What is the Heine Transformation Formula for q-hypergeometric series?

[alyafey22]

Prove the following

\(\displaystyle {}_2\phi_1 (a,b;c;q,z) =\frac{(b;q)_\infty (az;q)_\infty}{(c;q)_\infty (z;q)_\infty} {}_2\phi_1 \left(\frac{c}{b},z;az;q,b \right)\)​

 

[Aaron Bex]

Proof:
Let
{}_2\phi_1 (a,b;c;q,z) = \sum_{n=0}^{\infty} \frac{(a;q)_n (b;q)_n}{(q;q)_n (c;q)_n} (z;q)^n

and

{}_2\phi_1 \left(\frac{c}{b},z;az;q,b \right)= \sum_{n=0}^{\infty} \frac{(\frac{c}{b};q)_n (z;q)_n}{(q;q)_n (az;q)_n} (b;q)^n

Multiply both sides by the common denominator and apply the q-Pochhammer identity:

\frac{(b;q)_\infty (az;q)_\infty}{(c;q)_\infty (z;q)_\infty} \sum_{n=0}^{\infty} \frac{(\frac{c}{b};q)_n (z;q)_n (b;q)_n}{(q;q)_n (az;q)_n} (b;q)^n