Lerch zeta function

In mathematics, the Lerch zeta-function, sometimes called the HurwitzLerch zeta-function, is a special function that generalizes the Hurwitz zeta-function and the polylogarithm. It is named after the Czech mathematician Mathias Lerch .

Definition

The Lerch zeta-function is given by

A related function, the Lerch transcendent, is given by

The two are related, as

Integral representations

An integral representation is given by

for

A contour integral representation is given by

for

where the contour must not enclose any of the points

A Hermite-like integral representation is given by

for

and

for

Special cases

The Hurwitz zeta-function is a special case, given by

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

The Legendre chi function is a special case, given by

The Riemann zeta-function is given by

The Dirichlet eta-function is given by

Identities

For λ rational, the summand is a root of unity, and thus may be expressed as a finite sum over the Hurwitz zeta-function.

Various identities include:

and

and

Series representations

A series representation for the Lerch transcendent is given by

(Note that is a binomial coefficient.)

The series is valid for all s, and for complex z with Re(z)<1/2. Note a general resemblance to a similar series representation for the Hurwitz zeta function.

A Taylor's series in the first parameter was given by Erdélyi. It may be written as the following series, which is valid for

B. R. Johnson (1974). "Generalized Lerch zeta-function". Pacific J. Math. 53 (1): 189–193. 

If s is a positive integer, then

where is the digamma function.

A Taylor series in the third variable is given by

where is the Pochhammer symbol.

Series at a = -n is given by

A special case for n = 0 has the following series

where is the polylogarithm.

An asymptotic series for

for and

for

An asymptotic series in the incomplete Gamma function

for

Software

The Lerch transcendent is implemented as LerchPhi in Maple.

References

External links

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