# Help interpret notation $\sum_{j_1+j_2+\cdots+j_m=n}a^{j_1}_1a^{j_2}_2\cdots a_m^{j_m}$

I read somewhere that this sum can be written as: $$\sum_{r+s=n}a_rb_s=\sum_{r=0}^na_rb_{n-r}\tag1$$ This means to create all possible orders of $$(r,s)$$ and add these together.

Now, my question is how do you write this summation in terms of the RHS above:

## $$\sum_{j_1+j_2+\cdots+j_m=n}a^{j_1}_1a^{j_2}_2\cdots a_m^{j_m}=?\tag2$$

• It just means that you sum the value of $a_1^{j_1}\dots$ for all ordered lists $j_1,\dots,j_m$ such that $j_1+\dots+j_m=n$. – Benedict Randall Shaw Nov 3 '18 at 13:25
• @BenedictRandallShaw yes, but I want the notation to be (or $(2)$ be written) like the RHS of $(1)$ – John Glenn Nov 3 '18 at 13:27
• That's going to take multiple summation signs. Are you OK with that? – Arthur Nov 3 '18 at 13:30
• @JohnGlenn Why do you want that? Curiosity? To help program it? For an assignment? Would you allow multiple summation symbols with an ellipsis between them or do you just want a single one (or any fixed number?)? All of this context would help. – Mark S. Nov 3 '18 at 13:31
• @Arthur It only requires multiple summation signs if you restrict yourself to using the exponents as the indices of summation. – Mark S. Nov 3 '18 at 16:39

For $$m=3$$,
$$\sum_{r=0}^n\sum_{s=0}^{n-r}a_rb_s c_{n-r-s}.$$
For $$m=4$$,
$$\sum_{r=0}^n\sum_{s=0}^{n-r}\sum_{t=0}^{n-r-s}a_rb_s c_td_{n-r-s-t}.$$