GeneralizedPolylog and MultiPolylog represent the function class consisting of generalized polylogarithms, multiple polylogarithms, harmonic polylogarithms, hyperlogarithms, and related functions.

The generalized polylogarithm is defined recursively, as the iterated integral

$\mathrm{GeneralizedPolylog}\left(\left[a_1,\mathrm{...},a_w\right]\,x\right)={\int }_0^x\frac{\mathrm{GeneralizedPolylog}\left(\left[a_2,\mathrm{...},a_w\right]\,y\right)}{y-a_1}\ⅆy$

The recursion stops, as

$\mathrm{GeneralizedPolylog}\left(\left[\right]\,x\right)=1$

For all the a[i] indices being zero, an alternative definition is used, as

$\mathrm{GeneralizedPolylog}\left(\left[\underset{\text{w times}}{\underbrace{0,\mathrm{...},0}}\right]\,x\right)=\frac{{\ln \left(x\right)}^n}{n!}$

The multiple polylogarithm, on the other hand, represent the sum form over $i_1\>i_2\>..\.\>\mathrm{i\_\_n} \>0$

$\mathrm{MultiPolylog}\left(m_1,\mathrm{...},m_n\,z_1,\mathrm{...},z_n\right)=\sum _i \prod _{}^{}\left(\frac{z_1^{i_1}}{i_1^{m_1}},\mathrm{...},\frac{z_n^{i_n}}{i_n^{m_n}}\right)$

and the analytic continuation thereof outside its convergent region, which is given by the restrictions

$\prod _{j=1}^n{a}_j$

$\left|z_1\right|<1,\left|z_1{z}_2\right|<1, \dots \&comma;\left|\prod _{i=1}^n{z}_i\right|<1$

The relation between GeneralizedPolylog and MultiPolylog is given as

$\mathrm{GeneralizedPolylog}\left(\left[\underset{m_1\text{times}}{\underbrace{0,\mathrm{...},0}}\&comma;a_1\&comma;\underset{m_2\text{times}}{\underbrace{0,\mathrm{...},0}}\&comma;a_2\&comma; \dots \&comma;\underset{m_n\text{times}}{\underbrace{0,\mathrm{...},0}}\&comma;a_n\right]\&comma;x\right)={\left(-1\right)}^n\mathrm{MultiPolylog}\left(\left[m_1+1,\mathrm{...},m_n+1\right]\&comma;\left[\frac{x}{a_1}\&comma;\frac{a_1}{a_2},\mathrm{...},\frac{a_{n-1}}{a_n}\right]\right)$

The generalized polylogarithm and related functions show up in high energy physics, where scattering amplitudes and other observables in quantum field theories, often are given in terms of this class of functions when calculated with high precision, i.e. beyond the leading order in perturbative expansion used in the Feynman diagrammatic expansion.

Initialization: Set the display of special functions in output to typeset mathematical notation (textbook notation):

$\mathrm{Typesetting}:-\mathrm{EnableTypesetRule}\left(\mathrm{Typesetting}:-\mathrm{SpecialFunctionRules}\right)\&colon;$

Functions such as ln, polylog and MultiZeta may appear as special cases of the generalized polylogarithms

$\mathrm{\%GeneralizedPolylog}\left(\left[0\right]\&comma;x\right)\&equals;\mathrm{GeneralizedPolylog}\left(\left[0\right]\&comma;x\right)\&semi;$

$\mathrm{GeneralizedPolylog}\left(\left[0\right]\&comma;x\right)=\ln \left(x\right)$

$\mathrm{\%GeneralizedPolylog}\left(\left[0\&comma;0\&comma;0\&comma;0\&comma;1\right]\&comma;x\right)\&equals;\mathrm{GeneralizedPolylog}\left(\left[0\&comma;0\&comma;0\&comma;0\&comma;1\right]\&comma;x\right)\&semi;$

$\mathrm{GeneralizedPolylog}\left(\left[0\&comma;0\&comma;0\&comma;0\&comma;1\right]\&comma;x\right)=-{\mathrm{Li}}_5\left(x\right)$

$\left(\mathrm{\%MultiPolylog}\&equals;\mathrm{MultiPolylog}\right)\left(\left[2\&comma;3\&comma;4\&comma;5\right]\&comma;\left[1\&comma;1\&comma;1\&comma;1\right]\right)\&semi;$

$\mathrm{MultiPolylog}\left(\left[2\&comma;3\&comma;4\&comma;5\right]\&comma;\left[1\&comma;1\&comma;1\&comma;1\right]\right)=\mathrm{MultiZeta}\left(2\&comma;3\&comma;4\&comma;5\right)$

$\left(\mathrm{\%MultiPolylog} \&equals; \mathrm{MultiPolylog}\right)\left(\left[2\&comma;1\&comma;1\right]\&comma;\left[1\&comma;-1\&comma;-1\right]\right)$

$\mathrm{MultiPolylog}\left(\left[2\&comma;1\&comma;1\right]\&comma;\left[1\&comma;\mathrm{-1}\&comma;\mathrm{-1}\right]\right)=\frac{{\ln \left(2\right)}^2{\mathrm{\pi }}^2}{8}-\frac{7{\mathrm{\pi }}^4}{288}+3{\mathrm{Li}}_4\left(\frac{1}{2}\right)+\frac{{\ln \left(2\right)}^4}{8}$

$\left(\mathrm{\%MultiPolylog} \&equals; \mathrm{MultiPolylog}\right)\left(\left[2\&comma;1\right]\&comma;\left[1\&comma;x\right]\right)$

$\mathrm{MultiPolylog}\left(\left[2\&comma;1\right]\&comma;\left[1\&comma;x\right]\right)={\mathrm{Li}}_2\left(1-x\right)\ln \left(1-x\right)-{\mathrm{Li}}_3\left(x\right)-2{\mathrm{Li}}_3\left(1-x\right)+2\mathrm{\zeta }\left(3\right)$

$\left(\mathrm{\%GeneralizedPolylog}\&equals;\mathrm{GeneralizedPolylog}\right)\left(\left[0\&comma;1\&comma;1\right]\&comma;x\right)$

$\mathrm{GeneralizedPolylog}\left(\left[0\&comma;1\&comma;1\right]\&comma;x\right)=-{\mathrm{Li}}_3\left(1-x\right)+{\mathrm{Li}}_2\left(1-x\right)\ln \left(1-x\right)+\frac{\ln \left(x\right){\ln \left(1-x\right)}^2}{2}+\mathrm{\zeta }\left(3\right)$

$\mathrm{GeneralizedPolylog}\left(\left[0.23-1.78\&ast;I\&comma;1.99\&plus;3.33\&ast;I\&comma;0.77\&plus;0.09\&ast;I\right]\&comma;1.35-1.01\&ast;I\right)$

$\mathrm{GeneralizedPolylog}\left(\left[0.23-1.78\mathrm{I}\&comma;1.99+3.33\mathrm{I}\&comma;0.77+0.09\mathrm{I}\right]\&comma;1.35-1.01\mathrm{I}\right)$

$\mathrm{GeneralizedPolylog}\left(\left[\left(0.23-1.78\mathrm{I}\right)z\&comma;\left(1.99+3.33\mathrm{I}\right)z\&comma;\left(0.77+0.09\mathrm{I}\right)z\right]\&comma;\left(1.35-1.01\mathrm{I}\right)z\right)$

$\mathrm{GeneralizedPolylog}\left(\left[\left(0.23-1.78\mathrm{I}\right)z\&comma;\left(1.99+3.33\mathrm{I}\right)z\&comma;\left(0.77+0.09\mathrm{I}\right)z\right]\&comma;\left(1.35-1.01\mathrm{I}\right)z\right)$

$\mathrm{evalf}\left[8\right]\left(\mathrm{eval}\left(\&equals;\&comma;\left[z\&equals;1.91-0.39 I\right]\right)\right)$

$0.013040566+0.21053300\mathrm{I}=0.013040566+0.21053300\mathrm{I}$

$\mathrm{GeneralizedPolylog}\left(\left[0.23-1.78\mathrm{I}\right]\&comma;1.35-1.01\mathrm{I}\right)\mathrm{GeneralizedPolylog}\left(\left[1.99+3.33\mathrm{I}\&comma;0.77+0.09\mathrm{I}\right]\&comma;1.35-1.01\mathrm{I}\right)$

$\left(\mathrm{-0.2780299456}-1.097010462\mathrm{I}\right)\mathrm{GeneralizedPolylog}\left(\left[1.99+3.33\mathrm{I}\&comma;0.77+0.09\mathrm{I}\right]\&comma;1.35-1.01\mathrm{I}\right)$

$\mathrm{GeneralizedPolylog}\left(\left[0.23-1.78\mathrm{I}\&comma;1.99+3.33\mathrm{I}\&comma;0.77+0.09\mathrm{I}\right]\&comma;1.35-1.01\mathrm{I}\right)+\mathrm{GeneralizedPolylog}\left(\left[1.99+3.33\mathrm{I}\&comma;0.23-1.78\mathrm{I}\&comma;0.77+0.09\mathrm{I}\right]\&comma;1.35-1.01\mathrm{I}\right)+\mathrm{GeneralizedPolylog}\left(\left[1.99+3.33\mathrm{I}\&comma;0.77+0.09\mathrm{I}\&comma;0.23-1.78\mathrm{I}\right]\&comma;1.35-1.01\mathrm{I}\right)$

$\mathrm{GeneralizedPolylog}\left(\left[0.23-1.78\mathrm{I}\&comma;1.99+3.33\mathrm{I}\&comma;0.77+0.09\mathrm{I}\right]\&comma;1.35-1.01\mathrm{I}\right)+\mathrm{GeneralizedPolylog}\left(\left[1.99+3.33\mathrm{I}\&comma;0.23-1.78\mathrm{I}\&comma;0.77+0.09\mathrm{I}\right]\&comma;1.35-1.01\mathrm{I}\right)+\mathrm{GeneralizedPolylog}\left(\left[1.99+3.33\mathrm{I}\&comma;0.77+0.09\mathrm{I}\&comma;0.23-1.78\mathrm{I}\right]\&comma;1.35-1.01\mathrm{I}\right)$

$\mathrm{evalf}\left[6\right]\left(\&equals;\right)$

$0.264849+0.438022\mathrm{I}=0.264849+0.438022\mathrm{I}$

$\mathrm{\%MultiPolylog}\left(\left[2\right]\&comma;\left[0.98-0.11\mathrm{I}\right]\right)\mathrm{\%MultiPolylog}\left(\left[3\right]\&comma;\left[2.77-1.04\mathrm{I}\right]\right)$

$\mathrm{MultiPolylog}\left(\left[2\right]\&comma;\left[0.98-0.11\mathrm{I}\right]\right)\mathrm{MultiPolylog}\left(\left[3\right]\&comma;\left[2.77-1.04\mathrm{I}\right]\right)$

$\mathrm{\%MultiPolylog}\left(\left[2\&comma;3\right]\&comma;\left[0.98-0.11\mathrm{I}\&comma;2.77-1.04\mathrm{I}\right]\right)+\mathrm{\%MultiPolylog}\left(\left[3\&comma;2\right]\&comma;\left[2.77-1.04\mathrm{I}\&comma;0.98-0.11\mathrm{I}\right]\right)+\mathrm{\%MultiPolylog}\left(\left[5\right]\&comma;\left[\left(0.98-0.11\mathrm{I}\right)\left(2.77-1.04\mathrm{I}\right)\right]\right)$

$\mathrm{MultiPolylog}\left(\left[2\&comma;3\right]\&comma;\left[0.98-0.11\mathrm{I}\&comma;2.77-1.04\mathrm{I}\right]\right)+\mathrm{MultiPolylog}\left(\left[3\&comma;2\right]\&comma;\left[2.77-1.04\mathrm{I}\&comma;0.98-0.11\mathrm{I}\right]\right)+\mathrm{MultiPolylog}\left(\left[5\right]\&comma;\left[2.6002-1.3239\mathrm{I}\right]\right)$

$\mathrm{evalf}\left[4\right]\left(\mathrm{value}\left(\&equals;\right)\right)$

$2.809-4.448\mathrm{I}=2.809-4.448\mathrm{I}$

[1] A.B.Goncharov. "Multiple polylogarithms, cyclotomy and modular complexes", Math Res.Letters. Vol. 5 (1998): 497-516.
[2] Jens Vollinga, Stefan Weinzierl.  "Numerical evaluation of multiple polylogarithms", Comput.Phys.Commun. Vol. 167 (2005): 23 pp.
[3] H. Frellesvig, D. Tommasini, C. Wever.  "On the reduction of generalized polylogarithms to Li_n and Li_22 and on the evaluation thereof", JHEP 1603 (2016): 35pp

The GeneralizedPolylog command was introduced in Maple 2018.

The MultiPolylog command was introduced in Maple 2018.