Lerch transcendent

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

The Lerch zeta function is given by:

:$L(\backslash lambda,\; s,\; \backslash alpha)\; =\; \backslash sum\_\{n=0\}^\backslash infty$

\frac { e^{2\pi i\lambda n}} {(n+\alpha)^s}=\Phi(e^{2\pi i\lambda}, s,\alpha)

The Hurwitz zeta function is the special case{{harvnb|Guillera|Sondow|2008|p=248–249}}

:$\backslash zeta(s,\backslash alpha)\; =\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{1\}\{(n+\backslash alpha)^s\}\; =\; \backslash Phi(1,s,\backslash alpha)$

The polylogarithm is another special case:

:$\backslash textrm\{Li\}\_s(z)\; =\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{z^n\}\{n^s\}\; =z\backslash Phi(z,s,1)$

The Riemann zeta function is a special case of both of the above:

:$\backslash zeta(s)\; =\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{1\}\{n^s\}\; =\; \backslash Phi(1,s,1)$

The Dirichlet eta function:

:$\backslash eta(s)\; =\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{(-1)^\{n-1\}\}\{n^s\}\; =\; \backslash Phi(-1,s,1)$

The Dirichlet beta function:

:$\backslash beta(s)\; =\; \backslash sum\_\{k=0\}^\backslash infty\; \backslash frac\{(-1)^\{k\}\}\{(2k+1)^s\}\; =\; 2^\{-s\}\backslash Phi(-1,s,\backslash tfrac12)$

The Legendre chi function:

:$\backslash chi\_s(z)\; =\; \backslash sum\_\{k=0\}^\backslash infty\; \backslash frac\{z^\{2k+1\}\}\{(2k+1)^s\}=\; \backslash frac\; \{z\}\{2^s\}\; \backslash Phi(z^2,s,\backslash tfrac12)$

The inverse tangent integral:{{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}}

:$\backslash textrm\{Ti\}\_s(z)=\; \backslash sum\_\{k=0\}^\backslash infty\; \backslash frac\{(-1)^k\; z^\{2k+1\}\}\{(2k+1)^s\}=\backslash frac\{z\}\{2^s\}\backslash Phi(-z^2,s,\backslash tfrac12)$

The polygamma functions for positive integers n:The polygamma function has the series representation

$\backslash psi^\{(m)\}(z)\; =\; (-1)^\{m+1\}\backslash ,\; m!\; \backslash sum\_\{k=0\}^\backslash infty\; \backslash frac\{1\}\{(z+k)^\{m+1\}\}$

which holds for integer values of {{math|m > 0}} and any complex {{mvar|z}} not equal to a negative integer.{{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}}

:$\backslash psi^\{(n)\}(\backslash alpha)=\; (-1)^\{n+1\}\; n!\backslash Phi\; (1,n+1,\backslash alpha)$

The Clausen function:{{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}}

:$\backslash text\{Cl\}\_2(z)=\; \backslash frac\{ie^\{-iz\}\}2\; \backslash Phi(e^\{-iz\},2,1)-\backslash frac\{ie^\{iz\}\}2\; \backslash Phi(e^\{iz\},2,1)$

The Lerch transcendent has an integral representation:

:$$

\Phi(z,s,a)=\frac{1}{\Gamma(s)}\int_0^\infty

\frac{t^{s-1}e^{-at}}{1-ze^{-t}}\,dt

The proof is based on using the integral definition of the gamma function to write

:$\backslash Phi(z,s,a)\backslash Gamma(s)$

= \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}

and then interchanging the sum and integral. The resulting integral representation converges for $z\; \backslash in\; \backslash Complex\; \backslash setminus\; [1,\backslash infty),$ Re(s) > 0, and Re(a) > 0. This analytically continues $\backslash Phi(z,s,a)$ 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.{{harvnb|Bateman|Erdélyi|1953|p=27}}{{harvnb|Guillera|Sondow|2008|loc=Lemma 2.1 and 2.2}}

A contour integral representation is given by

:$$

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

where C is a Hankel contour counterclockwise around the positive real axis, not enclosing any of the points $t\; =\; \backslash log(z)\; +\; 2k\backslash pi\; i$ (for integer k) which are poles of the integrand. The integral assumes Re(a) > 0.{{harvnb|Bateman|Erdélyi|1953|p=28}}

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

:

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

:$\backslash Re(a)>0.$

Similar representations include

:$$

\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,

and

:$\backslash Phi(-z,s,a)=\; \backslash frac\{1\}\{2a^s\}\; +\; \backslash int\_\{0\}^\{\backslash infty\}\backslash frac\{\backslash cos(t\backslash log\; z)\backslash sin\backslash Big(s\backslash arctan\backslash tfrac\{t\}\{a\}\backslash Big)\; -\; \backslash sin(t\backslash log\; z)\backslash cos\backslash Big(s\backslash arctan\backslash tfrac\{t\}\{a\}\backslash Big)\}\{\backslash big(a^2\; +\; t^2\backslash big)^\{\backslash frac\{s\}\{2\}\}\; \backslash sinh\backslash pi\; t\; \}\backslash ,dt,$

holding for positive z (and more generally wherever the integrals converge). Furthermore,

:$\backslash Phi(e^\{i\backslash varphi\},s,a)=L\backslash big(\backslash tfrac\{\backslash varphi\}\{2\backslash pi\},\; s,\; a\backslash big)=\; \backslash frac\{1\}\{a^s\}\; +\; \backslash frac\{1\}\{2\backslash Gamma(s)\}\backslash int\_\{0\}^\{\backslash infty\}\backslash frac\{t^\{s-1\}e^\{-at\}\backslash big(e^\{i\backslash varphi\}-e^\{-t\}\backslash big)\}\{\backslash cosh\{t\}-\backslash cos\{\backslash varphi\}\}\backslash ,dt,$

The last formula is also known as Lipschitz formula.

For λ rational, the summand is a root of unity, and thus $L(\backslash lambda,\; s,\; \backslash alpha)$ may be expressed as a finite sum over the Hurwitz zeta function. Suppose $\backslash lambda\; =\; \backslash frac\{p\}\{q\}$ with $p,\; q\; \backslash in\; \backslash Z$ and $q\; >\; 0$. Then $z\; =\; \backslash omega\; =\; e^\{2\; \backslash pi\; i\; \backslash frac\{p\}\{q\}\}$ and $\backslash omega^q\; =\; 1$.

:$\backslash Phi(\backslash omega,\; s,\; \backslash alpha)\; =\; \backslash sum\_\{n=0\}^\backslash infty$

\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)

Various identities include:

:$\backslash Phi(z,s,a)=z^n\; \backslash Phi(z,s,a+n)\; +\; \backslash sum\_\{k=0\}^\{n-1\}\; \backslash frac\; \{z^k\}\{(k+a)^s\}$

and

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

and

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

A series representation for the Lerch transcendent is given by

:$\backslash Phi(z,s,q)=\backslash frac\{1\}\{1-z\}$

\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 $\backslash tbinom\{n\}\{k\}$ 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.{{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}}

A Taylor series in the first parameter was given by Arthur Erdélyi. It may be written as the following series, which is valid for{{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}}

:

:$$

\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]

If n 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 $\backslash psi(n)$ is the digamma function.

A Taylor series in the third variable is given by

:

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

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

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

for

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

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

The representation as a generalized hypergeometric function is{{cite journal|first1=J. E.|last1=Gottschalk|first2=E. N. |last2=Maslen| title=Reduction formulae for generalized hypergeometric functions 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}}

:$$

\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).

The polylogarithm function $\backslash mathrm\{Li\}\_n(z)$ is defined as

:$\backslash mathrm\{Li\}\_0(z)=\backslash frac\{z\}\{1-z\},\; \backslash qquad\; \backslash mathrm\{Li\}\_\{-n\}(z)=z\; \backslash frac\{d\}\{dz\}\; \backslash mathrm\{Li\}\_\{1-n\}(z).$

Let

:$$

\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}

For and $z\; \backslash in\; \backslash Omega\_\{a\}$, an asymptotic expansion of $\backslash Phi(z,s,a)$ for large $a$ and fixed $s$ and $z$ is given by

:$$

\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})

for $N\; \backslash in\; \backslash mathbb\{N\}$, where $(s)\_n\; =\; s\; (s+1)\backslash cdots\; (s+n-1)$ is the Pochhammer symbol.{{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 }}

Let

:$f(z,x,a)\; \backslash equiv\; \backslash frac\{1-(z\; e^\{-x\})^\{1-a\}\}\{1-z\; e^\{-x\}\}.$

Let $C\_\{n\}(z,a)$ be its Taylor coefficients at $x=0$. Then for fixed $N\; \backslash in\; \backslash mathbb\{N\},\; \backslash Re\; a\; >\; 1$ and $\backslash Re\; s\; >\; 0$,

:$$

\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),

as $\backslash Re\; a\; \backslash to\; \backslash infty$.{{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 }}

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].

{{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 |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 |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 ₂ψ₂ | 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 | 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