Editing Polylogarithm (section)

==Polylogarithm ladders==

[[Leonard Lewin (telecommunications engineer)|Leonard Lewin]] discovered a remarkable and broad generalization of a number of classical relationships on the polylogarithm for special values. These are now called ''polylogarithm ladders''. Define <math>\rho = \tfrac{1}{2} (\sqrt{5}-1)</math> as the reciprocal of the [[golden ratio]]. Then two simple examples of dilogarithm ladders are

<math display="block">\operatorname{Li}_2(\rho^6) = 4 \operatorname{Li}_2(\rho^3) + 3 \operatorname{Li}_2(\rho^2) - 6 \operatorname{Li}_2(\rho) + \tfrac {7}{30} \pi^2</math>

given by {{harvs | txt | author-link= Harold Scott MacDonald Coxeter | last= Coxeter | year= 1935}} and

<math display="block">\operatorname{Li}_2(\rho) = \tfrac{1}{10} \pi^2 - \ln^2\rho</math>

given by [[John Landen|Landen]]. Polylogarithm ladders occur naturally and deeply in [[K-theory]] and [[algebraic geometry]]. Polylogarithm ladders provide the basis for the rapid computations of various mathematical constants by means of the [[BBP algorithm]] {{harv|Bailey|Borwein|Plouffe|1997}}.