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

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.

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. Suppose with and . Then and .

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. [1]

A Taylor 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. doi:10.2140/pjm.1974.53.189.

If n 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

Asymptotic expansion

The polylogarithm function is defined as

Let

For and , an asymptotic expansion of for large and fixed and is given by

for , where is the Pochhammer symbol.[2]

Let

Let be its Taylor coefficients at . Then for fixed and ,

as .[3]

Software

The Lerch transcendent is implemented as LerchPhi in Maple and Mathematica, and as lerchphi in mpmath and SymPy.

References

  1. "The Analytic Continuation of the Lerch Transcendent and the Riemann Zeta Function". Retrieved 28 April 2020.
  2. Ferreira, Chelo; López, José L. (October 2004). "Asymptotic expansions of the Hurwitz–Lerch zeta function". Journal of Mathematical Analysis and Applications. 298 (1): 210–224. doi:10.1016/j.jmaa.2004.05.040.
  3. Cai, Xing Shi; López, José L. (10 June 2019). "A note on the asymptotic expansion of the Lerch's transcendent". Integral Transforms and Special Functions. 30 (10): 844–855. arXiv:1806.01122. doi:10.1080/10652469.2019.1627530. S2CID 119619877.
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