Limits in double sum under redefinition of indices 
*

*Suppose I have a double sum of the form: $$ \sum_{n=1}^{\infty} \sum_{k=-\infty}^{\infty} p(k)q(n+k) $$ How do the limits change under the redefinition: $n+k \rightarrow n$ where $n, k$ take integer values?


*Is there any way at all to write the resulting double sum as: $$\sum_{n=1}^{\infty} \sum_{k=\text{something}} p(k) q(n) $$ I might be missing something trivial but any help would be much appreciated.
 A: Your difficulty likely stems from the fact that you're using the same symbol $n$ for two different purposes. If you would like the final sum to involve $n$, as a first step rewrite the sum using a new symbol, say
$$
\sum_{m=1}^{\infty} \sum_{k=-\infty}^{\infty} p(k)\, q(m+k)
$$
Now, you want $n = m + k$, so $m = n - k$. The problem is that the index of the outer sum will now depend on $k$. In particular, $m=1$ becomes $n = 1 + k$, and $m \to \infty$ becomes $n \to \infty$. If the original sum still converges with the limits reversed, then we obtain:
\begin{align}
\sum_{m=1}^{\infty} \sum_{k=-\infty}^{\infty} p(k)\, q(m+k) 
&= \sum_{k=-\infty}^{\infty} \sum_{m=1}^{\infty} p(k)\, q(m+k) \\
&= \sum_{k=-\infty}^{\infty} \sum_{n=1+k}^{\infty} p(k)\, q(n) 
\end{align}

Alternatively, we can leave the summation in the order its given, and initially rename $k$, say as $j$, so we have
$$
\sum_{n=1}^{\infty} \sum_{j=-\infty}^{\infty} p(j)\, q(n+j).
$$
Now, set $k = n + j$, so $j = k - n$, and
$$
\sum_{n=1}^{\infty} \sum_{j=-\infty}^{\infty} p(j)\, q(m+j) 
= \sum_{n=1}^{\infty} \sum_{k=-\infty}^{\infty} p(k - n)\, q(k).
$$
This avoids having to swap the order of summation, but your summands are not as simple. So this is likely not any more useful for you.
