The Lerch Phi function is defined as follows:

This definition is valid for  or . By analytic continuation, it is extended to the whole complex -plane for each value of  and .

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

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

LerchPhi(1,a,v) = Zeta(0,a,v).  If , it is also true that limit(LerchPhi(z,a,v),z=1) = Zeta(0,a,v). If , this limit does not exist.

If , 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.

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