Page 26 - Textos de Matemática Vol. 34

16 LU´IS DANIEL DE ABREU
respectively. The critical difference is the presence of the qn2 in the numerator of n2
Fν(z;q) while in Gν(z;q) there is a q 2 . The main instrument to be used in the proofs is the Euler formulae:
(2.3)
and (2.4)
n=0
(q;q) z,z∈C n
(−z;q)∞=
∞ q n(n−1) 2n
1 ∞ zn
(z;q) = (q;q) ,|z|<1.
∞ n=0 n
Theorem 2.1. For z ∈ C and ν > 0 the following connection formula holds:
∞ p(p−1)+j (j −1)+ν 2 −ν (i+p)
(2.5) 1 = (−qν+n+1;−qn+1;q)∞ (−qν+1; −q; q)n (−qν+1; −q; q)∞
ν A (q) 1 q 2 ν− ν (3)− 2
+(ν+1)j p+j−ν
z;q).
2 q 2 2 Jν (q 2z;q )= Aν(q ) p,j=0
Jν (2q 2
(2)
(q;q)p(q;q)j Proof. From the definitions (1.4) and (1.5) the relation
follows. This allows to rewrite Gν (z; q2) as follows
∞ Gν(z;q2) =
n=0
(−1)n
qn2 (q2ν+2; q2; q2)n
z2n
∞
= (−1)n
qn2
z2n
(2.6) =
Now, from Euler’s formula (2.3),
(−1)n ∞
n=0 1
(qν+1; q; q)n(−qν+1; −q; q)n
∞ (−qν+1;−q;q)∞ n=0
qn2 (−qν+n+1; −qn+1; q) (qν+1;q;q)n
∞ z2n.
ν +n+1
1 Gν(z;q ) = (−qν+1;−q;q)
(ν +n+1)j +(n+1)p
(2.7) (−q
Substituting this on (2.6) produces the identity
;−q
(q;q) (q;q) q p j
.
n+1 ;q)∞ =
q p(p−1)+j(j−1) 2
2
j,p=0
∞

∞ j,p=0
q p(p−1)+j(j−1) 2
(q;q) (q;q) q p j
(ν+1)j+p
j+p
Fν(q 2 z;q).