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Concept
Lerch zeta function
Summary
In mathematics, the Lerch zeta function, sometimes called the Hurwitz–Lerch zeta function, is a special function that generalizes the Hurwitz zeta function and the polylogarithm. It is named after Czech mathematician Mathias Lerch, who published a paper about the function in 1887.
The Lerch zeta function is given by
A related function, the Lerch transcendent, is given by
The transcendent only converges for any real number , where:
or
and .
The two are related, as
The Lerch transcendent has an integral representation:
The proof is based on using the integral definition of the Gamma function to write
and then interchanging the sum and integral. The resulting integral representation converges for Re(s) > 0, and Re(a) > 0. This analytically continues to z outside the unit disk. The integral formula also holds if z = 1, Re(s) > 1, and Re(a) > 0; see Hurwitz zeta function.
A contour integral representation is given by
where C is a Hankel contour counterclockwise around the positive real axis, not enclosing any of the points (for integer k) which are poles of the integrand. The integral assumes Re(a) > 0.
A Hermite-like integral representation is given by
for
and
for
Similar representations include
and
holding for positive z (and more generally wherever the integrals converge). Furthermore,
The last formula is also known as Lipschitz formula.
The Lerch zeta function and Lerch transcendent generalize various special functions.
The Hurwitz zeta function is the special case
The polylogarithm is another special case:
The Riemann zeta function is a special case of both of the above:
Other special cases include:
The Dirichlet eta function:
The Dirichlet beta function:
The Legendre chi function:
The polygamma function:
For λ rational, the summand is a root of unity, and thus may be expressed as a finite sum over the Hurwitz zeta function. Suppose with and . Then and .
Various identities include:
and
and
A series representation for the Lerch transcendent is given by
(Note that is a binomial coefficient.
Official source
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