(z/(1-q))2Phi1(q,q;q^2;z)=Sum z^k/(1-q^k)

Theorem

The following formula holds: $$\dfrac{z}{1-z} {}_2\phi_1(q,q;q^2;z) = \displaystyle\sum_{k=1}^{\infty} \dfrac{z^k}{1-q^k},$$ where ${}_2\phi_1$ denotes basic hypergeometric phi.

References

• 1953: Harry Bateman: Higher Transcendental Functions Volume I ... (previous) ... (next): $4.8 (6)$ (typo in text: text has sum beginning at $k=0$)