compute a double series

Question

I am trying to simplify this one to a single summation form:

$\sum_\limits{n=0}^{\infty}\sum_\limits{j=0}^{\infty}\frac{(-1)^{n+j-1}}{j!n!}\sigma^n y^{n+j-1}$

I have no idea of how to do that,maybe someone can give me some help or hint?thanks!

Answer 1

Both series are the Taylor series of $e^{x}$. Then $$\sum_{n\geq0}\sum_{j\geq0}\frac{\left(-1\right)^{n+j-1}}{n!j!}\sigma^{n}y^{n+j-1}=\sum_{n\geq0}\frac{\left(-\sigma y\right)^{n}}{n!}\sum_{j\geq0}\frac{\left(-y\right)^{j-1}}{j!}=\frac{e^{-\sigma y}e^{-y}}{y}=\frac{e^{-y\left(\sigma+1\right)}}{y}.$$ You can write it as a single series using again the Taylor series $$\frac{e^{-y\left(\sigma+1\right)}}{y}=\frac{1}{y}\sum_{k\geq0}\frac{\left(-1\right)^{k}y^{k}\left(\sigma+1\right)^{k}}{k!}.$$