Lerch zeta function

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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 the Czech mathematician Mathias Lerch [1].

DefinitionEdit

The Lerch zeta function is given by

 

A related function, the Lerch transcendent, is given by

 

The two are related, as

 

Integral representationsEdit

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 casesEdit

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

 

IdentitiesEdit

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 representationsEdit

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 expansionEdit

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]

SoftwareEdit

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

ReferencesEdit

  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.

External linksEdit