
2
.
Definition 4 For q < 3 and ξ = (μ, σ ) ∈ R × (0, ∞), the q-Gaussian measure
with location parameter μ and scale parameter σ on R is an absolutely continuous probability measure with respect to the one-dimensional Lebesgue measure with
Radon–Nikodym derivative
p q (x; ξ) = p q (x; μ, σ ) :=
1
Z q σ
Rexp q
−
1
3 − q
x − μ
σ
2
.
We call p q (x; ξ) = p q (x; μ, σ ) the q-Gaussian density with location parameter μ
and scale parameter σ .