Lerch zeta function

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 .

Definition

The Lerch zeta-function is given by

$L(\lambda, \alpha, s) = \sum_{n=0}^\infty \frac { \exp (2\pi i\lambda n)} {(n+\alpha)^s}.$

A related function, the Lerch transcendent, is given by

$\Phi(z, s, \alpha) = \sum_{n=0}^\infty \frac { z^n} {(n+\alpha)^s}.$

The two are related, as

$\,\Phi(\exp (2\pi i\lambda), s,\alpha)=L(\lambda, \alpha,s).$

Integral representations

An integral representation is given by

$\Phi(z,s,a)=\frac{1}{\Gamma(s)}\int_0^\infty \frac{t^{s-1}e^{-at}}{1-ze^{-t}}\,dt$

for

$\Re(a)>0\wedge\Re(s)>0\wedge z<1\vee\Re(a)>0\wedge\Re(s)>1\wedge z=1.$

A contour integral representation is given by

$\Phi(z,s,a)=-\frac{\Gamma(1-s)}{2\pi i}\int_0^{(+\infty)} \frac{(-t)^{s-1}e^{-at}}{1-ze^{-t}}\,dt$

for

$\Re(a)>0\wedge\Re(s)<0\wedge z<1$

where the contour must not enclose any of the points $t=\log(z)+2k\pi i,k\in Z.$

A Hermite-like integral representation is given by

$\Phi(z,s,a)= \frac{1}{2a^s}+ \int_0^\infty \frac{z^t}{(a+t)^s}\,dt+ \frac{2}{a^{s-1}} \int_0^\infty \frac{\sin(s\arctan(t)-ta\log(z))}{(1+t^2)^{s/2}(e^{2\pi at}-1)}\,dt$

for

$\Re(a)>0\wedge |z|<1$

and

$\Phi(z,s,a)=\frac{1}{2a^s}+ \frac{\log^{s-1}(1/z)}{z^a}\Gamma(1-s,a\log(1/z))+ \frac{2}{a^{s-1}} \int_0^\infty \frac{\sin(s\arctan(t)-ta\log(z))}{(1+t^2)^{s/2}(e^{2\pi at}-1)}\,dt$

for

$\Re(a)>0.$

Special cases

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

$\,\zeta(s,\alpha)=L(0, \alpha,s)=\Phi(1,s,\alpha).$

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

$\,\textrm{Li}_s(z)=z\Phi(z,s,1).$

The Legendre chi function is a special case, given by

$\,\chi_n(z)=2^{-n}z \Phi (z^2,n,1/2).$

The Riemann zeta-function is given by

$\,\zeta(s)=\Phi (1,s,1).$

The Dirichlet eta-function is given by

$\,\eta(s)=\Phi (-1,s,1).$

Identities

For λ rational, the summand is a root of unity, and thus $L(\lambda, \alpha, s)$ may be expressed as a finite sum over the Hurwitz zeta-function.

Various identities include:

$\Phi(z,s,a)=z^n \Phi(z,s,a+n) + \sum_{k=0}^{n-1} \frac {z^k}{(k+a)^s}$

and

$\Phi(z,s-1,a)=\left(a+z\frac{\partial}{\partial z}\right) \Phi(z,s,a)$

and

$\Phi(z,s+1,a)=-\,\frac{1}{s}\frac{\partial}{\partial a} \Phi(z,s,a).$

Series representations

A series representation for the Lerch transcendent is given by

$\Phi(z,s,q)=\frac{1}{1-z} \sum_{n=0}^\infty \left(\frac{-z}{1-z} \right)^n \sum_{k=0}^n (-1)^k \binom{n}{k} (q+k)^{-s}.$

(Note that $\tbinom{n}{k}$ 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

$|\log(z)|<2 \pi;s\neq 1,2,3,\dots; a\neq 0,-1,-2,\dots$

$\Phi(z,s,a)=z^{-a}\left[\Gamma(1-s)\left(-\log (z)\right)^{s-1} +\sum_{k=0}^\infty \zeta(s-k,a)\frac{\log^k (z)}{k!}\right]$

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

If s is a positive integer, then

$\Phi(z,n,a)=z^{-a}\left\{ \sum_{{k=0}\atop k\neq n-1}^ \infty \zeta(n-k,a)\frac{\log^k (z)}{k!} +\left[\psi(n)-\psi(a)-\log(-\log(z))\right]\frac{\log^{n-1}(z)}{(n-1)!}\right\},$

where $\psi(n)$ is the digamma function.

A Taylor series in the third variable is given by

$\Phi(z,s,a+x)=\sum_{k=0}^\infty \Phi(z,s+k,a)(s)_{k}\frac{(-x)^k}{k!};|x|<\Re(a),$

where $(s)_{k}$ is the Pochhammer symbol.

Series at a = -n is given by

$\Phi(z,s,a)=\sum_{k=0}^n \frac{z^k}{(a+k)^s} +z^n\sum_{m=0}^\infty (1-m-s)_{m}\operatorname{Li}_{s+m}(z)\frac{(a+n)^m}{m!};\ a\rightarrow-n$

A special case for n = 0 has the following series

$\Phi(z,s,a)=\frac{1}{a^s} +\sum_{m=0}^\infty (1-m-s)_m \operatorname{Li}_{s+m}(z)\frac{a^m}{m!}; |a|<1,$

where $\operatorname{Li}_s(z)$ is the polylogarithm.

An asymptotic series for $s\rightarrow-\infty$

$\Phi(z,s,a)=z^{-a}\Gamma(1-s)\sum_{k=-\infty}^\infty [2k\pi i-\log(z)]^{s-1}e^{2k\pi ai}$

for $|a|<1;\Re(s)<0 ;z\notin (-\infty,0)$ and

$\Phi(-z,s,a)=z^{-a}\Gamma(1-s)\sum_{k=-\infty}^\infty [(2k+1)\pi i-\log(z)]^{s-1}e^{(2k+1)\pi ai}$

for $|a|<1;\Re(s)<0 ;z\notin (0,\infty).$

An asymptotic series in the incomplete Gamma function

$\Phi(z,s,a)=\frac{1}{2a^s}+ \frac{1}{z^a}\sum_{k=1}^\infty \frac{e^{-2\pi i(k-1)a}\Gamma(1-s,a(-2\pi i(k-1)-\log(z)))} {(-2\pi i(k-1)-\log(z))^{1-s}}+ \frac{e^{2\pi ika}\Gamma(1-s,a(2\pi ik-\log(z)))}{(2\pi ik-\log(z))^{1-s}}$

for $|a|<1;\Re(s)<0.$

Software

The Lerch transcendent is implemented as LerchPhi in Maple.

References

• 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, Inc. ISBN 0-12-384933-0. LCCN 2014010276. ISBN 978-0-12-384933-5.
• 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 $\scriptstyle{\mathfrak K}(w,x,s) = \sum_{k=0}^\infty {e^{2k\pi ix} \over (w+k)^s}$", Acta Mathematica (in French) 11 (1–4): 19–24, doi:10.1007/BF02612318, JFM 19.0438.01, MR 1554747.

External links

• 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.