1 Introduction

A celebrated theorem of Pólya [Reference Pólya13] states that the simple symmetric random walk on $\mathbb {Z}^d$ is recurrent in dimensions 1 and 2, but transient in dimension $d\geq 3$ . A fundamental problem is to calculate the return probability, which we denote by $p(d)$ , for $d\geq 3$ (see [Reference Finch4, Section 5.9] for an overview of some of the contributions to this problem). Exact formulae for $p(3)$ , the return probability in dimension 3, were given in [Reference Domb2, Reference McCrea and Whipple9, Reference Watson14], and this line of research culminated in the simple closed-form formula

$$ \begin{align*}p(3)=1-1/u(3)=0.3405373295\ldots,\end{align*} $$

where

(1.1) $$ \begin{align} u(3)&=\frac{\sqrt{6}}{32\pi^3}\Gamma\bigg(\frac{1}{24}\bigg)\,\Gamma\bigg(\frac{5}{24}\bigg)\,\Gamma\bigg(\frac{7}{24}\bigg)\,\Gamma\bigg(\frac{11}{24}\bigg) \end{align} $$

(1.2) $$\begin{align} &=\frac{\sqrt{3}-1}{32\pi^3}\bigg[\Gamma\bigg(\frac{1}{24}\bigg)\,\Gamma\bigg(\frac{11}{24}\bigg)\bigg]^2 \\&=1.5163860591\cdots. \nonumber \end{align}$$

Equation (1.1) was derived in [Reference Glasser and Zucker6], but the value printed there included an erroneous additional factor of $384\pi $ , as pointed out in [Reference Zucker16]. The simplified expression (1.2) was obtained in [Reference Joyce and Zucker7] through an application of a Gamma function identity given in [Reference Borwein and Zucker1] to (1.1). The numerical values presented above for $p(3)$ and $u(3)$ are given in [Reference Finch4, page 322], whilst numerical evaluations for $p(d)$ for $4\leq d\leq 8$ are tabulated in [Reference Finch4, page 323].

More generally, in dimension $d\geq 3$ , [Reference Gaunt, Nadarajah and Pogány5] derived a closed-form expression for the return probability $p(d)$ in terms of the Lauricella function of type C: for $d\geq 3$ ,

$$ \begin{align*} p(d)=1-1/u(d), \end{align*} $$

where

(1.3) $$ \begin{align} u(d) = F_C^{(d)} \bigg( 1, \frac12; 1, \ldots, 1; \frac {1}{d^2}, \ldots, \frac {1}{d^2} \bigg), \end{align} $$

and the Lauricella function $F_C$ (see [Reference Exton3, Reference Lauricella8]) is defined by

$$ \begin{align*} \displaystyle F_C^{(d)} ( a, b; c_1, \ldots, c_d; x_1, \ldots, x_d ) = \sum_{k_1 \geq 0} \cdots \sum_{k_d \geq 0} \frac {(a)_{k_1 + \cdots + k_d} (b)_{k_1 + \cdots + k_d}} {( c_1 )_{k_1} \cdots ( c_d )_{k_d}} \frac {x_1^{k_1} \cdots x_d^{k_d}} {k_1! \cdots k_d!}, \end{align*} $$

with $(v)_0=1$ and $(v)_k = v (v + 1) \cdots (v + k - 1)$ , $k\geq 1$ , denoting the Pochhammer symbol.

Now that a closed-form formula for the return probability for general $d\geq 3$ has been achieved in terms of a known special function, a natural further question is whether one can obtain closed-form expressions in terms of a simpler special function in a dimension greater than 3. In this note, we address this problem by obtaining closed-form expressions for the return probability in dimension 4 in terms of the ${}_{q+1}F_q$ generalised hypergeometric function, defined via the power series

(1.4) $$ \begin{align} {}_{q+1}F_q\bigg({a_1,\ldots,a_{q+1} \atop b_1,\ldots,b_q} \; \bigg| \;x\bigg)=\sum_{k=0}^\infty\frac{(a_1)_k\cdots(a_{q+1})_k}{(b_1)_k\cdots(b_q)_k}\frac{x^k}{k!}, \end{align} $$

which is absolutely convergent for $|x|<1$ , and is also absolutely convergent for ${|x|=1}$ if $\mathrm {Re}(\sum _{i=1}^qb_i-\sum _{j=1}^{q+1}a_j)>0$ (see [Reference Olver, Lozier, Boisvert and Clark12, Section 16.2]). Our formulae for the return probability offer a considerable simplification of the formula for $p(4)$ obtained from simply setting $d=4$ in the general formula (1.3), since the generalised hypergeometric function is defined through a single infinite series, and is a more elementary and widely used special function than the Lauricella function of type C.