Lerch transcendent

The Lerch transcendent $\Phi$ is defined for $|z|<1$ and $a \in \mathbb{C} \setminus \{ 0,-1,-2,\ldots\}$ by $$\Phi(z,s,a)=\displaystyle\sum_{k=0}^{\infty} \dfrac{z^k}{(a+k)^s}.$$

Properties

Lerch transcendent polylogarithm
Relationship between Lerch transcendent and Lerch zeta
Dirichlet beta in terms of Lerch transcendent
Legendre chi in terms of Lerch transcendent
Li2(z)=zPhi(z,2,1)

References

• 1953: Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G. Tricomi: Higher Transcendental Functions Volume I ... (previous) ... (next): $\S 1.11 (1)$