Yesterday I wrote a post that looked at the hyperbolic tangent sum

x oplus y = tanhleft(tanh^{-1}x + tanh^{-1} yright) = frac{x + y}{ 1 + xy}

for x and y strictly between −1 and 1. This sum arises when adding velocities in special relativity. The post ended with a description of the expression for

x_1 oplus x_2 oplus x_3 oplus cdots oplus x_n

in terms of elementary symmetric polynomials but did not offer a proof. This post will give a proof and show why elementary symmetric polynomials arise naturally.

We start by noting

tanh x = frac{exp(x) - exp(-x)}{exp(x) + exp(-x)} = frac{exp(2x) -1}{exp(2x) + 1}

and

tanh^{-1} x = frac{1}{2} log frac{1+x}{1-x}

Then

begin{align*} bigoplus_{i=1}^n x_i &= tanhleft( sum_{i=1}^n tanh^{-1}(x_i) right) \ &= tanhleft( frac{1}{2} sum_{i=1}^n log frac{1+x_i}{1 - x_i} right) \ & = tanhleft( frac{1}{2} sum_{i=1}^n log(1 + x_i) - frac{1}{2} sum_{i=1}^n log(1 - x_i)right) \ &= frac{prod_{i=1}^n (1 + x_i) - prod_{i=1}^n (1 - x_i)}{prod_{i=1}^n (1 + x_i)+prod_{i=1}^n (1 - x_i)} end{align*}

Now

prod_{i=1}^n (1 + x_i) = e_0 + e_1 + e_2 + cdots + e_n

where

begin{align*} e_0 &= 1 \ e_1 &= sum_{i=1}^n x_i \ e_2 &= sum_{i=1}^n sum_{j = i+1}^n x_ix_j \ phantom{e_1} &phantom{=}cdots \ e_n &= x_1x_2x_3 cdots x_n end{align*}

are the elementary symmetric polynomials.

We get e0 by choosing the 1 term from each binomial in the product. We get e1 by choosing the 1 term from all but one of the binomials and choosing an x as the remaining term. We get e2 by choosing the 1 term from n − 2 of the binomials and choosing x‘s from the two remaining terms, and so on. Finally, we get en by choosing an x from each binomial.

Similarly

prod_{i=1}^n (1 - x_i) = e_0 - e_1 + e_2 - cdots +(-1)^n e_n

Therefore

begin{align*} frac{prod_{i=1}^n (1 + x_i) - prod_{i=1}^n (1 - x_i)}{prod_{i=1}^n (1 + x_i)+prod_{i=1}^n (1 - x_i)} &= left. sum_{i text{ odd}} e_i,,middle/sum_{i text{ even}} e_i right. end{align*}

because the even terms cancel out in the numerator and the odd terms cancel out in the denominator.

In words, the hyperbolic tangent sum of multiple arguments is the ratio of the sums of the odd and even elementary symmetric polynomials in the arguments.

Elementary symmetric polynomials enter the derivation because they are what you get when you expand products of (1 + xi).

The post Tanh and elementary symmetric polynomials first appeared on John D. Cook.

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