Notation for special summations

[epkid08]

Is there special notation for a function like this:
$$g(x_1)=\sum^k_jf(x)$$

$$g(x_2)=\sum^p_k\sum^k_jf(x)$$

$$g(x_3)=\sum^s_p\sum^p_k\sum^k_jf(x)$$

If so, what would g(x_n) be?

 

[epkid08]

I forgot to add, a function like this:

$$f(x_n)=\sum^{\sum^{\sum^{\sum^{\sum}}}}...$$

(the limit of each sum is another sum; fallowing the pattern in the above post of course)

 

[Ben Niehoff]

For a small, finite number of indexes, sometimes I see the shorthand

$$\sum_{i,j,k} f(x)$$

This usually occurs when all of the indexes run from 0 to the same limit N.

If you have a variable number of indexes, then you can "index the indexes" as follows:

$$S_n = \sum_{i_1 \dots i_n} f(x)$$

Oh, wait, I see you want each index to run up to the previous index. For a small number of indexes, you can do

$$S_3 = \sum^N_{i > j > k} a_{ijk}(x)$$

And in general, you could probably write

$$S_n = \sum^N_{i_1 > \dots > i_n} a_{i_1 \dots i_n}(x)$$