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 Czech mathematician Mathias Lerch, who published a paper about the function in 1887.

Definition

The Lerch zeta function is given by

:$L(\lambda, s, \alpha) = \sum_^\infty \frac .$

A related function, the Lerch transcendent, is given by

:$\Phi(z, s, \alpha) = \sum_^\infty \frac .$

The two are related, as

:$\,\Phi(e^, s,\alpha)=L(\lambda, s, \alpha).$

Integral representations

The Lerch transcendent has an integral representation:
:$\Phi(z,s,a)=\frac\int_0^\infty \frac\,dt$
The proof is based on using the integral definition of the Gamma function to write
:$\Phi(z,s,a)\Gamma(s) = \sum_^\infty \frac \int_0^\infty x^s e^ \frac = \sum_^\infty \int_0^\infty t^s z^n e^ \frac$
and then interchanging the sum and integral. The resulting integral representation converges for z \in \Complex \setminus analytically continues $\Phi(z,s,a)$ 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.

A contour integral representation is given by
:$\Phi(z,s,a)=-\frac \int_C \frac\,dt$
where ''C'' is a Hankel contour counterclockwise around the positive real axis, not enclosing any of the points $t = \log(z) + 2k\pi i$ (for integer ''k'') which are poles of the integrand. The integral assumes Re(''a'') > 0.

Other integral representations

A Hermite-like integral representation is given by

:$\Phi(z,s,a)= \frac+ \int_0^\infty \frac\,dt+ \frac \int_0^\infty \frac\,dt$
for
:$\Re(a)>0\wedge , z, <1$
and
:$\Phi(z,s,a)=\frac+ \frac\Gamma(1-s,a\log(1/z))+ \frac \int_0^\infty \frac\,dt$
for
:$\Re(a)>0.$

Similar representations include

:$\Phi(z,s,a)= \frac + \int_^\frac\,dt,$

and

:$\Phi(-z,s,a)= \frac + \int_^\frac\,dt,$

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

:$\Phi(e^,s,a)=L\big(\tfrac, s, a\big)= \frac + \frac\int_^\frac\,dt,$

The last formula is also known as ''Lipschitz formula''.

Special cases

The Lerch zeta function and Lerch transcendent generalize various special functions.

The Hurwitz zeta function is the special case
:$\zeta(s,\alpha) = L(0, s, \alpha) = \Phi(1,s,\alpha) = \sum_^\infty \frac.$
The polylogarithm is another special case:
:$\textrm_s(z)=z\Phi(z,s,1) = \sum_^\infty \frac.$
The Riemann zeta function is a special case of both of the above:
:$\zeta(s) = \Phi(1,s,1) = \sum_^\infty \frac.$

Other special cases include:
* The Dirichlet eta function:
::$\eta(s) = \Phi(-1,s,1) = \sum_^\infty \frac$
* The Dirichlet beta function:
::$\beta(s) = 2^ \Phi(-1,s,1/2) = \sum_^\infty \frac$
* The Legendre chi function:
::$\chi_s(z)=2^z \Phi(z^2,s,1/2) = \sum_^\infty \frac$
* The Polygamma function:
::$\psi^(\alpha)= (-1)^ n!\Phi (1,n+1,\alpha)$

Identities

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

:$\Phi(\omega, s, \alpha) = \sum_^\infty \frac = \sum_^ \sum_^\infty \frac = \sum_^ \omega^m q^ \zeta(s,\frac)$

Various identities include:
:$\Phi(z,s,a)=z^n \Phi(z,s,a+n) + \sum_^ \frac$

and

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

and

:$\Phi(z,s+1,a)=-\frac\frac \Phi(z,s,a).$

Series representations

A series representation for the Lerch transcendent is given by

:$\Phi(z,s,q)=\frac \sum_^\infty \left(\frac \right)^n \sum_^n (-1)^k \binom (q+k)^.$
(Note that $\tbinom$ 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 series in the first parameter was given by Erdélyi. It may be written as the following series, which is valid for
:$\left, \log(z)\ < 2 \pi;s\neq 1,2,3,\dots; a\neq 0,-1,-2,\dots$
:
\Phi(z,s,a)=z^\left Gamma(1-s)\left(-\log (z)\right)^
+\sum_^\infty \zeta(s-k,a)\frac\right

If ''n'' is a positive integer, then
:$\Phi(z,n,a)=z^\left\,$
where $\psi(n)$ is the digamma function.

A Taylor series in the third variable is given by
:$\Phi(z,s,a+x)=\sum_^\infty \Phi(z,s+k,a)(s)_\frac;, x, <\Re(a),$
where $(s)_$ is the Pochhammer symbol.

Series at ''a'' = −''n'' is given by
:$\Phi(z,s,a)=\sum_^n \frac +z^n\sum_^\infty (1-m-s)_\operatorname_(z)\frac;\ a\rightarrow-n$

A special case for ''n'' = 0 has the following series
:$\Phi(z,s,a)=\frac +\sum_^\infty (1-m-s)_m \operatorname_(z)\frac; , a, <1,$
where $\operatorname_s(z)$ is the polylogarithm.

An asymptotic series for $s\rightarrow-\infty$
:\Phi(z,s,a)=z^\Gamma(1-s)\sum_^\infty k\pi i-\log(z)e^

for $, a, <1;\Re(s)<0 ;z\notin (-\infty,0)$
and
:
\Phi(-z,s,a)=z^\Gamma(1-s)\sum_^\infty
2k+1)\pi i-\log(z)e^

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

An asymptotic series in the incomplete gamma function
:$\Phi(z,s,a)=\frac+ \frac\sum_^\infty \frac + \frac$
for $, a, <1;\Re(s)<0.$

The representation as a generalized hypergeometric function is
:$\Phi(z,s,\alpha)=\frac_F_s\left(\begin 1,\alpha,\alpha,\alpha,\cdots\\ 1+\alpha,1+\alpha,1+\alpha,\cdots\\ \end\mid z\right).$

Asymptotic expansion

The polylogarithm function $\mathrm_n(z)$ is defined as
:$\mathrm_0(z)=\frac, \qquad \mathrm_(z)=z \frac \mathrm_(z).$
Let
:$\Omega_ \equiv\begin \mathbb\setminus[1,\infty) & \text \Re a > 0, \\ & \text \Re a \le 0. \end$
For $, \mathrm(a), <\pi, s \in \mathbb$ and $z \in \Omega_$, an asymptotic expansion of $\Phi(z,s,a)$ for large $a$ and fixed $s$ and $z$ is given by
:$\Phi(z,s,a) = \frac \frac + \sum_^ \frac \frac +O(a^)$
for $N \in \mathbb$, where $(s)_n = s (s+1)\cdots (s+n-1)$ is the Falling and rising factorials, Pochhammer symbol.

Let
:$f(z,x,a) \equiv \frac.$
Let $C_(z,a)$ be its Taylor coefficients at $x=0$. Then for fixed $N \in \mathbb, \Re a > 1$ and $\Re s > 0$,
:$\Phi(z,s,a) - \frac = \sum_^ C_(z,a) \frac + O\left( (\Re a)^+a z^ \right),$
as $\Re a \to \infty$.

Software

The Lerch transcendent is implemented as LerchPhi i
Maple
an
Mathematica
and as lerchphi i

an

References

* .
* . (See § 1.11, "The function Ψ(''z'',''s'',''v'')", p. 27)
*
* . (Includes various basic identities in the introduction.)
* .
* .
* .

External links

* .
* Ramunas Garunkstis,
Home Page
' (2005) ''(Provides numerous references and preprints.)''
*
*
*
* {{dlmf, id=25.14 , title=Lerch's Transcendent

Zeta and L-functions