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 .
The Lerch zeta function is given by
A related function, the Lerch transcendent, is given by
The two are related, as
An integral representation is given by
A contour integral representation is given by
where the contour must not enclose any of the points
A Hermite-like integral representation is given by
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
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:
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 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
An asymptotic series in the incomplete gamma function
The Lerch transcendent is implemented as LerchPhi in Maple.
- Apostol, T. M. (2010), "Lerch's Transcendent", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248.
- Bateman, H.; Erdélyi, A. (1953), Higher Transcendental Functions, Vol. I (PDF), New York: McGraw-Hill. (See § 1.11, "The function Ψ(z,s,v)", p. 27)
- Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) [October 2014]. "9.55.". In Zwillinger, Daniel; Moll, Victor Hugo. Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (8 ed.). Academic Press. ISBN 978-0-12-384933-5. LCCN 2014010276.
- Guillera, Jesus; Sondow, Jonathan (2008), "Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent", The Ramanujan Journal, 16 (3): 247–270, arXiv:math.NT/0506319 , doi:10.1007/s11139-007-9102-0, MR 2429900. (Includes various basic identities in the introduction.)
- Jackson, M. (1950), "On Lerch's transcendent and the basic bilateral hypergeometric series 2ψ2", J. London Math. Soc., 25 (3): 189–196, doi:10.1112/jlms/s1-25.3.189, MR 0036882.
- Laurinčikas, Antanas; Garunkštis, Ramūnas (2002), The Lerch zeta-function, Dordrecht: Kluwer Academic Publishers, ISBN 978-1-4020-1014-9, MR 1979048.
- Lerch, Mathias (1887), "Note sur la fonction ", Acta Mathematica (in French), 11 (1–4): 19–24, doi:10.1007/BF02612318, JFM 19.0438.01, MR 1554747.
- Aksenov, Sergej V.; Jentschura, Ulrich D. (2002), C and Mathematica Programs for Calculation of Lerch's Transcendent.
- Ramunas Garunkstis, Home Page (2005) (Provides numerous references and preprints.)
- Ramunas Garunkstis, Approximation of the Lerch Zeta Function (PDF)
- S. Kanemitsu, Y. Tanigawa and H. Tsukada, A generalization of Bochner's formula, (undated, 2005 or earlier)
- Weisstein, Eric W. "Lerch Transcendent". MathWorld.
- "§25.14, Lerch's Transcendent". NIST Digital Library of Mathematical Functions. National Institute of Standards and Technology. 2010. Retrieved 28 January 2012.