LOG#161. Polylogia flashes(III).

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In the third post of this series I will write more fantastic identities related to our friends, the polylogs!

(1)   \begin{equation*} Li_s(z)=\sum_{k=1}^\infty \dfrac{z^k}{k^s}=z+\dfrac{z^2}{2^s}+\dfrac{z^3}{3^s}+\cdots \forall z,\vert z\vert<1 \end{equation*}

and by analytic continuation that equation can be extended to all \vert z\vert>1. In fact

(2)   \begin{equation*} Li_{s+1}(z)=\int_0^z\dfrac{Li_s(t)}{t}dt \end{equation*}

such as Im(Li_s(z))=-\dfrac{\pi \mu^{s-1}}{\Gamma (s)} \forall z\geq 1, since we define

(3)   \begin{equation*} Im(Li_s(z+i\delta))=\dfrac{\pi \mu^{s-1}}{\Gamma (s)} \end{equation*}

and the equations

(4)   \begin{equation*} z\dfrac{\partial Li_s(z)}{\partial z}=Li_{s-1}(z) \end{equation*}

(5)   \begin{equation*} \dfrac{\partial Li_s(z)}{\partial \ln z}=Li_{s-1}(z) \end{equation*}

(6)   \begin{equation*} \dfrac{\partial Li_s(e^\mu)}{\partial \mu}=Li_{s-1}(e^\mu) \end{equation*}

Duplication formula for the polylogarithm:

(7)   \begin{equation*} \boxed{Li_s(-z)+Li_s(z)=2^{1-s}Li_s(z^2)} \end{equation*}

Connection with the Kummer’s function can be established

(8)   \begin{equation*} \Lambda_n(z)=\int_0^z\dfrac{\log^{n-1}\vert t\vert}{1+t}dt \end{equation*}

(9)   \begin{equation*} \Lambda_n(z)+\Lambda_n(-z)=2^{1-n}\Lambda_n(-z^2) \end{equation*}

and thus

(10)   \begin{equation*} Li_n(z)=Li_n(1)+\sum_{k=1}^{n-1}(-1)^{k-1}\dfrac{\log^k\vert z\vert}{k!}Li_{n-k}+\dfrac{(-1)^{n-1}}{(n-1)!}\left[\Lambda_n(-1)-\Lambda_n(-z)\right] \end{equation*}

We also have

(11)   \begin{equation*} \sum_{m=0}^{p-1}Li_s(ze^{2\pi i\frac{m}{p}})=p^{1-s}Li_s(z^p) \end{equation*}

Some extra values of (negative) integer polylogs that are rational functions or logarithms

(12)   \begin{equation*} Li_0(z)=\dfrac{z}{1-z} \end{equation*}

(13)   \begin{equation*} Li_1(z)=-\ln (1-z) \end{equation*}

(14)   \begin{equation*} Li_{-1}(z)=\dfrac{z}{(1-z)^2} \end{equation*}

(15)   \begin{equation*} Li_{-2}(z)=\dfrac{z(1+z)}{(1-z)^3} \end{equation*}

(16)   \begin{equation*} Li_{-3}(z)=\dfrac{z(1+4z+z^2)}{(1-z)^4} \end{equation*}

(17)   \begin{equation*} Li_{-4}(z)=\dfrac{z(1+z)(1+10z+z^2)}{(1-z)^5} \end{equation*}

And more generally, we have the general formulae

(18)   \begin{equation*} \boxed{Li_{-n}(z)=\left(z\dfrac{\partial}{\partial z}\right)^n\left(\dfrac{z}{1-z}\right)} \end{equation*}

(19)   \begin{equation*} Li_{-n}(z)=\left[\dfrac{\partial}{\partial \log (z)}\right]^n\left(\dfrac{z}{1-z}\right)=\sum_{k=0}^nk!S(n+1,k+1)\left(\dfrac{z}{1-z}\right)^{k+1} \end{equation*}

and where S(n,k) are the Stirling numbers of the second kind.

 

(20)   \begin{equation*} Li_{-n}(z)=(-1)^{n+1}\sum_{k=0}^nk!S(n+1,k+1)\left(-\dfrac{1}{1-z}\right)^{k+1} \end{equation*}

Furthermore,

(21)   \begin{equation*} Li_{-n}=\dfrac{1}{(1-z)^{n+1}}\sum_{k=0}^{n-1}\langle\begin{matrix}n\\ k\end{matrix}\rangle z^{n-k} \end{equation*}

and where \langle\begin{matrix}n\\ k\end{matrix}\rangle are the eulerian numbers.

We write now some interesting values of the polylog you will love too

    \[Li_1(\frac{1}{2})=\ln (2)\]

    \[Li_2(\frac{1}{2})=\dfrac{\pi^2}{12}-\dfrac{1}{2}\left(\ln 2\right)^2\]

    \[Li_3(\frac{1}{2})=\dfrac{1}{6}\ln^32-\dfrac{1}{12}\pi^2\ln (2)+\dfrac{7}{8}\zeta (3)\]

    \[Li_4(\frac{1}{2}))\dfrac{\pi^4}{360}-\dfrac{1}{24}\ln^42+\dfrac{\pi^2}{24}\ln^22-\dfrac{1}{2}\zeta(\overline{3},\overline{1})\]

and where

    \[\zeta(\overline{3},\overline{1})=\sum_{m>n>0}(-1)^{m+n}m^{-3}n^{-1}\]

    \[Li_s(e^{2\pi i\frac{m}{p}})=p^{-s}\sum_{k=1}^{p}e^{2\pi i m\frac{k}{p}}\zeta (s,\frac{k}{p})\]

with m=1,2,\ldots,p-1.

The polylog and other functions can be also be related, as we have seen:

(22)   \begin{equation*} Li_s(1)=\zeta(s),Re(s)>1 \end{equation*}

(23)   \begin{equation*} Li_s(-1)=-\eta(s) \end{equation*}

(24)   \begin{equation*} Li_s(\pm i)=2^{-s}\eta(s)\pm i\beta(s) \end{equation*}

where \beta(s) is the Dirichlet beta function. The complete Fermi-Dirac integral is also polylogarithmic

    \[F_s(\mu)=-Li_{s+1}(-e^{\mu})\]

The incomplete polylog is also interesting

    \[Li_s(z)=Li_s(0,z)\]

    \[Li_s(b,z)=\dfrac{1}{\Gamma (z)}\int_b^{\infty}\dfrac{x^{s-1}dx}{\frac{e^x}{z}-1}\]

    \[Li_s(b,z)=\sum_{k=1}^\infty\dfrac{z^k}{k^s}\dfrac{\Gamma (s,kb)}{\Gamma (s)}\]

and the incomplete gamma function is defined by

    \[\Gamma (s,x)=\int_x^\infty t^{s-1}{e^{-t}}dt\]

    \[\gamma (s,x)=\int_0^xt^{s-1}e^{-t}dt\]

    \[Li_s(z)=z\Psi(z,s,1)\]

(25)   \begin{equation*} i^{-s}Li_s(e^{2\pi i x})+i^sLi_s(e^{-2\pi i x})=\dfrac{(2\pi)^s}{\Gamma(s)}\zeta (1-s,x) \end{equation*}

with 0\leq Re(x)<1, Im(x)\geq 0, 0<Re(x)\leq 1Im(x)<0. Moreover,

(26)   \begin{equation*} Li_s(z)=\dfrac{\Gamma (1-s)}{(2\pi)^{1-s}}\left[i^{1-s}\zeta\left((1-s,\dfrac{1}{2}+\dfrac{\ln(-z)}{2\pi i}\right)+i^{s-1}\zeta\left(1-s,\dfrac{1}{2}-\dfrac{\ln (-z)}{2\pi i}\right)\right] \end{equation*}

There is also a formula called inversion formula

(27)   \begin{equation*} Li_s(z)+(-1)^sLi_s(\frac{1}{z})=\dfrac{(2\pi i)^s}{\Gamma (s)}\zeta (1-s,\frac{1}{2}+\dfrac{\ln (-z)}{2\pi i} \end{equation*}

\forall s\in C and for z\notin (1,\infty)

(28)   \begin{equation*} Li_s(z)+(-1)^sLi_s\left(\dfrac{1}{z}\right)=\dfrac{(2\pi i)^s}{\Gamma (s)}\left(\zeta (1-s,\dfrac{1}{2}-\dfrac{\ln(-1/z)}{2\pi i}\right) \end{equation*}

The expression

    \[\zeta (1-n, x)=-\dfrac{B_n(x)}{n}\]

implies that

(29)   \begin{equation*} Li_n(e^{2\pi i x})+(-1)^nLi_n(e^{-2\pi i x})=-\dfrac{(2\pi i)^n}{n!}B_n(x) \end{equation*}

with

0\leq Re(x)<1 if Im(z)\geq 0 and 0<Re(x)\leq 1 if Im(x)<0. The following feynmanity (nullity) identity holds as well

(30)   \begin{equation*} \boxed{Li_{-n}(z)+(-1)^nLi_{-n}\left(\dfrac{1}{z}\right)=0} \end{equation*}

\forall n=1,2,3,\ldots and n=0,\pm 1,\pm 2,\ldots

(31)   \begin{equation*} Li_n(z)+(-1)^nLi_n(1/z)=-\dfrac{(2\pi i)^n}{n!}B_n\left(\dfrac{1}{2}+\dfrac{\ln (-z)}{2\pi i}\right) \end{equation*}

for z\notin (0,1) and

(32)   \begin{equation*} Li_n(z)+(-1)^nLi_n(1/z)=-\dfrac{(2\pi i)^n}{n!}B_n\left(\dfrac{1}{2}-\dfrac{\ln (-1/z)}{2\pi i}\right) \end{equation*}

for z\notin (1,\infty). Polylogs and Clausen functions are related (we already saw this before)

(33)   \begin{equation*} Li_s(e^{\pm i\theta})=Ci_s(\theta)\pm iSi_s(\theta) \end{equation*}

The inverse tangent integral is related to polylogs

(34)   \begin{equation*} Ti_s(z)=\dfrac{1}{2i}\left[Li_s(iz)-Li_s(-iz)\right] \end{equation*}

(35)   \begin{equation*} Ti_0(z)=\dfrac{z}{1+z^2} \end{equation*}

(36)   \begin{equation*} Ti_2(z)=\int_0^z\dfrac{\arctan(t)}{t}dt \end{equation*}

(37)   \begin{equation*} Ti_{n+1}(z)=\int_0^z\dfrac{Ti_n(t)}{t}dt \end{equation*}

The Legendre chi function \chi_s(z) is also related to polylogs

(38)   \begin{equation*} \chi_s(z)=\dfrac{1}{2}(Li_s(z)-Li_s(-z)) \end{equation*}

The incomplete zeta function or Debye functions are polylogs as well

(39)   \begin{equation*} Z_n(z)=\dfrac{1}{(n-1)!}\int_z^\infty \dfrac{t^{n-1}}{e^t-1}dt,\forall n=1,2,3,\ldots \end{equation*}

(40)   \begin{equation*} Li_n(e^\mu)=\sum_{k=0}^{n-1}Z_{n-k}(-\mu)\dfrac{\mu^k}{k!},\forall n=1,2,3,\ldots \end{equation*}

(41)   \begin{equation*} Z_n(z)=\sum_{k=0}^{n-1}Li_{n-k}(e^{-k})\dfrac{z^k}{k!},\forall n=1,2,3,\ldots \end{equation*}

Now, we will write some polylog integrals

(42)   \begin{equation*} Li_s(z)=\dfrac{1}{\Gamma (s)}\int_0^\infty\dfrac{t^{s-1}}{\frac{e^{t}}{z}-1}dt \end{equation*}

This last integral converges if Re(s)>0 \forall z\in R and z\geq 1. It is the Bose-Einstein distribution!

The Fermi-Dirac integrals read

(43)   \begin{equation*} -Li_s(-z)=\dfrac{1}{\Gamma (s)}\int_0^\infty\dfrac{t^{s-1}}{\frac{e^{t}}{z}+1}dt \end{equation*}

For Re(s)<0 and \forall z excepting z\in R and z\geq 0 we have

    \[Li_s(z)=\int_0^\infty\dfrac{t^{-s}\sin(s\pi/2-t\ln (-z))}{\sinh (\pi t)}dt\]

and

(44)   \begin{equation*} Li_s(e^{\mu})=-\dfrac{\Gamma (1-s)}{2\pi i}\oint_H\dfrac{(-t)^{s-1}}{e^{t-\mu}-1}dt \end{equation*}

with residue equal to

    \[R=\dfrac{i}{2\pi}=\Gamma(1-s)(-\mu)^{s-1}\]

See you in the next polylog post!

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