128
H. Matsuzoe and A. Takatsu
6.1 q-Logarithmic Functions and Their Refinements
6.1.1 Definitions
For q ∈ R, we set χ q : (0, ∞) → (0, ∞) by
χ q (s) := s
q
.
We define a strictly increasing function ln q : (0, ∞) → R by
ln q (t) :=
t
1
1
χ q (s)
ds
and we denote by exp q the inverse function of ln q : (0, ∞) → ln q (0, ∞). The functions ln q and exp q are called the q-logarithmic function and the q-exponential function, respectively. We observe that
d
dt
ln q (t) =
1
χ q (t)
= t
−q
for t ∈ (0, ∞),
d
dτ
exp q (τ ) = χ q (exp q (τ )) = exp q (τ )
q
for τ ∈ ln q (0, ∞).
It holds for q ∈ R that χ q (1) = 1 and ln q (1) = 0.
Remark 1 (1) For q = 1, we have that
ln 1 (t) = log(t)
for t ∈ (0, ∞),
ln 1 (0, ∞) = R,
exp 1 (τ ) = exp(τ )
for τ ∈ R.
(2) For q = 1, we have that
ln q (t) =
t
1−q
− 1
1 − q
for t ∈ (0, ∞),
ln q (0, ∞) =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
−∞,
1
q − 1
if q > 1,
−
1
1 − q
, ∞
if q < 1,
exp q (τ ) = {1 + (1 − q)τ }
1
1−q
for τ ∈ ln q (0, ∞).