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

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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.

Proof

References