Legendre chi in terms of Lerch transcendent

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Theorem

The following formula holds: $$\chi_{\nu}(z)=2^{-\nu} z \Phi \left(z^2,\nu,\dfrac{1}{2} \right),$$ where $\chi$ denotes Legendre chi and $\Phi$ denotes the Lerch transcendent.

Proof

References