Polylogarithm

The function

$$\mathop{\rm Li}\nolimits _n(z)\equiv \sum_{k=1}^\infty {z^k\over k^n},$$ (1)

Also known as Jonquière's Function. (Note that the Notation $\mathop{\rm Li}\nolimits (z)$ is also used for the Logarithmic Integral.) The polylogarithm arises in Feynman Diagram integrals, and the special case $n=2$ is called the Dilogarithm. The polylogarithm of Negative Integer order arises in sums of the form

$$\sum_{k=1}^\infty k^n r^k=\mathop{\rm Li}\nolimits _{-n}(r) ... ...}} \sum_{i=1}^n \left\langle{n\atop i}\right\rangle{} r^{n-i},$$ (2)

where $\left\langle{n\atop i}\right\rangle{}$ is an Eulerian Number.

The polylogarithm satisfies the fundamental identities

$$-\ln(1-2^{-n})=\mathop{\rm Li}\nolimits _1(2^{-n})$$ (3)

$$\mathop{\rm Li}\nolimits _s(-1)=-(1-2^{1-s})\zeta(s),$$ (4)

where $\zeta(s)$ is the Riemann Zeta Function. The derivative is therefore given by

$${d\over ds} \mathop{\rm Li}\nolimits _s(-1)=-2^{1-s}\zeta(s)\ln 2-(1-2^{1-s})\zeta'(s),$$ (5)

or in the special case $s=0$, by

$$\left[{{d\over ds} \mathop{\rm Li}\nolimits _s(-1)}\right]_{... ...tyle{1\over 2}}\ln(2\pi)=\ln\left({\sqrt{2\over\pi}\,}\right).$$ (6)

This latter fact provides a remarkable proof of the Wallis Formula.

The polylogarithm identities lead to remarkable expressions. Ramanujan gave the polylogarithm identities

$$\mathop{\rm Li}\nolimits _2({\textstyle{1\over 3}})-{\textst... ...)={\textstyle{1\over 18}}\pi^2-{\textstyle{1\over 6}}(\ln 3)^2$$ (7)

$$\mathop{\rm Li}\nolimits _2(-{\textstyle{1\over 2}})+{\texts... ...tstyle{1\over 2}}(\ln 2)^2-{\textstyle{1\over 3}}(\ln 3)^2\eno$$ (8)

$$\mathop{\rm Li}\nolimits _2({\textstyle{1\over 4}})+{\textst... ...}}\pi^2+2\ln 2\ln 3-2(\ln 2)^2-{\textstyle{2\over 3}}(\ln 3)^2$$ (9)

$$\mathop{\rm Li}\nolimits _2(-{\textstyle{1\over 3}})-{\texts... ...=-{\textstyle{1\over 18}}\pi^2+{\textstyle{1\over 6}}(\ln 3)^2$$ (10)

$$\mathop{\rm Li}\nolimits _2(-{\textstyle{1\over 8}})+\mathop... ...ver 9}})=-{\textstyle{1\over 2}}(\ln {\textstyle{9\over 8}})^2$$ (11)

(Berndt 1994), and Bailey et al. show that

$$\pi^2=36\mathop{\rm Li}\nolimits _2({\textstyle{1\over 2}})-... ...ver 8}})+6\mathop{\rm Li}\nolimits _2({\textstyle{1\over 64}})$$ (12)

$$12\mathop{\rm Li}\nolimits _2({\textstyle{1\over 2}})=\pi^2-6(\ln 2)^2$$ (13)

$${\textstyle{35\over 2}}\zeta(3)-\pi^2\ln 2=36\mathop{\rm Li}... ...over 8}})+\mathop{\rm Li}\nolimits _3({\textstyle{1\over 64}})$$ (14)

$$2(\ln 2)^3-7\zeta(3)=-24\mathop{\rm Li}\nolimits _3({\textst... ...over 8}})-\mathop{\rm Li}\nolimits _3({\textstyle{1\over 64}})$$ (15)

$$10(\ln 2)^3-2\pi^2\ln 2=-48\mathop{\rm Li}\nolimits _3({\tex... ...er 8}})-3\mathop{\rm Li}\nolimits _3({\textstyle{1\over 64}}),$$ (16)

and

${\mathop{\rm Li}\nolimits _m({\textstyle{1\over 64}})\over 6^{m-1}}-{\mathop{\r... ...thop{\rm Li}\nolimits _m({\textstyle{1\over 2}})\over 9}-{5(-\ln 2)^m\over 9m!}$
$+{\pi^2(-\ln 2)^{m-2}\over 54(m-2)!}-{\pi^4(-\ln 2)^{m-4}\over 486(m-4)!}-{403\zeta(5)(-\ln 2)^{m-5}\over 1296(m-5)!} =0.\quad$	(17)

No general Algorithm is know for the integration of polylogarithms of functions.

See also Dilogarithm, Eulerian Number, Legendre's Chi-Function, Logarithmic Integral, Nielsen-Ramanujan Constants

References

Bailey, D.; Borwein, P.; and Plouffe, S. ``On the Rapid Computation of Various Polylogarithmic Constants.'' http://www.cecm.sfu.ca/~pborwein/PAPERS/P123.ps.

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 323-326, 1994.

Lewin, L. Polylogarithms and Associated Functions. New York: North-Holland, 1981.

Lewin, L. (Ed.). Structural Properties of Polylogarithms. Providence, RI: Amer. Math. Soc., 1991.

Nielsen, N. Der Euler'sche Dilogarithms. Leipzig, Germany: Halle, 1909.