Changing order of summation including a min in the summation

Question

Lets say I have the following expression:

$$ h(x) = \sum_{k=1}^n \sum_{v=1}^{\min\{k,j\}} \frac{(-1)^{n-k}k!}{(k-v)!} {n \brack k}f(x)^{k-v} B_{n,v}^f(x) $$

Now my goal is to have the $v$ summation come before the $k$ summation, how would I go about doing this?

Answer 1

$h(x) = \sum_{k=1}^n \sum_{v=1}^{\min\{k,j\}} \frac{(-1)^{n-k}k!}{(k-v)!} {n \brack k}f(x)^{k-v} B_{n,v}^f(x) $

If $n \le j$, since $k \le n$, the sums become $\sum_{k=1}^n \sum_{v=1}^{k} = \sum_{v=1}^{n} \sum_{k= v}^n $.

If $n > j$, the sums could be split into two parts like this:

\begin{align} \sum_{k=1}^n \sum_{v=1}^{\min\{k,j\}} &=\sum_{k=1}^j \sum_{v=1}^{k} +\sum_{k=j+1}^n \sum_{v=1}^{j}\\ &= \sum_{v=1}^{j}\sum_{k=v}^j +\sum_{v=1}^{j} \sum_{k=j+1}^n \end{align}