Документ взят из кэша поисковой машины. Адрес оригинального документа : http://theory.sinp.msu.ru/~smirnov/HPL.pdf Дата изменения: Sun Nov 2 13:43:01 2008 Дата индексирования: Mon Oct 1 19:53:54 2012 Кодировка:

Summing up series with nested sums
n

Si (n) =
j =1 n

1 , Sik (n) = i j

n j =1 n j =1

Sk (j ) , i j Sklm (j ) ji

Sikl (n) =
j =1

Skl (j ) , Siklm (n) = i j

E.g., with one index:
(n) = S1 (n - 1) - E ,
(k )

(n) = (-1)k k !(Sk

+1

(n - 1) - (k +1)) ,

k = 1, 2 ,... ,

SUMMER XSummer

[J.A.M. Vermaseren'00 ] [S. Moch and P. Uwer'00 ]
­ p.52


Harmonic polylogarithms (HPL) Ha1 ,a2 ,...,an (x) H (a1 ,a2 ,... ,an ; x), with ai = 1, 0, -1
[E. Remiddi & J.A.M. Vermaseren'00]

are generalizations of the usual polylogarithms Lia (z ) and Nielsen polylogarithms Sa,b (z )
x

H (a1 ,a2 ,... ,an ; x) =

where f (±1; t) = 1/(1 t), f (0; t) = 1/t,
H (±1; x) = ln(1 x), H (0; x) = ln x,

0

f (a1 ; t)H (a2 ,... ,an ; t)dt,

with ai = 1, 0, -1. HPL are implemented in Mathematica
[D. Maitre'06]

­ p.53