Probability to return to the origin for a uniform random walk

Question

Consider a uniform random walk on $\mathbb{R}$, with stepsize chosen uniformly from the interval $(-1,1)$. The random walk start at $x=0$. Denote by $\rho_p dx$ the probability that the random walk returns to the interval $(0,dx)$ after $p$ steps, for infinitesimally small $dx$. For $p=1$ and for $p=2$ one has $\rho_p=1/2$, for $p=3$ one has $\rho_p=3/8$, for large $p$ the diffusion approximation gives $\rho_p=(2\pi p/3)^{-1/2}$.

Q: Is $\rho_p$ known exactly for any $p$?

Motivation: this would give the small-$a$ limit of the box integral in https://mathoverflow.net/a/331993/11260 , $\lim_{a\rightarrow 0}a^{1-p}f_p(a)=2^p\rho_p$, see also fedja's comment in https://math.stackexchange.com/q/3233160/87355

Answer 1

Looking at fedja's comment that you mentioned, we can formalize your description as follows: $$\rho_p=g_p(0),$$ where $g_p$ is the pdf of $$T_p:=\sum_{i=1}^p V_i=2S_p-p,$$ $S_p:=\sum_{i=1}^p U_i$, the $U_i$'s are i.i.d. random variables (r.v.'s) uniformly distributed on $[0,1]$, and $V_i:=2U_i-1$, so that the $V_i$'s are i.i.d. r.v.'s uniformly distributed on $[-1,1]$.

Note that $g_p(t)=\frac12\,f_p(\frac{p+t}2)$ for real $t$, where $f_p$ is the pdf of $S_p$. So, by the Irwin–Hall formula, $$\rho_p=g_p(0)=\frac12\,f_p\Big(\frac p2\Big) =\frac1{2(p-1)!}\sum_{k=0}^{\lfloor p/2\rfloor}(-1)^{k-1}\binom pk\Big(\frac p2-k\Big)^{p-1}.$$

In particular, the respective values of $\rho_p$ for $p=1,2,3,4,5$ are $\frac{1}{2},\frac{1}{2},\frac{3}{8},\frac{1}{3},\frac{115}{384}$.