Functional equations satisfied by both sine and tangent functions.

Question

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.

1. 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.
2. 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?

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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).

Answer 1

Recently, while doing research on Heronian spherical triangles, I made an interesting observation about my identity $\texttt{id2_8_1_4}$ which I added over the summer of $2024$. The context is a sequence of numbers such that if $w_0,w_1,w_2,w_3,w_4$ are any five consecutive terms of the sequence then

$$ 0 = (w_4 w_3-w_1 w_0)(w_3 w_1-w_2 w_2) + (w_4 w_0-w_3 w_1)(w_2 w_1-w_3 w_2) \tag1 $$

This is also true if the indices of the five terms form an arithmetic progression. The two products expanded yields the eight terms of my identity

\\ {} 2 vars with 8 terms highest degree 1 of total 4 [JE,RF]
{ id2_8_1_4 = +(a-b-b)*(a-b)*(a-b)*(a+b) -(a-b-b)*(a-b)*a*a
              -(a-b-b)*(a-b)*a*(a+b+b) +(a-b-b)*a*(a+b)*(a+b+b)
              +(a-b)*(a-b)*a*(a+b) -(a-b)*a*(a+b)*(a+b)
              -(a-b)*(a+b)*(a+b)*(a+b+b) +a*a*(a+b)*(a+b+b) ; } 

The functional equation associated with this identity has the property that the reciprocal of any solution is also a solution. Suppose that we drop the assumption that $f(-x)=-f(x)$. Then the functional equation has solutions $f(x) = k_1x+k_2$ where $k_2\ne0$ is now allowed. Furthermore, it has not only sine and tangent solutions, but also cosine solutions and their reciprocals. For example, $f(x) = k_1\sec(k_2+k_3x)$. Also, all twelve Jacobi elliptic functions are solutions. I think it is a truly remarkable functional equation. This now answers my first question in the affirmative (assuming that $f(-x)=-f(x)$ is not required and $f(x)$ is allowed to be a formal Laurent series with negative powers of $x$).

1. Do identities exist with solutions aside from the Jacobi functions mentioned?

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At the end of Jan 2025 I found another identity similar to my older $\texttt{id2_8_1_4}$. It is

\\ {} 2 vars with 24 terms highest degree 1 of total 4 [JE,RF,ZS]
{ id2_24_1_4 =
 -(a-b-b-b)*(a-b-b)*(a-b)*(a+b)
 +(a-b-b-b)*(a-b-b)*(a-b)*(a+b+b)
 +(a-b-b-b)*(a-b-b)*a*a
 -(a-b-b-b)*(a-b-b)*a*(a+b+b)
 -(a-b-b-b)*(a-b-b)*a*(a+b+b+b)
 +(a-b-b-b)*(a-b-b)*(a+b)*(a+b+b+b)
 -(a-b-b-b)*(a-b)*a*a
 +(a-b-b-b)*(a-b)*a*(a+b)
 +(a-b-b-b)*(a-b)*a*(a+b+b+b)
 -(a-b-b-b)*(a-b)*(a+b+b)*(a+b+b+b)
 -(a-b-b-b)*a*(a+b)*(a+b+b+b)
 +(a-b-b-b)*a*(a+b+b)*(a+b+b+b)
 +(a-b-b)*(a-b)*a*(a+b)
 -(a-b-b)*(a-b)*a*(a+b+b)
 -(a-b-b)*a*a*(a+b)
 +(a-b-b)*a*(a+b)*(a+b+b)
 +(a-b-b)*a*(a+b+b)*(a+b+b+b)
 -(a-b-b)*(a+b)*(a+b+b)*(a+b+b+b)
 +(a-b)*a*a*(a+b+b)
 -(a-b)*a*(a+b)*(a+b+b)
 -(a-b)*a*(a+b)*(a+b+b+b)
 +(a-b)*(a+b)*(a+b+b)*(a+b+b+b)
 +a*a*(a+b)*(a+b+b+b)
 -a*a*(a+b+b)*(a+b+b+b)
; }

The factors are seven consecutive terms of an arithmetic progression. I found it by studying the sequence of $x$ coordinates of multiples of a point of an elliptic curve. Any seven consecutive terms of these sequences satisfy a homogeneous quartic polynomial relation which comes from the new identity. This also implies that the Weierstrass $\wp$ function satisfies the corresponding functional equation.

When I first considered the corresponding functional equations of my identities it seemed natural to restrict the functions as in equation $(2)$ of my question. With the examples of Weierstrass $\wp$ and $\zeta$ functions I now consider it better to expand the functions allowed to meromorphic ones also.