Addition theorems for Dixon functions

The last couple blog posts have been about Dixon elliptic functions, functions which are analogous in some ways to sine and cosine functions. Whereas sine and cosine satisfy a Pythagorean identity

$\sin^2(z) + \cos^2(z) = 1$

the Dixon functions sm and cm satisfy what you might call a Fermat identity

$\text{sm}^3(z) + \text{cm}^3(z) = 1$

alluding to Fermat’s last theorem.

The functions sm and cm also satisfy addition identities, but these look very different than the addition identities for sine and cosine.

$\begin{align*} \text{sm}(x + y) &= \frac { \text{sm}^2(x)\,\text{cm}(y)- \text{cm}(x)\,\text{sm}^2(y)} { \text{sm}(x)\,\text{cm}^2(y)- \text{cm}^2(x)\,\text{sm}(y)} \\ & \\ \text{cm}(x + y) &= \frac { \text{sm}(x)\,\text{cm}(x)- \text{sm}(y)\,\text{cm}(y)} { \text{sm}(x)\,\text{cm}^2(y)- \text{cm}^2(x)\,\text{sm}(y)} \\ \end{align*}$

Once you’ve seen the binomial theorem and the addition identities for trig functions, you might come away with the impression that it is common to be able to simply relate the value of a function at x + y to its values at x and at y. It is not.

There are only three classes of functions that satisfy addition theorems. (See this post for a precise definition of what is meant by an addition theorem.) And once you’ve seen the binomial theorem and the sum angle identities, you’ve seen representatives of two of the three classes. The three classes of functions with addition theorems for functions of z are

Rational functions of z
Rational functions of exp(λz)
Elliptic functions of z

The binomial theorem is an example of the first category and sum angle identities are examples of he second category (with λ = i). Dixon functions are examples of the third category.