Page 27 - Textos de Matemática Vol. 34

CONNECTING BASIC BESSEL FUNCTIONS 17
To rewrite the last identity in terms of the q-Bessel functions, observe that from (2.1) and (2.2),
G (z;q2)= 1 1 J(3)(q−1z;q2) ν2−1ν2
and
Aν(q)(q 2z)ν
1 1 J(2)(2qp+j−ν z;q).
ν 2
F (qi+p z;q) =
2 i+p−ν ν 2 Aν(q )(2q 2 z)ν
This establishes the result.
Now the inverse problem will be studied. The function Jν

(z; q) will be written Theorem 2.2. For z ∈ C and ν > 0, the following relation holds
as a linear combination of Jν (z; q)’s.
ν(j+p−1) A (q2)
q 2 ν J(2)(z; q)=(−qν+1;−q; q)
∞
∞ (−1)p+j
q(ν+1)j+p j+p−1 J(3)(q 2
z; q2).
(3)
(2)
Aν(q) ν
Proof. Rewrite Fν (z; q) in the following way:
(q;q)p(q;q)j ν
=
∞ qn2 (−qν+1; −q; q)
n=0 (qν+1; q; q)n(−qν+1; −q; q)n
∞ Fν (z; q) =
qn2 (qν+1; q; q)n
(−1)n
p,j =0
z2n (−1)n n z2n
n=0
∞
q n 2 z 2 n
(q2ν+2; q2; q2)n(−qν+n+1; −qn+1; q)∞
(−1)n n=0
q n 2 z 2 n (q2ν+2; q2; q2)n
(−qν+1; −q; q)n.
=
Using (2.5) in this last expression, it becomes
∞
(2.8) (−1)n
(−qν+1; −q; q)∞.
n=0
Now, by Euler’s formula (2.4),
1
; −q
∞
(−1)p+j
q j (ν +1)+p+n(j +p) (q; q)p(q; q)j
= Substitution of this in (2.8) produces the identity
∞
(2.10) Fν (z; q) = (−qν+1; −q; q)∞ (−1)p+j
.
(2.9) ν+n+1 (−q
n+1
; q)∞
j,p=0
q j (ν +1)+p (q; q)p(q; q)j
j +p Gν (zq 2
; q2).
j,p=0
This, together with (2.1) and (2.2), establishes the result.