The functional equation identity, (assuming also $\,f(-x)=-f(x)\,$ for all $\,x$),
$$ f(a)f(b)f(a\!-\!b) + f(b)f(c)f(b\!-\!c) + f(c)f(a)f(c\!-\!a) + f(a\!-\!b)f(b\!-\!c)f(c\!-\!a) = 0 \tag{1}$$
for all $\,a,b,c\,$ has solutions $f(x)=k_1\sin(k_2\,x)$ and $f(x)=k_1\tan(k_2\,x)\,$ with $\,k_1,k_2\,$ complex constants.
As a limiting case of both $\,\sin\,$ and $\,\tan,\,$ $\,f(x)=k_1x\,$ is also a solution, and the simplest.
I am looking only for non-zero solutions that have a formal power series expansion. That is,
$$ f(x) = a_1 \frac{x^1}{1!} + a_3 \frac{x^3}{3!} + a_5 \frac{x^5}{5!} + a_7 \frac{x^7}{7!}+ \cdots \tag{2}$$
is the exponential generating function for the sequence $\,(0,a_1,0,a_3,0,a_5,0,\dots).\,$ For a solution of the above functional equation, if $\,a_1=0\,$ then $\,f(x)\equiv 0.\,$ Otherwise, $\,a_1\ne 0\,$ and $\,a_3,a_5\,$ can be arbitrary while the rest of the coefficients are determined uniquely. I used Mathematica to compute the first few coefficients. I found, for example,
$$ a_7 \!=\! \frac{11 a_1 a_3 a_5\!-\!10 a_3^3}{a_1^2}, \;\;a_9 \!=\! \frac{21 a_1^2 a_5^2 \!+\!60 a_1 a_3^2 a_5 \!-\!80 a_3^4}{a_1^3},\;\;\dots. \tag{3}$$
I know of $18$ similar identities for $\,\sin\,$ and $\,\tan\,$ (including this one) in three or more variables. They have some common features as follows.
- Each is an irreducible homogeneous polynomial equated to zero where each monomial term in the polynomial is a product of factors each of which is of the form $\,f(x)\,$ where $\,x\,$ is a variable or an integer linear combination of variables.
- I also require that $\,f(x) = k_1x\,$ is a solution in which case the functional equation is a homogeneous algebraic identity.
As a non-example, the similar looking non-homogeneous functional equation
$$ f(a\!-\!b)\!+\!f(b\!-\!c)\!+\!f(c\!-\!a)\!-\! f(a\!-\!b)f(b\!-\!c)f(c\!-\!a)\!=\!0 \tag{4}$$
has only the non-zero solutions $\,f(x) = \tan(k_2x),\; k_2\ne0 \,$ and thus, does not qualify.
I am interested in those which are satisfied by both $f=\sin$ and $f=\tan$.
In all but one of the identities of this kind that I know of, they are also satisfied by the familfy of functions $\,f(x)=k_1\text{sn}(k_2\,x|m),\,$ where sn is a Jacobi elliptic function as well as two other related elliptic function sc, sd. The one exception is for an identity with Jacobi Zeta and Epsilon function solutions. This leads to two natural questions.
1. Do identities exist with solutions aside from the Jacobi functions mentioned?
2. Do identities exist with only sine and tangent solutions?
NOTE: Perhaps it would be easier to understand a specialization case. Suppose there is only one variable $\,a.\,$ Consider the polynomial ring $\,\mathbb{Z}[f_1,f_2,f_3,\dots].\,$ In the first functional equation $(1)$ replace $\,b\,$ with $\,2a,\,$ and $\,c\,$ with $\,-2a\,$ to get the equation
$$ f(a)f(3a)f(4a)-f(2a)^2f(4a)+f(a)f(2a)f(3a)-f(a)^2f(2a) = 0.\tag{5} $$
The polynomial equation associated with this equation is
$$ f_1f_3f_4-f_2^2f_4+f_1f_2f_3-f_1^2f_2 = 0 \tag{6}$$
where $\,f_n:=f(na).$ This single polynomial equation also has solutions $\,f(x)=k_1\text{sn}(k_2\,x|m)\,$ and seems to be the simplest such equation for the Jacobi sn function. There are an infinite number of other equations which come from specializing the first functional equation $(1)$. I conjecture that there is some kind of basis for the ideal of all such equations. The issues raised here are similar to the ones for my "Dedekind Eta-function Identities" list and studied by Ralf Hemmecke in his 2018 article "Construction of all polynomial relations among Dedekind eta functions of level N".
NOTE: The 18 identities I refer to are in my file Special Algebraic Identities (ident04.txt) along with hundreds of special algebraic identities (also available via the Wayback Machine).
