100
G. Vinod
g(x)D x f (x)d(qx) = f (x)g(x) −
f (qx)D x g(x)d(qx)
(47)
It is helpful to introduce the dilation operator ˆ
Q:
ˆ
Q f (x) = f (qx)
(48)
ˆ
Qx
m
= q
m x
m
= e
m ln q x
m
=
1 +
m ln q
1!
+
(m ln q)
2
2!
+ · · ·
x
m
=
1 +
(ln q)x∂ x
1!
+
((ln q)
2
(x∂ x )
2
2!
+ · · ·
x
m
= e
(ln q)x∂ x x
m
= q
x∂ x x
m
(49)
This can be generalised as
q
±ax∂ x f (x) = f (q
±a x)
(50)
With the help of the dilation operator, the action of the q-difference operator can be
written as
D x f (x) =
q
x∂ x f (x) − f (x)
x(q − 1)
(51)
The following q-commutation relation holds:
[D x , x] q = 1
( 5 2 )
Also it can be shown that
D x , q
x∂ x
q
= 0
(53)
and
D x , q
−x∂ x
q −1 = 0
( 5 4 )
Generalisation of the above results is
D x , q
ax∂ x
q a = 0; a ∈ Q
(55)
D
n
x , q
ax∂ x
q na = 0; n ∈ Z, a ∈ Q
(56)