6
$\begingroup$

Some will recognize this as similar to a question I asked before, but I want to ask it without the trigonometry.

Let $e_k$ be the $k$th-degree elementary symmetric polynomial in $x_1,x_2,x_3,\ldots$. If $k$ is more than the number of $x$s, then $e_k$ is the sum of no terms and is $0$. From one POV, the following Pythagorean identities are as elementary as anything not in the high-school curriculum: $$ (e_0+e_2+e_4+\cdots)^2 - (e_1+e_3+e_5+\cdots)^2 = (1-x_1^2)(1-x_2^2)(1-x_3^2)\cdots $$ $$ (e_0-e_2+e_4-\cdots)^2 + (e_1-e_3+e_5-\cdots)^2 = (1+x_1^2)(1+x_2^2)(1+x_3^2)\cdots $$ So are these "known" in the sense of being in refereed publications one could cite? And if not, are they "known" in the sense that some people see them mentioned or explicitly used from time to time?

(And if there should happen to be infinitely many $x$s, could this still be considered only algebra by thinking of these as a sort of "formal" series?)

Improve this question
$\endgroup$
1
  • $\begingroup$ What's the complete homogeneous symmetric function analogue? I might recognize that more easily. I get the feeling this is standard or at least a routine application of something standard in symmetric function theory. The right hand sides look like the squares of Cauchy kernels. $\endgroup$
    – Alexander Woo
    Commented Feb 13, 2013 at 0:42

1 Answer 1

Reset to default
10
$\begingroup$

These are both simple corollaries of $$\sum_{k\geq 0} t^ke_k(x_1,x_2,\dots)=\prod_{k\geq 0}(1+tx_k).$$

There is a typo in both your identities. They should read, $$(e_0+e_2+\cdots)^2-(e_1+e_3+\cdots)^2=(e_0+e_1+e_2+\cdots)(e_0-e_1+e_2-\cdots)$$ $$=\prod_{k\geq 0}(1+x_k)\prod_{k\geq 0}(1-x_k)=(1-x_1^2)(1-x_2^2)\cdots,$$ and $$(e_0-e_2+e_4-\cdots)^2+(e_1-e_3+e_5-\cdots)^2=(\mathfrak {Re}[\prod_{k\geq 0}(1+ix_k)])^2+(\mathfrak{Im}[\prod_{k\geq 0}(1+ix_k)])^2$$ $$=(1+x_1^2)(1+x_2^2)\cdots$$

Improve this answer
$\endgroup$
4
  • $\begingroup$ Can you cite a book or refereed paper that states the identity of which these are corollaries? $\endgroup$
    – Michael Hardy
    Commented Feb 13, 2013 at 3:55
  • 1
    $\begingroup$ Look at chapter 1 in Macdonald's book on symmetric functions. Though, that identity is very close to the definition of the elementary symmetric polynomials :) $\endgroup$
    – Gjergji Zaimi
    Commented Feb 13, 2013 at 3:59
  • $\begingroup$ The odd thing is that when the $x$s are sines and the first identity generalizes $1-\sin^2\theta=\cos^2\theta$, or the $x$s are tangents and the second identity generalizes $1+\tan^2\theta=\sec^2\theta$, the validity of the identities in no way depends on the fact that the circle is parametrized by arc length rather than in some other way, but yet the way I found these identities was by deriving them from an identity one side of which was $\tan(\theta_1+\theta_2+\theta_3+\cdots)$. $\endgroup$
    – Michael Hardy
    Commented Feb 13, 2013 at 4:01
  • $\begingroup$ To be more specific, this is Equation 2.2 in Section 2 of Chapter I, found on p. 19 of the 2nd edition. $\endgroup$
    – Alexander Woo
    Commented Feb 13, 2013 at 22:33

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .