I am attempting to find a closed form solution or a nice series for $x^{x+1}=(x+1)^{x}$. First of all I looked at $a^b=b^a$. Fixing a, this means finding out when $a^{1/a} = x^{1/x}$. $f(x)=x^{1/x}$ is an interesting function. It has a maximum at e. (Proof due to Jakub Steiner.) $x^{1/x}$ goes to 1 as x goes to infinity. Based on that, I can conclude that $a^{1/a} = x^{1/x}$ has non-trivial solutions when x > 1 and that when a > e the non-trivial solution is between 1 and $e^{1/e}$.

I thought this was pretty neat, but it isn't helping me to solve the original problem. So, I would like to know is there a name, given a, for the value of b (not equal to a) such that $a^b = b^a$? For example, we could call b the "switch-ponent" of a but there must be a better name than that. Anyone know what these are called or maybe a name for the original problem?