Parity between TeX `sum` and `prod` snippets

[kiritofeng]

I've noticed that the snippet for sum is

snippet sum "sum" w
\sum_{n=${1:1}}^{${2:\infty}} ${3:a_n z^n}
endsnippet

while the snippet for product is

snippet prod "product" w
\prod_{${1:n=${2:1}}}^{${3:\infty}} ${4:${VISUAL}} $0
endsnippet

Would it be possible to get parity between the two, so the new sum snippet reads either

snippet sum "sum" w
\sum_{${1:n=${2:1}}^{${3:\infty}} ${4:${VISUAL}} $0
endsnippet

or (less preferably)

snippet sum "sum" w
\sum_{${1:n=${2:1}}^{${3:\infty}} ${4:a_n z^n} $0
endsnippet

should you want to keep a_n z^n as the default?
In particular, this also has the side benefit of making it easier to change the iteration variables for a sum.