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{\displaystyle x=0}

${\displaystyle x=0}$

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Taylor coefficient

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Wikitext context	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]] [http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Lerch.html]. ==Definition== The Lerch zeta function is given by :<math>L(\lambda, \alpha, s) = \sum_{n=0}^\infty \frac { e^{2\pi i\lambda n}} {(n+\alpha)^s}.</math> A related function, the '''Lerch transcendent''', is given by :<math>\Phi(z, s, \alpha) = \sum_{n=0}^\infty \frac { z^n} {(n+\alpha)^s}.</math> The two are related, as :<math>\,\Phi(e^{2\pi i\lambda}, s,\alpha)=L(\lambda, \alpha,s).</math> ==Integral representations== An integral representation is given by :<math> \Phi(z,s,a)=\frac{1}{\Gamma(s)}\int_0^\infty \frac{t^{s-1}e^{-at}}{1-ze^{-t}}\,dt</math> for :<math>\Re(a)>0\wedge\Re(s)>0\wedge z<1\vee\Re(a)>0\wedge\Re(s)>1\wedge z=1. </math> A [[contour integral]] representation is given by :<math> \Phi(z,s,a)=-\frac{\Gamma(1-s)}{2\pi i}\int_0^{(+\infty)} \frac{(-t)^{s-1}e^{-at}}{1-ze^{-t}}\,dt</math> for :<math>\Re(a)>0\wedge\Re(s)<0\wedge z<1</math> where the contour must not enclose any of the points <math>t=\log(z)+2k\pi i,k\in Z.</math> A Hermite-like integral representation is given by :<math> \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 </math> for :<math>\Re(a)>0\wedge \|z\|<1 </math> and :<math> \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 </math> for :<math>\Re(a)>0. </math> Similar representations include :<math> \Phi(z,s,a)= \frac{1}{2a^s} + \int_{0}^{\infty}\frac{\cos(t\log z)\sin\Big(s\arctan\tfrac{t}{a}\Big) - \sin(t\log z)\cos\Big(s\arctan\tfrac{t}{a}\Big)}{\big(a^2 + t^2\big)^{\frac{s}{2}} \tanh\pi t }\,dt, </math> and :<math> \Phi(-z,s,a)= \frac{1}{2a^s} + \int_{0}^{\infty}\frac{\cos(t\log z)\sin\Big(s\arctan\tfrac{t}{a}\Big) - \sin(t\log z)\cos\Big(s\arctan\tfrac{t}{a}\Big)}{\big(a^2 + t^2\big)^{\frac{s}{2}} \sinh\pi t }\,dt, </math> holding for positive ''z'' (and more generally wherever the integrals converge). Furthermore, :<math> \Phi(e^{i\varphi},s,a)=L\big(\tfrac{\varphi}{2\pi},a,s\big)= \frac{1}{a^s} + \frac{1}{2\Gamma(s)}\int_{0}^{\infty}\frac{t^{s-1}e^{-at}\big(e^{i\varphi}-e^{-t}\big)}{\cosh{t}-\cos{\varphi}}\,dt, </math> The last formula is also known as ''Lipschitz formula''. ==Special cases== The [[Hurwitz zeta function]] is a special case, given by :<math>\,\zeta(s,\alpha)=L(0, \alpha,s)=\Phi(1,s,\alpha).</math> The [[polylogarithm]] is a special case of the Lerch Zeta, given by :<math>\,\textrm{Li}_s(z)=z\Phi(z,s,1).</math> The [[Legendre chi function]] is a special case, given by :<math>\,\chi_n(z)=2^{-n}z \Phi (z^2,n,1/2).</math> The [[Riemann zeta function]] is given by :<math>\,\zeta(s)=\Phi (1,s,1).</math> The [[Dirichlet eta function]] is given by :<math>\,\eta(s)=\Phi (-1,s,1).</math> ==Identities== For λ rational, the summand is a [[root of unity]], and thus <math>L(\lambda, \alpha, s)</math> may be expressed as a finite sum over the Hurwitz zeta-function. Suppose <math>\lambda = \frac{p}{q}</math> with <math>p, q \in \mathbb{Z}</math> and <math>q > 0</math>. Then <math>z = \omega = e^{2 \pi i \frac{p}{q}}</math> and <math>\omega^q = 1</math>. :<math>\Phi(\omega, s, \alpha) = \sum_{n=0}^\infty \frac {\omega^n} {(n+\alpha)^s} = \sum_{m=0}^{q-1} \sum_{n=0}^\infty \frac {\omega^{qn + m}}{(qn + m + \alpha)^s} = \sum_{m=0}^{q-1} \omega^m q^{-s} \zeta(s,\frac{m + \alpha}{q})</math> Various identities include: :<math>\Phi(z,s,a)=z^n \Phi(z,s,a+n) + \sum_{k=0}^{n-1} \frac {z^k}{(k+a)^s}</math> and :<math>\Phi(z,s-1,a)=\left(a+z\frac{\partial}{\partial z}\right) \Phi(z,s,a)</math> and :<math>\Phi(z,s+1,a)=-\,\frac{1}{s}\frac{\partial}{\partial a} \Phi(z,s,a).</math> ==Series representations== A series representation for the Lerch transcendent is given by :<math>\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}.</math> (Note that <math>\tbinom{n}{k}</math> 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. <ref name="benorin"> {{cite web \| url=https://www.physicsforums.com/insights/the-analytic-continuation-of-the-lerch-and-the-zeta-functions/ \| title=The Analytic Continuation of the Lerch Transcendent and the Riemann Zeta Function \| access-date=28 April 2020 }} </ref> A [[Taylor series]] in the first parameter was given by [[Arthur Erdélyi\|Erdélyi]]. It may be written as the following series, which is valid for :<math>\|\log(z)\|<2 \pi;s\neq 1,2,3,\dots; a\neq 0,-1,-2,\dots</math> :<math> \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] </math> {{cite journal\|author=B. R. Johnson\|title=Generalized Lerch zeta-function\|journal=Pacific J. Math.\|volume=53\|number=1\|date=1974\|pages=189–193\|doi=10.2140/pjm.1974.53.189\|doi-access=free}} If n is a positive integer, then :<math> \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\}, </math> where <math>\psi(n)</math> is the [[digamma function]]. A [[Taylor series]] in the third variable is given by :<math> \Phi(z,s,a+x)=\sum_{k=0}^\infty \Phi(z,s+k,a)(s)_{k}\frac{(-x)^k}{k!};\|x\|<\Re(a), </math> where <math>(s)_{k}</math> is the [[Pochhammer symbol]]. Series at ''a'' = -''n'' is given by :<math> \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 </math> A special case for ''n'' = 0 has the following series :<math> \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, </math> where <math>\operatorname{Li}_s(z)</math> is the [[polylogarithm]]. An [[asymptotic series]] for <math>s\rightarrow-\infty</math> :<math> \Phi(z,s,a)=z^{-a}\Gamma(1-s)\sum_{k=-\infty}^\infty [2k\pi i-\log(z)]^{s-1}e^{2k\pi ai} </math> for <math>\|a\|<1;\Re(s)<0 ;z\notin (-\infty,0) </math> and :<math> \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} </math> for <math>\|a\|<1;\Re(s)<0 ;z\notin (0,\infty). </math> An asymptotic series in the [[incomplete gamma function]] :<math> \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}} </math> for <math>\|a\|<1;\Re(s)<0.</math> == Asymptotic expansion == The polylogarithm function <math>\mathrm{Li}_n(z)</math> is defined as :<math> \mathrm{Li}_0(z)=\frac{z}{1-z}, \qquad \mathrm{Li}_{-n}(z)=z \frac{d}{dz} \mathrm{Li}_{1-n}(z). </math> Let :<math> \Omega_{a} \equiv \begin{cases} \mathbb{C}\setminus[1,\infty) & \text{if } \Re a > 0, \\ {z \in \mathbb{C}, \|z\|<1} & \text{if } \Re a \le 0. \end{cases} </math> For <math>\|\mathrm{Arg}(a)\|<\pi, s \in \mathbb{C}</math> and <math>z \in \Omega_{a}</math>, an asymptotic expansion of <math>\Phi(z,s,a)</math> for large <math>a</math> and fixed <math>s</math> and <math>z</math> is given by :<math> \Phi(z,s,a) = \frac{1}{1-z} \frac{1}{a^{s}} + \sum_{n=1}^{N-1} \frac{(-1)^{n} \mathrm{Li}_{-n}(z)}{n!} \frac{(s)_{n}}{a^{n+s}} +O(a^{-N-s}) </math> for <math>N \in \mathbb{N}</math>, where <math>(s)_n = s (s+1)\cdots (s+n-1)</math> is the [[Falling and rising factorials\|Pochhammer symbol]].<ref>{{cite journal \|last1=Ferreira \|first1=Chelo \|last2=López \|first2=José L. \|title=Asymptotic expansions of the Hurwitz–Lerch zeta function \|journal=Journal of Mathematical Analysis and Applications \|date=October 2004 \|volume=298 \|issue=1 \|pages=210–224 \|doi=10.1016/j.jmaa.2004.05.040}}</ref> Let :<math> f(z,x,a) \equiv \frac{1-(z e^{-x})^{1-a}}{1-z e^{-x}}. </math> Let <math>C_{n}(z,a)</math> be its Taylor coefficients at <math>x=0</math>. Then for fixed <math>N \in \mathbb{N}, \Re a > 1</math> and <math>\Re s > 0</math>, :<math> \Phi(z,s,a) - \frac{\mathrm{Li}_{s}(z)}{z^{a}} = \sum_{n=0}^{N-1} C_{n}(z,a) \frac{(s)_{n}}{a^{n+s}} + O\left( (\Re a)^{1-N-s}+a z^{-\Re a} \right), </math> as <math>\Re a \to \infty</math>.<ref>{{cite journal \|last1=Cai \|first1=Xing Shi \|last2=López \|first2=José L. \|title=A note on the asymptotic expansion of the Lerch's transcendent \|journal=Integral Transforms and Special Functions \|date=10 June 2019 \|volume=30 \|issue=10 \|pages=844–855 \|doi=10.1080/10652469.2019.1627530\|arxiv=1806.01122 \|s2cid=119619877 }}</ref> ==Software== The Lerch transcendent is implemented as LerchPhi in [http://www.maplesoft.com/support/help/Maple/view.aspx?path=LerchPhi Maple] and [https://functions.wolfram.com/ZetaFunctionsandPolylogarithms/LerchPhi/ Mathematica], and as lerchphi in [http://mpmath.org/doc/current/functions/zeta.html#lerch-transcendent mpmath] and [https://docs.sympy.org/latest/modules/functions/special.html#sympy.functions.special.zeta_functions.lerchphi SymPy]. ==References== {{Reflist}} * {{dlmf \| id= 25.14 \| first= T. M. \| last= Apostol \| title= Lerch's Transcendent}}. * {{citation \| first1= H. \| last1= Bateman \| author1-link= Harry Bateman \| first2= A. \| last2= Erdélyi \| author2-link= Arthur Erdélyi \| title= Higher Transcendental Functions, Vol. I \| year= 1953 \| location= New York \| publisher= McGraw-Hill \| url=http://apps.nrbook.com/bateman/Vol1.pdf}}. (See § 1.11, "The function Ψ(''z'',''s'',''v'')", p. 27) * {{cite book \|author-first1=Izrail Solomonovich \|author-last1=Gradshteyn \|author-link1=Izrail Solomonovich Gradshteyn \|author-first2=Iosif Moiseevich \|author-last2=Ryzhik \|author-link2=Iosif Moiseevich Ryzhik \|author-first3=Yuri Veniaminovich \|author-last3=Geronimus \|author-link3=Yuri Veniaminovich Geronimus \|author-first4=Michail Yulyevich \|author-last4=Tseytlin \|author-link4=Michail Yulyevich Tseytlin \|author-first5=Alan \|author-last5=Jeffrey \|editor-first1=Daniel \|editor-last1=Zwillinger \|editor-first2=Victor Hugo \|editor-last2=Moll \|translator=Scripta Technica, Inc. \|title=Table of Integrals, Series, and Products \|publisher=Academic Press \|date=2015 \|orig-year=October 2014 \|edition=8 \|language=English \|isbn=978-0-12-384933-5 \|lccn=2014010276 <!-- \|url=https://books.google.com/books?id=NjnLAwAAQBAJ \|access-date=2016-02-21-->\|title-link=Gradshteyn and Ryzhik \|chapter=9.55.}} * {{citation \| first1= Jesus \| last1= Guillera \| first2= Jonathan \| last2= Sondow \| arxiv= math.NT/0506319 \| mr = 2429900 \| title= Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent \| journal= The Ramanujan Journal \| volume= 16 \| year= 2008 \| pages= 247–270 \| issue= 3 \| doi= 10.1007/s11139-007-9102-0\| s2cid= 119131640 }}. (Includes various basic identities in the introduction.) * {{citation \| first= M. \| last= Jackson \| title= On Lerch's transcendent and the basic bilateral hypergeometric series <sub>2</sub>''ψ''<sub>2</sub> \| year= 1950 \| journal= J. London Math. Soc. \| volume= 25 \| issue= 3 \| pages= 189–196 \| doi= 10.1112/jlms/s1-25.3.189 \| mr= 0036882}}. * {{citation \| first1= F. \| last1= Johansson \| first2= Ia. \| last2= Blagouchine \| arxiv= 1804.01679 \| mr = 3925487 \| title= Computing Stieltjes constants using complex integration \| journal= Mathematics of Computation \| volume= 88 \| year= 2019 \| pages= 1829–1850 \| issue= 318 \| doi= 10.1090/mcom/3401\| s2cid= 4619883 }}. * {{citation \| first1= Antanas \| last1= Laurinčikas \| first2= Ramūnas \| last2= Garunkštis \| title= The Lerch zeta-function \| publisher= Kluwer Academic Publishers \| location= Dordrecht \| year= 2002 \| isbn= 978-1-4020-1014-9 \| mr= 1979048}}. * {{citation \| first= Mathias \| last= Lerch \| authorlink= Mathias Lerch \| title= Note sur la fonction <math>\scriptstyle{\mathfrak K}(w,x,s) = \sum_{k=0}^\infty {e^{2k\pi ix} \over (w+k)^s}</math> \| language= French \| year= 1887 \| journal= Acta Mathematica \| volume= 11 \| issue= 1–4 \| pages= 19–24 \| doi= 10.1007/BF02612318 \| mr= 1554747 \| jfm= 19.0438.01\| s2cid= 121885446 \| url= https://zenodo.org/record/1681743 \| doi-access= free }}. ==External links== * {{citation \| first1= Sergej V. \| last1= Aksenov \| first2= Ulrich D. \| last2= Jentschura \| year=2002 \| url= http://aksenov.freeshell.org/lerchphi.html \| title= C and Mathematica Programs for Calculation of Lerch's Transcendent}}. * Ramunas Garunkstis, ''[http://www.mif.vu.lt/~garunkstis Home Page]'' (2005) ''(Provides numerous references and preprints.)'' * Ramunas Garunkstis, ''[http://www.mif.vu.lt/~garunkstis/preprintai/approx.pdf Approximation of the Lerch Zeta Function]'' (PDF) * S. Kanemitsu, Y. Tanigawa and H. Tsukada, ''[http://www.iisc.ernet.in/nias/HRJ/vol27/Ktt.pdf A generalization of Bochner's formula]{{dead link\|date=December 2017 \|bot=InternetArchiveBot \|fix-attempted=yes }}'', (undated, 2005 or earlier) * {{MathWorld \| urlname= LerchTranscendent \| title= Lerch Transcendent}} * {{cite web \| url= http://dlmf.nist.gov/25.14 \|title= §25.14, Lerch's Transcendent \|date= 2010 \|work= NIST Digital Library of Mathematical Functions \|publisher= National Institute of Standards and Technology \|accessdate= 28 January 2012 }} [[Category:Zeta and L-functions]]
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