Non- linear recurrent relation (exponential term)

Question

Is there any solution to this recurrent relation: $X_n=\alpha-e^{-\beta X_{n-1}}$, $X_0=0$,

$\alpha>1$ and $\beta>0$

Answer 1

We would likely expect no closed form solution to exist in terms of standard functions; the main reason for this is because, in the special case where $\alpha=0$ and $\beta=1$, we have $$X_0=0$$ $$X_1=-1$$ $$X_2=-e$$ $$X_3=-e^{e}$$ $$X_3=-e^{e^e}$$ and so on. This is essentially the same as tetration, but that causes an issue, since there's no particularly well-accepted way to define tetration in any closed form (for instance, there are various ways to choose smooth functions that satisfy the recurrence).

If we want to be particularly rigorous, the rate of growth that this function possesses is a problem for any elementary function I can think of. If we considered the smallest set of functions $F$ in $\mathbb{N\rightarrow N}$ such that

• Any composition of a pair of functions in $F$ is in $F$.
• Any function of the form $g(n)=\sum_{i=0}^{m(n)}f(n)$ for $f,m\in F$ has $g\in F$. Similarly for products.
• Any power or quotient of two functions in $F$ is in $F$.
• The identity, logarithm, factorial, tec. - the sort of functions you'd encounter in a simple calculus class - are in $F$.

So, basically, $F$ is the set of functions which have some form in terms of algebraic operations, summations, and "elementary" functions. So, stuff like $$f(n)=\sum_{i=0}^{2^n}e^{n!}$$ would be in this class. The problem is that no such function can grow faster that $O(n^{n^{\cdot^{\cdot^{\cdot^{n}}}}})=O(^{k}n)$ where the latter notation is tetration (i.e. $n$ raised to its own power, $n$ times) for some finite $k$ (this is provable with structural induction - it's true of the elementary functions, and any "legal" combination of two functions satisfying the condition*), which is a smaller class of growth than what $X$ exhibits, which is $O(^{n}k)$, which, for large enough $n$ grows arbitrarily large power towers. Thus, the sequence grows too quickly to have a closed form in elementary terms.

(*Example: If $f(n)\in O(^{k}n)$, then $f(n)!\leq f(n)^{f(n)}$ so $f(n)!\in O(^{2k}n)$. You could do this with every sort of combination of functions I list, noting that if the given $k$ is finite for $f$, no transformation of $f$ makes it infinite/non-existant - so, unless we start with some "elementary" function that is in a big enough growth class, we'll never create one by combining old ones)

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The above tells us that it is difficult to nest exponents; that is, there's no particular "short-cut" to calculating power towers. However, we could extend this lane of thought (though doing so rigorously eludes me): Why should we be able find a shortcut to calculate $X_n$? Its definition still uses a power tower. Sure, there are plenty of particular cases in which the power tower can be simplified (e.g. if $\alpha=1$ and $\beta=0$), but, to some degree, the number of times we use an exponent in the definition of a variable shouldn't be possible to simplify in general. What if we considered the class of numbers $C_n(x,\alpha,\beta)$ which could be written as an elementary function $f(x,\alpha,\beta)$ employing at most $n$ exponents (or functions like $\sin$ and $\cos$ that essentially invoke an exponent). If $X$ could be written in closed form, then somehow, there would be some $n$ such that $X$ never left $C_n(0,\alpha,\beta)$ for any choice of $\alpha$ and $\beta$ - that somehow, there is always a relation between $X_{n+1}$ and some function using less exponents. This would basically imply that there is no purpose in applying the exponent more than $n$ times, since everything beyond there would be expressible in other ways. This result appears far too strong to be true.

We can also note that every function defined globally in elementary terms is analytic. If a nice elementary function could be given as a closed form, and was analytic, why would we expect it to work at some locations, but not others, despite there never being a singularity?

Any proof will be difficult though; there are cases where $X_n$ is writable in closed form - clearly, any conditions making $X$ eventually periodic will yield a closed form. However, I doubt any other conditions exist where $X$ is writable in closed form, because otherwise we need to deal with nesting exponents.