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In Sato's theory, the following formal delta function is defined:

$\delta(\lambda,z)=\frac{1}{\lambda}\sum_{n=-\infty}^\infty(\frac{z}{\lambda})^n=\frac{1}{z}\frac{1}{1-\lambda/z}+\frac{1}{\lambda}\frac{1}{1-z/\lambda}$

Given a function $f(z)=\sum a_iz^i$,

$f(\lambda)\delta(\lambda,z)=f(z)\delta(\lambda,z)$.

I want to know the properties as many as possible. Or useful references are welcome to be provided. Thanks!

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The formal delta function obeys the usual properties that the Dirac delta function does, but relative to the pairing defined by the residue. For instance, $$ \mathrm{Res}_z f(z)\delta(z,w) = f(w)$$ for any formal distribution $f(z)$.

This and more can be found in Kac's Vertex algebras for beginners, particularly Proposition 2.1 in §2.1.

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  • $\begingroup$ thanks for the answer, which is very helpful. I want to know whether the following relation is right, $\delta(-\lambda,z+\mu-\lambda)=???\delta(-\mu,z)$ Thanks again. $\endgroup$
    – Jack Cheng
    Commented May 6, 2010 at 12:29
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    $\begingroup$ I think so. Although formally $\delta(x,y)$ is not a "function" of $x-y$, for all practical purposes it is. $\endgroup$
    – José Figueroa-O'Farrill
    Commented May 6, 2010 at 13:25
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In addition to Kac's excellent book I thought I'd mention Vertex algebras and algebraic curves by Ed Frenkel and me, which goes a little more into formal delta functions, D-modules and the like --- see section 1.1 and chapter 19 (2nd edition) in particular.

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