Draft:Multiple polylogarithm

In mathematics, the multiple polylogarithm is multivariable generalization of the polylogarithm. For special cases of it's arguments, the multiple polylogarithm reduces to the normal polylogarithm.

Definitions
The multiple polylogarithms have numerous definitions. Not all definitions are equivalent, but they are all related. Just like the polylogarithms, the multiple polylogarithm can be defined as either a recursive integral, or a convergant power series.

Recursive Integral

Define $$I_{\gamma}(a_0;;z) = 1$$ and for $$n>0 $$,

$$I_{{\gamma}({a_0}\to{z})}(a_0;a_1,a_2, \ldots, a_n ; z) = \int_{{\gamma}({a_0}\to{z})} \frac{d\xi}{\xi - a_n}I_{{\gamma}({a_0}\to{\xi})}(a_0;a_1, a_2, \ldots, a_{(n-1)} ; \xi) $$.

Where $${\gamma}({a_0}\to{z})$$ denotes a path from $$a_0$$ to $$z$$, and $${{\gamma}({a_0}\to{\xi})}$$ denotes travelling along that same path to a midway point $$\xi$$. Often the subscript specifying the path is dropped.

Convergant Power Series

$$\mathrm{Li}_{m_1,\ldots,m_k}(z_1,\ldots,z_k)=\sum_{0<n_1<n_2<\cdots<n_k}\frac{z_1^{n_1}z_2^{n_2}\cdots z_k^{n_k}}{n_1^{m_1}n_2^{m_2}\cdots n_k^{m_k}}$$.

We note that this power series definition allows us a natural generalization of the known identity between the classical polylogarithm and the Riemann zeta function, $$\left(\text{Li}_{n}(1)= \zeta(n)\right)$$, by invoking the multiple zeta function:

$$\mathrm{Li}_{m_1,\ldots,m_k}(1,\ldots,1)=\zeta(m_1,\ldots,m_k)$$.

Properties
The recursive integral definition for integration beginning at a base-point $$a_0$$ can be broken up into sums and products of integrations beginning at $$0$$. For example:

$$ I_{\gamma_1}(a_0; a_1, a_2; a_3) = I_{\gamma_2}(0; a_1, a_2; a_3) - I_{\gamma_3}(0; a_1, a_2; a_0) - I_{\gamma_3}\left[ I_{\gamma_2 }(0; a_2; a_3) - I_{\gamma_3}(0; a_2; a_0) \right]. $$

Where $$\gamma_1$$ is a path going $$(a_0 \to a_3)$$, $$\gamma_2$$ is from $$(0 \to a_3)$$, $$\gamma_3$$ is from $$(0 \to a_0)$$, and the loop formed by traversing $$\gamma_1, -\gamma_2,\text{ then }\gamma_3$$ does not contain $$a_1$$ or $$a_2$$.