Theorem 3.1.4 Liouville’s Identity.
Let \(F(x,y,z)\) defined on integer triples \((x,y,z)\in\Z\) such that:
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\(F\) is odd in the variable \(x\text{:}\)\begin{equation*} F(-x,y,z)=-F(x,y,z) \, . \end{equation*}
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\(F\) is even in the pair \((y,z)\text{:}\)\begin{equation*} F(x,-y,-z)=F(x,y,z) \, . \end{equation*}
For every \(n\in\N\text{,}\) Liouville’s identity states:
\begin{equation*}
\sum_{a^2+bc=n}{\big[ 2F(c^2a,a+b,2a+2bc) \big]} = F(b + c, a, b-c) + 2T_1(n) + 2T_2(n)
\end{equation*}
where the summation is over all ordered triples \((a,b,c)\) of integers satisfying \(a^2+bc=n\) and
\begin{gather*}
T_1(n)=\sum_{j=1}^{n-1}F(j,j,-j) \quad \text{(with } j \text{ is odd)} \, ,\\
T_2(n)=\sum_{j=1}^{n-1}F(2,j,2j) \quad \text{(with } j \text{ is odd)} \, .
\end{gather*}
Remark 3.1.5.
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Discrete symmetry-based transformation formula.
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Leverages the parity properties of \(F\) to simplify sums over all possible solutions to sums over a subset of possible solutions of the quadratic Diophantine equations.
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Used by Josiah Gibbs in his derivation of Lioville’s theorem.
Warning 3.1.6.
Make sure to fact-check this statement.
