An arithmetic formula of Liouville (Q2433588)

In his famous series of eighteen articles, Liouville presented without proof numerous arithmetic formulas. These formulas are summarized in Volume 2 (Chapter XI) of [\textit{L. E. Dickson}, History of the theory of numbers. Vol. I--III. Reprint of the 1920-1923 originals. New York, NY: Chelsea Publishing Co. (1966; Zbl 0958.11500)]. The present authors are interested in the following identity. Let \(n\) be a positive odd integer, and let \(F\) be an odd complex-valued function on the set of integers. Then \[ \begin{aligned} \sum_{ax+by+cz=n\atop a, b, c, x, y, z\;\text{odd}} (F(a+b+c)+F(a-b-c)-F(a+b-c)-F(a-b+c)) ={1\over 8},\\ \sum_{d| n}(d^2-1)F(d)-3\sum_{ax<n\atop a, x\;\text{odd}} \sigma(o(n-ax))F(a),\text{ where }o(n)=n/2^s,\;2^s\| n. \end{aligned} \] The authors have not been able to locate a proof of this result in the literature, although they note that [\textit{P. S. Nazimoff}, Applications of the theory of elliptic functions to the theory of numbers. Chicago: University Bookstore (1928; JFM 54.0196.01)] indicates how an analytic proof can be given. The purpose of this paper is to present the first proof and an entirely elementary proof using only rearrangements of terms in finite sums. As an application of this formula the authors give a formula for the number of representations of a positive integer as the sum of twelve triangular numbers.