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A Hypothesis

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Post Sun Nov 13, 2016 5:09 am

Posts: 104
(Hypothesis) For integer $m\geq 1, l_j>0$ and $x_j\in (-1,1)$, then the following identity whether or not to established
\[\prod\limits_{j = 1}^m {{\rm Li}_{{l_j}}\left[ {{x_j}} \right]}=\sum\limits_{k = 0}^m {\sum\limits_{\left( {{i_1},{i_2}, \cdots ,{i_k}} \right) \in \left\{ {1,2, \cdots ,m} \right\},{i_1} < {i_2} < \cdots < {i_k}} {{{\left( { - 1} \right)}^{m - 1 - k}}S\left[ {\left. {\begin{array}{*{20}{c}}
{{l_{{i_1}}}, \cdots ,{l_{{i_k}}}} \\
{{x_{{i_1}}}, \cdots ,{x_{{i_k}}}} \\
\end{array}} \right|\begin{array}{*{20}{c}}
{\sum\limits_{j = 1}^m {{l_j}} - \sum\limits_{j = 1}^k {{l_{{i_j}}}} } \\
{{x_1} \cdots {x_m}/{x_{{i_1}}} \cdots {x_{{i_k}}}} \\
\end{array}} \right]} } \]

with $S\left[ {\left. {\begin{array}{*{20}{c}}
\emptyset \\
\emptyset \\
\end{array}} \right|\begin{array}{*{20}{c}}
l \\
x \\
\end{array}} \right]: = {\rm Li}_l\left[ x \right]$, where ${\rm Li}_p\left[ x \right] $ denotes the q-polylogarithm function defined by
\[{\rm Li}_p\left[ x \right]: = \sum\limits_{n = 1}^\infty {\frac{{{x^n}}}{{\left[ n \right]_q^p}}} ,\]
and the sums $S\left[ {\left. {\begin{array}{*{20}{c}}
{{l_1}, \cdots ,{l_{m - 1}}} \\
{{x_1}, \cdots ,{x_{m - 1}}} \\
\end{array}} \right|\begin{array}{*{20}{c}}
{{l_m}} \\
{{x_m}} \\
\end{array}} \right]$ are defined by
\[S\left[ {\left. {\begin{array}{*{20}{c}}
{{l_1}, \cdots ,{l_{m - 1}}} \\
{{x_1}, \cdots ,{x_{m - 1}}} \\
\end{array}} \right|\begin{array}{*{20}{c}}
{{l_m}} \\
{{x_m}} \\
\end{array}} \right]: = \sum\limits_{n = 1}^\infty {\frac{{{\zeta _n}\left[ {{l_1},{x_1}} \right]{\zeta _n}\left[ {{l_2},{x_2}} \right] \cdots {\zeta _n}\left[ {{l_{m - 1}},{x_{m - 1}}} \right]}}{{\left[ n \right]_q^l}}x_m^n}, \]
here the finite sums ${\zeta _n}\left[ {l,x} \right]$ is called the partial sum of q-polylogarithm function defined by
\[{\zeta _n}\left[ {l,x} \right] = \sum\limits_{j = 1}^n {\frac{{{x^j}}}{{\left[ j \right]_q^l}}} ,\]
and the non-negative integer $[n]_q$ is defined as
\[{\left[ n \right]_q}: = \frac{{1 - {q^n}}}{{1 - q}},\;0 < q < 1.\]
I have proved that when $m\le 5$ ,the hypothesis is hold. However, how to prove the general solution?

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