Z/(1-sqrt(q))2Phi1(q,sqrt(q);sqrt(q^3);z)=Sum z^k/(1-q^(k-1/2))

Theorem

The following formula holds: $$\dfrac{z}{1-\sqrt{q}} {}_2\phi_1 \left(q,\sqrt{q};\sqrt{q^3};z \right) = \displaystyle\sum_{k=1}^{\infty} \dfrac{z^k}{1-q^{k-\frac{1}{2}}},$$ where ${}_2\phi_1$ denotes basic hypergeometric phi.

References

• 1953: Harry Bateman: Higher Transcendental Functions Volume I ... (previous) ... (next): $4.8 (8)$