Polylogs

Polylogs, also known as generalized polylogarithms or multiple polylogarithms, are a class of special functions that generalize the familiar polylogarithm. The polylogarithm, denoted as Li_s(z), is defined by the power series sum_{k=1 to infinity} z^k / k^s. Polylogs extend this concept by introducing multiple arguments and weights.

The most common definition of a polylog involves a sequence of arguments z_1, z_2, ..., z_n and a sequence of positive integer weights k_1, k_2, ..., k_n. A polylog is then defined as the sum:

Sum_{i_1 > i_2 > ... > i_m >= 1} (z_1^{i_1} / i_1^{k_1}) * (z_2^{i_2} / i_2^{k_2}) * ... * (z_m^{i_m} / i_m^{k_m})

where the sum is over all possible indices. Simpler forms exist, such as the common definition involving a single argument z and multiple weights k_1, ..., k_n:

Li_{k_1, ..., k_n}(z) = Sum_{i_1 > i_2 > ... > i_n >= 1} z^{i_1} / (i_1^{k_1} * i_2^{k_2} * ... * i_n^{k_n})

When all weights are 1, this reduces to the standard polylogarithm Li_n(z). Polylogs appear in various areas of mathematics and physics, particularly in quantum field theory, combinatorics, and number theory. They are related to other special functions like the Riemann zeta function and have complex analytic properties and functional equations. Investigating their properties is an active area of research.