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Lerch zeta function

(Redirected from Lerch transcendant)

In mathematics, the Lerch zeta function is a special function that generalizes the Hurwitz zeta function and the polylogarithm. It is given by

$L(\lambda, \alpha, s) = \sum_{n=0}^\infty \frac { \exp (2\pi i\lambda n)} {(n+\alpha)^s}$

The Lerch zeta is related to the Lerch Transcendent, which is given by

$\Phi(z, s, \alpha) = \sum_{n=0}^\infty \frac { z^n} {(n+\alpha)^s}$

by

Φ(exp(2πiλ),s,α) = L(λ,α,s)

The Hurwitz zeta function is a special case, given by

ζ(s,α) = L(0,α,s) = Φ(1,s,α)

The polylogarithm is a special case of the Lerch Zeta, given by

Lis(x) = zΦ(z,s,1)

The Legendre chi function is a special case, given by

χn(z) = 2 - nzΦ(z2,n,1 / 2)