How to change index in summation

Question

Pease help me understand how they have changed index of summation from r to n here. If we take $$n = r-s$$ how n is changing from -$\infty $ to $\infty$

Answer 1

Take $f(x,t,s,r)= (-1)^s(x/2)^{r+s} t^{r-s}/(r!s!)$.

Change the order of summation, substitute $n=r+s$, then change the order again

$$\begin{align}\tag 1 g(x,t) &= \sum_{r=0}^{\infty} \sum_{s=0}^\infty f(x,t,s,r)\\&=\sum_{s=0}^\infty \sum_{r=0}^\infty f(x,t,s,r)\\&=\sum_{s=0}^\infty\sum_{n=-s}^\infty f(x,t,s,n+s)\\\tag 2&=\sum_{n=-\infty}^{\infty}\sum_{s=0}^\infty f(x,t,s,n+s)\end{align}$$

Answer 2

The possible values for $r$ are $0,1,2,\dots,\infty$, and the same for $s$. In special, when $r=0$, $n$ varies from $-\infty$ to $0$. When $r=1$, $n$ varies from $-\infty$ to $1$, and so on from $r\to\infty$.