The Lerch transcendent is generalization of the Hurwitz zeta function and polylogarithm function. Many
sums of reciprocal powers can be expressed in terms of
it. It is classically defined by

sometimes also denoted , for (or and ) and , , , ..., is implemented in the Wolfram
Language as LerchPhi[z,
s, a]. Note that the two are identical only for .

A series formula for valid on a larger domain in the complex -plane is given by

(3)

which holds for all complex and complex with (Guillera and Sondow 2005).

Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. "The Function ." §1.11
in Higher
Transcendental Functions, Vol. 1. New York: Krieger, pp. 27-31,
1981.Gradshteyn, I. S. and Ryzhik, I. M. "The Lerch Function
."
§9.55 in Tables
of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press,
p. 1029, 2000.Guillera, J. and Sondow, J. "Double Integrals
and Infinite Products for Some Classical Constants Via Analytic Continuations of
Lerch's Transcendent." 16 June 2005 http://arxiv.org/abs/math.NT/0506319.Tyagi,
S. "Double Exponential Method for Riemann Zeta, Lerch and Dirichlet -Functions." https://arxiv.org/abs/2203.02509.
7 Mar 2022.