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Is there a standard name or standard symbol for the binary operation that combines $x$ and $y$ to give $xy/(x+y)$, or equivalently $1/(1/x+1/y)$? (At least the expressions are equivalent if we ignore the case where $x$ or $y$ is 0.)

One possible name that comes to mind is "harmonic sum", but some might say that the harmonic sum of $x$ and $y$ should be defined as $1/x+1/y$.

Another possibility is "cosum" or "co-sum", since the relationship between this operation and ordinary addition is analogous to (and indeed tropicalizes to) the relationship between the operations min and max.

Anyway, in addition to knowing what to call this operation, I'd also like to know how to write it, if there is some existing notation for it.

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Harmonic mean, up to a constant factor. –  Mariano Suárez-Alvarez Jun 29 '13 at 4:18
Harmonic half-mean? –  Włodzimierz Holsztyński Jun 29 '13 at 6:08
Harmonic sum seems like a fine term by analogy with harmonic mean. –  Qiaochu Yuan Jun 29 '13 at 7:41
A good symbol would be $x\parallel y$, since $1/(\frac 1x+ \frac 1y)$ is the resistance of a parallel connection of resistances $x$ and $y$. –  Richard Stanley Jun 29 '13 at 13:16

2 Answers 2

$ x +^{-1} y $ seems like a good notation in that $x^{-1}+^{-1} y^{-1}=(x+y)^{-1} $, and $$ \frac{1}{x}\frac{1}{+}\frac{1}{y} = \frac{1}{x+y} $$ (is this the "Freshman's Dream" in another incarnation?)

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up vote 0 down vote accepted

I like Richard Stanley's suggested notation best, and I plan to use the term "harmonic sum" (unless someone points out existing notation and/or terminology).

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@Joel: Thanks for pointing out that Richard Stanley's suggested notation is already in, er, current use. :-) –  James Propp Jul 2 '13 at 12:40
The suggested notation has a lot of potential. –  Joel Reyes Noche Jul 2 '13 at 23:24
In LaTeX, you should use x\mathbin\parallel y instead of the more obvious x\parallel y, so that you get the spacing of a binary operator as opposed to a relation symbol (You can say \def\hsum{\mathbin\parallel} in your prelude, for example) –  Mariano Suárez-Alvarez Jul 8 '13 at 4:24
Probably this won't be an issue, but just so you're aware of it, in number theory the notation $p^k\mathbin\parallel n$ is often used to indicate that $p^k$ divides $n$, but $p^{k+1}$ does not divide $n$, i.e., it's a quick way to indicate that $k$ is $\text{ord}_p(n)$, the (normalized) $p$-adic valuation of $n$. –  Joe Silverman Aug 15 '13 at 2:32

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