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!

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!