Page 29 - Textos de Matemática Vol. 34

CONNECTING BASIC BESSEL FUNCTIONS 19
Using (3.1) and (3.2) to proceed as in the proof of Theorem 2.1, yields
1 ∞ qk(k−1) 24k2
G1 (z;q ) = 1 1 1 F1 (q2 z;q )
2
(−q;q2 )∞ k=0 (q2 ;q2 )k 2
1 ∞ qk(k+1) 24k2
and
The result follows from the definitions of the functions Sq(z), Cq(z), sin(z;q) and
G− 1 (z; q ) = 1 1 1 1 F− 1 (q 2 z; q ). 2 (−q2 ;q2 )∞ k=0 (q2 ;q2 )k 2
cos(z; q).
Analogously, Theorem 2.2 gives
Corollary 3.2. For every z ∈ C,

1∞ (−qn+1;q2)∞
k
qk (q2 ;q2 )k
k+3 sin(zq 2 4 ; q)
Sq (z) = (1 − q)
(−qz ;q )∞ k=0 22
2 2 (−1) (−qz ;q )∞ k=0
1
1
and
11∞k
(−qn+2;q2)∞
2 2 (−1)
q2
(q2 ;q2 )k
Cq(z) =
Proof. Setting ν = 1 and ν = −1 in (2.9) gives
k
k−3 cos(zq2 4 ;q).
(q2 ;q2 )k q ( n + 1 ) k
(−q
n+1 1 1
= (−1)k ;q2 ) k=0
1
1
1 ∞
q(n+1)k 1 1
and
n+1 1
In this cases, (2.10) becomes respectively
∞ (−q 2;q2) k=0
2 1 1
(−1)k
1∞ qk k
=
. (q2;q2)k
F1 (z;q) = (−qn+1;q2 ) (−1)k
G1 (zq2 ;q2) 2
∞11
2
F−1(z;q)=(−q 2
k=0
(q2 ;q2 )k
and
∞k n+1 1 k q2
2;q2)∞ (−1) 1 1
k 2 G−1(zq2;q ).
2
k=0
(q2 ;q2 )k
Acknowledgement: The idea of using Euler’s formula to connect different kinds of q−Bessel functions is due to the late Professor Joaquin Bustoz, who sug- gested it to the author while he was advising his Ph.D. thesis.