In a recent problem I was working through, I came across the following recurrence relation: $$ \text{K}_1(x,\ t;\ g) = 1,\quad x, t\in\Bbb{R}\quad g\in\text{C}^1 \\ \text{K}_{n+1}(x,\ t;\ g) = g'(t)\int_t^{g(x)}\text{K}_n(\sigma,\ g(t);\ g)\text{d}\sigma $$ where the given function parameter $g$ has at least one real fixed point $a$.
I'm looking for relatively simple functions $g$ such that this recurrence relation has a "nice" closed form formula for arbitrary $n$. Currently, the only thing I've found that results in anything even remotely simple is $g(x) = x$, which makes $$ \text{K}_{n+1}(x,\ t;\ g)=\frac{(x-t)^n}{n!} $$ Nothing else seems to result in anything simple; $x^k$, $e^x - 1$ , $\ln(1 + x)$, $\sin(x)$, $\cos(x)$, etc. Even $x^2$ results in $\text{K}_n$ being a polynomial in $x$ and $t$ of degree $6\left(2^n-\frac 12\right)$, but the coefficients on even only the $x^{6\left(2^n-\frac 12\right)}$ term don't appear to have any nice closed form (according to OEIS).
The problem seems to be related to finding functions $g$ where $g^{(-1)}(g(x))$ is relatively easy to calculate (with $g^{(-1)}=\int g$).
Any ideas on other functions $g$ that would result in a "nice" form for $\text{K}_n$? I know the idea of "nice" isn't well defined, but in this context it essentially means "has a closed form, not in terms of integrals, for arbitrary $n$ which is computable in a finite number of calculations" (i.e. no unevaluable infinite series)