Editing Lerch transcendent

{{Short description|Special mathematical function}}

In [[mathematics]], the '''Lerch transcendent''', is a [[special function]] that generalizes the [[Hurwitz zeta function]] and the [[polylogarithm]].  It is named after Czech mathematician [[Mathias Lerch]], who published a paper about a similar function in 1887.<ref>{{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 }}</ref> The Lerch transcendent, is given by:

:<math>\Phi(z, s, \alpha) = \sum_{n=0}^\infty
\frac { z^n} {(n+\alpha)^s}</math>.

It only converges for any real number <math>\alpha > 0</math>, where <math>|z| < 1</math>, or <math>\mathfrak{R}(s) > 1</math>, and <math>|z| = 1</math>.{{sfn|Guillera|Sondow|2008}}

==Special cases==
The Lerch transcendent is related to and generalizes various special functions.

The '''Lerch zeta function''' is given by:

:<math>L(\lambda, s, \alpha) = \sum_{n=0}^\infty
\frac { e^{2\pi i\lambda n}} {(n+\alpha)^s}=\Phi(e^{2\pi i\lambda}, s,\alpha)</math>

The [[Hurwitz zeta function]] is the special case<ref name="guillera-sondow-248-249">{{harvnb|Guillera|Sondow|2008|p=248–249}}</ref>
:<math>\zeta(s,\alpha) = \sum_{n=0}^\infty \frac{1}{(n+\alpha)^s} = \Phi(1,s,\alpha)</math>
The [[polylogarithm]] is another special case:<ref name="guillera-sondow-248-249" />
:<math>\textrm{Li}_s(z) = \sum_{n=1}^\infty \frac{z^n}{n^s} =z\Phi(z,s,1)</math>
The [[Riemann zeta function]] is a special case of both of the above:<ref name="guillera-sondow-248-249" />
:<math>\zeta(s) =\sum_{n=1}^\infty \frac{1}{n^s} = \Phi(1,s,1)</math>
The [[Dirichlet eta function]]:<ref name="guillera-sondow-248-249" />

:<math>\eta(s) = \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^s} = \Phi(-1,s,1)</math>

The [[Dirichlet beta function]]:<ref name="guillera-sondow-248-249" />

:<math>\beta(s) = \sum_{k=0}^\infty \frac{(-1)^{k}}{(2k+1)^s} = 2^{-s}\Phi(-1,s,\tfrac12)</math>

The [[Legendre chi function]]:<ref name="guillera-sondow-248-249" />

:<math>\chi_s(z) = \sum_{k=0}^\infty \frac{z^{2k+1}}{(2k+1)^s}= \frac {z}{2^s} \Phi(z^2,s,\tfrac12)</math>

The [[inverse tangent integral]]:<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Inverse Tangent Integral |url=https://mathworld.wolfram.com/InverseTangentIntegral.html |access-date=2024-10-13 |website=mathworld.wolfram.com |language=en}}</ref>

:<math>\textrm{Ti}_s(z)= \sum_{k=0}^\infty \frac{(-1)^k z^{2k+1}}{(2k+1)^s}=\frac{z}{2^s}\Phi(-z^2,s,\tfrac12) </math>

The [[polygamma function|polygamma functions]] for positive integers ''n'':<ref>The polygamma function has the series representation

<math>\psi^{(m)}(z) = (-1)^{m+1}\, m! \sum_{k=0}^\infty \frac{1}{(z+k)^{m+1}}</math>

which holds for integer values of {{math|''m'' > 0}} and any complex ''{{mvar|z}}'' not equal to a negative integer.</ref><ref>{{Cite web |last=Weisstein |first=Eric W. |title=Polygamma Function |url=https://mathworld.wolfram.com/PolygammaFunction.html |access-date=2024-10-14 |website=mathworld.wolfram.com |language=en}}</ref>

:<math>\psi^{(n)}(\alpha)= (-1)^{n+1} n!\Phi (1,n+1,\alpha)</math>

The [[Clausen function]]:<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Clausen Function |url=https://mathworld.wolfram.com/ClausenFunction.html |access-date=2024-10-14 |website=mathworld.wolfram.com |language=en}}</ref>

:<math>\text{Cl}_2(z)= \frac{ie^{-iz}}2 \Phi(e^{-iz},2,1)-\frac{ie^{iz}}2 \Phi(e^{iz},2,1)</math>

==Integral representations==

The Lerch transcendent has an integral representation:
:<math>
\Phi(z,s,a)=\frac{1}{\Gamma(s)}\int_0^\infty
\frac{t^{s-1}e^{-at}}{1-ze^{-t}}\,dt</math>
The proof is based on using the integral definition of the [[gamma function]] to write
:<math>\Phi(z,s,a)\Gamma(s)
= \sum_{n=0}^\infty \frac{z^n}{(n+a)^s} \int_0^\infty x^s e^{-x} \frac{dx}{x}
= \sum_{n=0}^\infty \int_0^\infty t^s z^n e^{-(n+a)t} \frac{dt}{t}</math>
and then interchanging the sum and integral. The resulting integral representation converges for <math>z \in \Complex \setminus [1,\infty),</math> Re(''s'') > 0, and Re(''a'') > 0. This [[analytic continuation|analytically continues]] <math>\Phi(z,s,a)</math> to ''z'' outside the [[unit disk]]. The integral formula also holds if ''z'' = 1, Re(''s'') > 1, and Re(''a'') > 0; see [[Hurwitz zeta function]].<ref>{{harvnb|Bateman|Erdélyi|1953|p=27}}</ref><ref>{{harvnb|Guillera|Sondow|2008|loc=Lemma 2.1 and 2.2}}</ref>

A [[contour integral]] representation is given by
:<math>
\Phi(z,s,a)=-\frac{\Gamma(1-s)}{2\pi i} \int_C \frac{(-t)^{s-1}e^{-at}}{1-ze^{-t}}\,dt</math>
where ''C'' is a [[Hankel contour]] counterclockwise around the positive real axis, not enclosing any of the points <math>t = \log(z) + 2k\pi i</math> (for integer ''k'') which are [[Pole (complex analysis)|poles]] of the integrand. The integral assumes Re(''a'') > 0.<ref>{{harvnb|Bateman|Erdélyi|1953|p=28}}</ref>

===Other integral representations===

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}, s, a\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''.

==Identities==
For λ rational, the summand is a [[root of unity]], and thus  <math>L(\lambda, s, \alpha)</math> may be expressed as a finite sum over the Hurwitz zeta function.  Suppose <math display="inline">\lambda = \frac{p}{q}</math> with <math>p, q \in \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 \left( s,\frac{m + \alpha}{q} \right) </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'')&lt;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 | date=27 April 2020 | access-date=28 April 2020}}</ref>

A [[Taylor series]] in the first parameter was given by [[Arthur Erdélyi]]. It may be written as the following series, which is valid for<ref>{{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}}</ref>
:<math>\left|\log(z)\right| < 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>

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>

The representation as a [[generalized hypergeometric function]] is<ref>{{cite journal|first1=J. E.|last1=Gottschalk|first2=E. N. |last2=Maslen| title=Reduction formulae for generalized [[hypergeometric function]]s of one variable|journal=J. Phys. A | year=1988| volume=21|issue=9|pages=1983–1998|doi=10.1088/0305-4470/21/9/015|bibcode=1988JPhA...21.1983G}}</ref>
:<math>
\Phi(z,s,\alpha)=\frac{1}{\alpha^s}{}_{s+1}F_s\left(\begin{array}{c}
1,\alpha,\alpha,\alpha,\cdots\\
1+\alpha,1+\alpha,1+\alpha,\cdots\\
\end{array}\mid z\right).
</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|doi-access=free }}</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.&nbsp;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 |editor-link2=Victor Hugo 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}}.

==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.)''
* {{cite journal|first1=Ramunas|last1=Garunkstis|url=http://www.mif.vu.lt/~garunkstis/preprintai/approx.pdf |title=Approximation of the Lerch Zeta Function|year=2004|journal=Lithuanian Mathematical Journal|volume=44|number=2|pages=140–144|doi=10.1023/B:LIMA.0000033779.41365.a5|s2cid=123059665 }}
* {{cite web|first1=S.|last1=Kanemitsu|first2=Y.|last2=Tanigawa|first3=H.|last3=Tsukada|url=https://hal.archives-ouvertes.fr/hal-02220916|title=A generalization of Bochner's formula|year=2015}} {{cite journal|first1=S.|last1=Kanemitsu|first2=Y.|last2=Tanigawa|first3=H.|last3=Tsukada|doi=10.46298/hrj.2004.150|journal=Hardy-Ramanujan Journal|title=A generalization of Bochner's formula|year=2004|volume=27|doi-access=free}}
* {{MathWorld | urlname= LerchTranscendent | title= Lerch Transcendent}}
* {{dlmf|id=25.14 |title=Lerch's Transcendent}}

[[Category:Zeta and L-functions]]