Lerch transcendent

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. The Lerch transcendent, is given by:

${\displaystyle \Phi (z,s,\alpha )=\sum _{n=0}^{\infty }{\frac {z^{n}}{(n+\alpha )^{s}}}}$.

It only converges for any real number ${\displaystyle \alpha >0}$, where ${\displaystyle |z|<1}$, or ${\displaystyle {\mathfrak {R}}(s)>1}$, and ${\displaystyle |z|=1}$.

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

The Lerch zeta function is given by:

${\displaystyle L(\lambda ,s,\alpha )=\sum _{n=0}^{\infty }{\frac {e^{2\pi i\lambda n}}{(n+\alpha )^{s}}}=\Phi (e^{2\pi i\lambda },s,\alpha )}$

The Hurwitz zeta function is the special case

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

The polylogarithm is another special case:

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

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

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

The Dirichlet eta function:

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

The Dirichlet beta function:

${\displaystyle \beta (s)=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(2k+1)^{s}}}=2^{-s}\Phi (-1,s,{\tfrac {1}{2}})}$

The Legendre chi function:

${\displaystyle \chi _{s}(z)=\sum _{k=0}^{\infty }{\frac {z^{2k+1}}{(2k+1)^{s}}}={\frac {z}{2^{s}}}\Phi (z^{2},s,{\tfrac {1}{2}})}$

The inverse tangent integral:

${\displaystyle {\textrm {Ti}}_{s}(z)=\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k+1}}{(2k+1)^{s}}}={\frac {z}{2^{s}}}\Phi (-z^{2},s,{\tfrac {1}{2}})}$

The polygamma functions for positive integers n:

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

The Clausen function:

${\displaystyle {\text{Cl}}_{2}(z)={\frac {ie^{-iz}}{2}}\Phi (e^{-iz},2,1)-{\frac {ie^{iz}}{2}}\Phi (e^{iz},2,1)}$

The Lerch transcendent has an integral representation:

${\displaystyle \Phi (z,s,a)={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }{\frac {t^{s-1}e^{-at}}{1-ze^{-t}}}\,dt}$

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

${\displaystyle \Phi (z,s,a)\Gamma (s)=\sum _{n=0}^{\infty }{\frac {z^{n}}{(n+a)^{s}}}\int _{0}^{\infty }x^{s}e^{-x}{\frac {dx}{x}}=\sum _{n=0}^{\infty }\int _{0}^{\infty }t^{s}z^{n}e^{-(n+a)t}{\frac {dt}{t}}}$

and then interchanging the sum and integral. The resulting integral representation converges for ${\displaystyle z\in \mathbb {C} \setminus [1,\infty ),}$Re(s) > 0, and Re(a) > 0. This analytically continues ${\displaystyle \Phi (z,s,a)}$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

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

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

A Hermite-like integral representation is given by

${\displaystyle \Phi (z,s,a)={\frac {1}{2a^{s}}}+\int _{0}^{\infty }{\frac {z^{t}}{(a+t)^{s}}}\,dt+{\frac {2}{a^{s-1}}}\int _{0}^{\infty }{\frac {\sin(s\arctan(t)-ta\log(z))}{(1+t^{2})^{s/2}(e^{2\pi at}-1)}}\,dt}$

for

${\displaystyle \Re (a)>0\wedge |z|<1}$

and

${\displaystyle \Phi (z,s,a)={\frac {1}{2a^{s}}}+{\frac {\log ^{s-1}(1/z)}{z^{a}}}\Gamma (1-s,a\log(1/z))+{\frac {2}{a^{s-1}}}\int _{0}^{\infty }{\frac {\sin(s\arctan(t)-ta\log(z))}{(1+t^{2})^{s/2}(e^{2\pi at}-1)}}\,dt}$

for

${\displaystyle \Re (a)>0.}$

Similar representations include

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

and

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

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

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

The last formula is also known as Lipschitz formula.

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

${\displaystyle \Phi (\omega ,s,\alpha )=\sum _{n=0}^{\infty }{\frac {\omega ^{n}}{(n+\alpha )^{s}}}=\sum _{m=0}^{q-1}\sum _{n=0}^{\infty }{\frac {\omega ^{qn+m}}{(qn+m+\alpha )^{s}}}=\sum _{m=0}^{q-1}\omega ^{m}q^{-s}\zeta \left(s,{\frac {m+\alpha }{q}}\right)}$

Various identities include:

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

and

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

and

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

A series representation for the Lerch transcendent is given by

${\displaystyle \Phi (z,s,q)={\frac {1}{1-z}}\sum _{n=0}^{\infty }\left({\frac {-z}{1-z}}\right)^{n}\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}(q+k)^{-s}.}$

(Note that ${\displaystyle {\tbinom {n}{k}}}$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 series in the first parameter was given by Arthur Erdélyi. It may be written as the following series, which is valid for

${\displaystyle \left|\log(z)\right|<2\pi ;s\neq 1,2,3,\dots ;a\neq 0,-1,-2,\dots }$

${\displaystyle \Phi (z,s,a)=z^{-a}\left[\Gamma (1-s)\left(-\log(z)\right)^{s-1}+\sum _{k=0}^{\infty }\zeta (s-k,a){\frac {\log ^{k}(z)}{k!}}\right]}$

If n is a positive integer, then

${\displaystyle \Phi (z,n,a)=z^{-a}\left\{\sum _{{k=0} \atop k\neq n-1}^{\infty }\zeta (n-k,a){\frac {\log ^{k}(z)}{k!}}+\left[\psi (n)-\psi (a)-\log(-\log(z))\right]{\frac {\log ^{n-1}(z)}{(n-1)!}}\right\},}$

where ${\displaystyle \psi (n)}$is the digamma function.

A Taylor series in the third variable is given by

${\displaystyle \Phi (z,s,a+x)=\sum _{k=0}^{\infty }\Phi (z,s+k,a)(s)_{k}{\frac {(-x)^{k}}{k!}};|x|<\Re (a),}$

where ${\displaystyle (s)_{k}}$is the Pochhammer symbol.

Series at a = −n is given by

${\displaystyle \Phi (z,s,a)=\sum _{k=0}^{n}{\frac {z^{k}}{(a+k)^{s}}}+z^{n}\sum _{m=0}^{\infty }(1-m-s)_{m}\operatorname {Li} _{s+m}(z){\frac {(a+n)^{m}}{m!}};\ a\rightarrow -n}$

A special case for n = 0 has the following series

${\displaystyle \Phi (z,s,a)={\frac {1}{a^{s}}}+\sum _{m=0}^{\infty }(1-m-s)_{m}\operatorname {Li} _{s+m}(z){\frac {a^{m}}{m!}};|a|<1,}$

where ${\displaystyle \operatorname {Li} _{s}(z)}$is the polylogarithm.

An asymptotic series for ${\displaystyle s\rightarrow -\infty }$

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

for ${\displaystyle |a|<1;\Re (s)<0;z\notin (-\infty ,0)}$and

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

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

An asymptotic series in the incomplete gamma function

${\displaystyle \Phi (z,s,a)={\frac {1}{2a^{s}}}+{\frac {1}{z^{a}}}\sum _{k=1}^{\infty }{\frac {e^{-2\pi i(k-1)a}\Gamma (1-s,a(-2\pi i(k-1)-\log(z)))}{(-2\pi i(k-1)-\log(z))^{1-s}}}+{\frac {e^{2\pi ika}\Gamma (1-s,a(2\pi ik-\log(z)))}{(2\pi ik-\log(z))^{1-s}}}}$

for ${\displaystyle |a|<1;\Re (s)<0.}$

The representation as a generalized hypergeometric function is

${\displaystyle \Phi (z,s,\alpha )={\frac {1}{\alpha ^{s}}}{}_{s+1}F_{s}\left({\begin{array}{c}1,\alpha ,\alpha ,\alpha ,\cdots \\1+\alpha ,1+\alpha ,1+\alpha ,\cdots \\\end{array}}\mid z\right).}$

The polylogarithm function ${\displaystyle \mathrm {Li} _{n}(z)}$is defined as

${\displaystyle \mathrm {Li} _{0}(z)={\frac {z}{1-z}},\qquad \mathrm {Li} _{-n}(z)=z{\frac {d}{dz}}\mathrm {Li} _{1-n}(z).}$

Let

${\displaystyle \Omega _{a}\equiv {\begin{cases}\mathbb {C} \setminus [1,\infty )&{\text{if }}\Re a>0,\\{z\in \mathbb {C} ,|z|<1}&{\text{if }}\Re a\leq 0.\end{cases}}}$

For ${\displaystyle |\mathrm {Arg} (a)|<\pi ,s\in \mathbb {C} }$and ${\displaystyle z\in \Omega _{a}}$, an asymptotic expansion of ${\displaystyle \Phi (z,s,a)}$for large ${\displaystyle a}$and fixed ${\displaystyle s}$and ${\displaystyle z}$is given by

${\displaystyle \Phi (z,s,a)={\frac {1}{1-z}}{\frac {1}{a^{s}}}+\sum _{n=1}^{N-1}{\frac {(-1)^{n}\mathrm {Li} _{-n}(z)}{n!}}{\frac {(s)_{n}}{a^{n+s}}}+O(a^{-N-s})}$

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

Let

${\displaystyle f(z,x,a)\equiv {\frac {1-(ze^{-x})^{1-a}}{1-ze^{-x}}}.}$

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

${\displaystyle \Phi (z,s,a)-{\frac {\mathrm {Li} _{s}(z)}{z^{a}}}=\sum _{n=0}^{N-1}C_{n}(z,a){\frac {(s)_{n}}{a^{n+s}}}+O\left((\Re a)^{1-N-s}+az^{-\Re a}\right),}$

as ${\displaystyle \Re a\to \infty }$.

The Lerch transcendent is implemented as LerchPhi in Maple and Mathematica, and as lerchphi in mpmath and SymPy.

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