general Lerch Phi function
LerchPhi(z, a, v)
The Lerch Phi function is defined as follows:
This definition is valid for z<1 or z=1and1<ℜ⁡a. 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<ℜ⁡a, it is also true that limit(LerchPhi(z,a,v),z=1) = Zeta(0,a,v). If ℜ⁡a≤1, this limit does not exist.
If 0≤ℜ⁡a, 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.
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