Page 24 - Textos de Matemática Vol. 34

14 LU´IS DANIEL DE ABREU
and
(1.5) (a; q)∞ = lim (a; q)n.
n→∞
The main reference in the theory of q−special functions is [8]. For k = 1, 2, 3 it is
easy to verify the relation
limJ(k)(z(1−q);q2)=J (z) q→1 ν ν
where Jν is the classical Bessel function defined, if ν > −1, by the series expansion
∞ (−1)n z ν +2n
n!Γ(ν + n + 1) 2 .
Thus, the three q−Bessel functions are q−analogues of the classical Bessel function.
The third Jackson q−Bessel function was also studied by [7] and often appears
in the literature as the Hahn-Exton q−Bessel function. In some contexts it is
natural to ask what is the best q−analogue. Comments on this have been made in
recent years, by Rahman [17], and by Koornwinder and Swarttouw [16]. They have
(2)
Jν (z) =
n=0
different points of view. Rahman finds the q−hypergeometric structure of Jν (z; q) (3)
nicer than Jν (z; q)’s. The other authors favor the third q−Bessel function because it is more suitable to Harmonic Analysis, both within or without the context of the
(3)
quantum groups. To see how Jν (z; q) appears in the theory of quantum groups, we (3) 2
quote [14]. In our recent work [1]-[3], Jν (z; q) was used to provide a basis for the L space associated to the Jackson q−integral and to build a q−Bessel analogue of the
Whittaker-Shannon-Koltenikov sampling theorem. This convenience of Jν (3)
for q−Harmonic analysis is due to the the q−Hankel theory satisfied by Jν [16] and to the well known orthogonality relation [15]
1
xJ(3)(qxj ; q2)J(3)(qxj ; q2)d x =
ν nν ν mν q
0
(1.6) =−1(1−q)qν−1J(3) (qj ;q2) d J(3)(qj ;q2)δ
2 ν+1nνdxνnνn,m
(3)
(z;q) (z;q)
with respect to Jackson’s q−integral
(1.7)
1 ∞ f (x) dqx = (1 − q)
f (qn) qn. 2
0
n=0 (3)
The notation jnν stands for the nth zero of Jν (z;q ). In [3] it was proved that jnν = q−n+εn , for a certain positive εn smaller than one. The results of [1] show that every function on a certain Hilbert space with reproducing kernel can be recovered from its values at the sampling points qjnν.