The Lerch Phi function is defined as follows:

This definition is valid for $\left|z\right|<1$ or $\left|z\right|=1\mathbf{and}1<\mathrm{\Re }\left(a\right)$. By analytic continuation, it is extended to the whole complex $z$-plane for each value of $a$ and $v$.

If $v$ and $a$ are positive integers, LerchPhi(z, a, v) has a branch cut in the $z$-plane along the real axis to the right of $z=1$, with a branch point at $z=1$.

If $a$ is a non-positive integer, LerchPhi(z, a, v) is a rational function of $z$ with a pole of order $1-a$ at $z=1$.

LerchPhi(1,a,v) = Zeta(0,a,v).  If $1<\mathrm{\Re }\left(a\right)$, it is also true that limit(LerchPhi(z,a,v),z=1) = Zeta(0,a,v). If $\mathrm{\Re }\left(a\right)\le 1$, this limit does not exist.

If $0\le \mathrm{\Re }\left(a\right)$ and $a$ is not an integer, LerchPhi(z, a, v) has an infinite singularity at each non-positive integer v.

If the coefficients of the series representation of a hypergeometric function are rational functions of the summation indices, then the hypergeometric function can be expressed as a linear sum of Lerch Phi functions.

If the parameters of the hypergeometric functions are rational, we can express the hypergeometric function as a linear sum of polylog functions.

$\mathrm{LerchPhi}\left(3\&comma;4\&comma;1\right)$

$\frac{\mathrm{polylog}\left(4\&comma;3\right)}{3}$

$\mathrm{LerchPhi}\left(0\&comma;7\&comma;4\right)$

$\frac{1}{16384}$

$\mathrm{LerchPhi}\left(4\&comma;0\&comma;3\right)$

$-\frac{1}{3}$

$\mathrm{LerchPhi}\left(z\&comma;a\&comma;1\right)$

$\frac{\mathrm{polylog}\left(a\&comma;z\right)}{z}$

$\mathrm{LerchPhi}\left(1\&comma;z\&comma;1\right)$

$\mathrm{\zeta }\left(z\right)$

$\mathrm{diff}\left(\mathrm{LerchPhi}\left(z\&comma;3\&comma;4\right)\&comma;z\right)$

$\frac{\mathrm{LerchPhi}\left(z\&comma;2\&comma;4\right)}{z}-\frac{4\mathrm{LerchPhi}\left(z\&comma;3\&comma;4\right)}{z}$

Erdelyi, A. Higher Transcendental Functions. McGraw-Hill, 1953.