Lerch zeta function

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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 [1].

Definition[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

The Lerch transcendent is implemented as LerchPhi in Maple.

References[edit]

External links[edit]