# Finding the Inverse of a Summation

I have seen more specific versions of this question but my question is more general. For any given summation does there exist an inverse. If not, how does one tell if the function has an inverse. Do these inverses always have closed forms (I imagine they do not)? How can one tell when a function such as a summation or its inverse has a closed form and if they do not how would one write them?

This question is motivated by a function I ran across of the form $f(x)=\sum_{a=1}^{x} \sum_{b=1}^{a} b^{b}$ that then required the use of its inverse. I have thus far failed to write the function or its inverse in closed form. Thanks for any insight you can provide!

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If the innermost sum of your $f(x)$ doesn't even seem to have a closed form, it seems even less likely that $f(x)$ itself has one... –  Guess who it is. May 10 '12 at 17:16
I agree that was my initial thought. So how would you write its inverse other then just writing $f^{-1}$ or even prove that it has one? –  Jesse Stern May 10 '12 at 17:19
Also, you are going to have a hard time coming up with a closed form for the inverse of $g(x)=x^x$, so already you are dealing with a "difficult" function on the inside of your sum. Not that it helps to get an inverse for $g$ to get an inverse for your $f$, just that these sorts of functions tend to be problematic. –  Thomas Andrews May 10 '12 at 17:33

For any given summation does there exist an inverse.

The summation of integers with variable upper limit of the sum produces a map $f:\mathbb N\to\mathbb Z$. In the special case under consideration, where all integers being summed are positive, we have a map $f:\mathbb N\to\mathbb N$ which is strictly increasing, and therefore injective. As such, it has an inverse $g=f^{-1}$, which is defined on the set $f(\mathbb N)$, the range of $f$.

Typically, the range of $f$ is not all of $\mathbb N$. For this reason, it is practical to relax the notion of inverse as follows (assuming $f$ is strictly increasing, as above): $$g(y)=\max\{x:f(x)\le y\} \tag1$$ This $g$ agrees with $f^{-1}$ where the latter is defined.

How can one tell when a function such as a summation or its inverse has a closed form

Easily: by drawing from the vast expertise contained in the OEIS and computer algebra systems. There is no realistic hope for explicit form of the inverse function (or the direct function) in your case.

if they do not how would one write them?

Just as a sum (or maximum/minimum as in (1)).

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