A particle lies on the real number line at the origin. For each step taken, the particle moves from its current position a distance (and direction) chosen equi-probably from range $[-1,r]$. However, if the particle would otherwise move left of the origin, the particle is set back at the origin before the next step.

Given $r$, can a closed expression be derived for the probability $P_{n,r}$ of the particle being at the origin after step $n$? Can an expression be derived for the centered variance $E_{n,r}(X^2)$?

This is a variation of a problem featured here, where instead of the particle "dying" upon crossing the origin it is simply set at the origin and allowed to continue. Also, I am asking about the probability of it simply being at the origin at the $n$th step, not whether it has at some point been there.

Being only a graduate, all I've managed to do is conjecture what the expression might be for particular cases. Inspired by the excellent answers to the original problem, I directly simulated the problem to find the probabilities and presumed they were given by a rational expression $\frac{N_r(n)}{(1+r)^n n!}$. Surprisingly, my results indicated that the numerator was likely an integer in these cases. My conjectures as to the numerator for those particular cases are:

• $r=0$: $N_0(n)=n!$ (Known precisely, since $P_{n,0} = 1$ for all $n$)
• $r=1$: $N_1(n) = (2n-1)!!$
• $r\rightarrow\infty$: $N_r(n)\rightarrow (n+1)^{n-1}$

Edit: The conjecture for the case when $r$ gets arbitrarily large has been changed. There was a typo. (I was considering its similarity to Cayley's Formula and typed that out instead!)

At this point I'm lost. Your consideration is appreciated.

Update: I've managed to derive a stronger conjecture for the expression for the "re-center" probability:

$$P_{n,r}=\left(\frac{a}{c}\right)^n\frac{\Gamma(1+\frac{d}{a}) \Gamma(1+\frac{b}{a}+n)}{\Gamma(1+\frac{b}{a}) \Gamma(1+\frac{d}{c}+n)}$$

I have made no progress in determining the dependence of the four variables $a$, $b$, $c$, $d$ on the parameter $r$, except in the obvious cases when $r=0$ and $r=1$. If this expression suggests any accessible combinatorial argument I'd be happy to hear it.

Second Update: I have a closed form conjecture for the "re-center" probability when $r \geq 1$:

$$P_{n,r}=\frac{\left(\frac{3r}{1+r}\right)_{n-1}}{(1+r)\left(2r\right)_{n-1}}, r\geq 1$$

This expression uses the rising Pochhammer symbol. It appears to fail for $r<1$ for whatever reason.

Third Update: I finally have a conjecture for the case when $r<1$:

$$P_{n,r}=\frac{\left(A_{-}\right)_{n-1}\left(A_{+}\right)_{n-1}}{(1+r)\left(B_{-}\right)_{n-1}\left(B_{+}\right)_{n-1}}$$

where

$$A_{\pm}=1+\frac{r+r^2\pm\sqrt{r(1+r)(r^2+9r-8)}}{4(1-r)(1+r)},B_{\pm}=1+\frac{r\pm\sqrt{9r^2-8r}}{4(1-r)}$$

Again, the rising Pochhammer symbol is used.