Prove known closed form for $\int_0^\infty e^{-x}I_0\left(\frac{x}{3}\right)^3\;dx$

I know that the following identity is correct, but I would love to see a derivation:

$$\int_0^\infty e^{-x}I_0\left(\frac{x}{3}\right)^3\;dx=\frac{\sqrt{6}}{32\pi^3}\Gamma\left(\frac{1}{24}\right)\Gamma\left(\frac{5}{24}\right)\Gamma\left(\frac{7}{24}\right)\Gamma\left(\frac{11}{24}\right)\tag{1}$$

where $I_0(x)$ is a modified Bessel function of the first kind. The form of the RHS makes me assume that the elliptic integral singular value $K(k_6)$ is somehow involved but I see no obvious way of transforming the LHS into an elliptic integral. My request is to see a way of evaluating this integral directly to show that it equals the RHS.

Background: This integral arises in the theory of the recurrence of random walks on a cubic lattice. Jonathan Novak elegantly shows here using generating functions that the recurrence probability of a random walk on $\mathbb{Z}^d$ is $p(d)=1-\frac{1}{u(d)}$ where:

$$u(d)=\int_0^\infty e^{-x}I_0\left(\frac{x}{d}\right)^d\;dx\tag{2}$$

The MathWorld page lists several ways of writing $u(3)$, including $(1)$, but in none of the references could I find a specific derivation of $(1)$. Most relevant discussions I found (e.g. here and here where elliptic integrals do appear) work from an alternative way of writing $u(3)$, namely:

$$u(3)=\frac{3}{2\pi^3}\int_{-\pi}^\pi\int_{-\pi}^\pi\int_{-\pi}^\pi\frac{dx\;dy\;dz}{3-\cos{x}-\cos{y}-\cos{z}}\tag{3}$$

rather than a representation like $(2)$. I believe $(3)$ comes from an alternative way of solving the random walk problem (e.g. see here) which I am not very familiar with. Accordingly, I am asking for a derivation of $(1)$ that works directly from the Bessel function integral, rather than considering other ways of calculating $u(3)$ (deriving the RHS of $(1)$ from $(3)$ would not answer my question). Although I appreciate that there may be simpler proofs of $(1)$ from the theory of random walks, I am wondering if there is a nice way of evaluating the integral in $(1)$ directly.

• Integral definition of the Bessel function seems like a way to go. $$I_0(y)=\frac{1}{\pi} \int_0^\pi e^{y \cos t}dt$$ – Yuriy S Jan 23 '17 at 8:57
• For example $$I_0 \left(\frac{x}{3} \right)^3=\frac{1}{\pi^3} \int_0^\pi \int_0^\pi \int_0^\pi e^{\frac{x}{3} (\cos t+\cos u+\cos v)}dtdudv$$ Now you can also see the relation to $(3)$ – Yuriy S Jan 23 '17 at 9:03
• Just to be clear $$\int_0^\infty e^{-x}I_0\left(\frac{x}{3}\right)^3\;dx=\frac{1}{\pi^3}\int_0^\infty \int_0^\pi \int_0^\pi \int_0^\pi e^{-\frac{x}{3} (3-\cos t-\cos u-\cos v)}dtdudvdx=$$ $$=\frac{3}{\pi^3} \int_0^\pi \int_0^\pi \int_0^\pi \frac{dt~du~dv}{3-\cos t-\cos u-\cos v}$$ – Yuriy S Jan 23 '17 at 9:13
• nice integral (+1) – tired Jan 23 '17 at 17:33
• Another form of the integral $$\int_0^\infty e^{-x}I_0\left(\frac{x}{3}\right)^3\;dx=\frac{12}{\pi^3} \int_0^1 \int_0^1 \int_0^1 \frac{dp~dq~ds}{(p^2+q^2+s^2)\sqrt{(1-p^2)(1-q^2)(1-s^2)}}$$ – Yuriy S Jan 23 '17 at 19:09

I think many identities are involved here, so let we outline a partial answer (for now). We have $$I_0(z) = \sum_{n\geq 0}\frac{z^{2n}}{4^n n!^2},\qquad \mathcal{L}(I_0(z))=\frac{\mathbb{1}_{s>1}}{\sqrt{s^2-1}}\tag{1}$$ and $$I_0(z)^2 = \sum_{n\geq 0}\frac{z^{2n}}{4^n}\sum_{k=0}^{n}\frac{1}{k!^2(n-k)!^2} = \sum_{n\geq 0}\frac{z^{2n}}{4^n n!^2}\sum_{k=0}^{n}\binom{n}{k}^2=\sum_{n\geq 0}\frac{z^{2n}\binom{2n}{n}}{4^n n!^2}\tag{2}$$ so: $$\mathcal{L}(I_0(z)^2) =\frac{2}{\pi}\,K\!\left(\frac{2}{s}\right)\mathbb{1}_{s>2}\tag{3}$$ and the connection with the complete elliptic integral of the first kind is less obscure. If we set $$A_n = \sum_{k=0}^{n}\frac{\binom{2k}{k}}{k!^2(n-k)!^2}\tag{4}$$ we have: $$I_0(z)^3 = \sum_{n\geq 0}\frac{z^{2n}}{4^n}A_n \tag{5}$$ and $$\begin{eqnarray*} \int_{0}^{+\infty}I_0(z)^3 e^{-3z}\,dz = \sum_{n\geq 0}\frac{(2n)!}{4^n 3^{2n+1}}\,A_n&=&\sum_{0\leq k\leq n}\frac{(2n)!\binom{2k}{k}}{4^n 3^{2n+1} k!^2 (n-k)!^2}\\&=&\sum_{k\geq 0}\frac{(2k)!}{k!^4}\sum_{n\geq k}\frac{(2n)!}{4^n 3^{2n+1}(n-k)!^2}.\tag{6}\end{eqnarray*}$$ The definitive connection with a singular value should now be a consequence of the identity $$K(k)^2 = \frac{\pi^2}{4}\sum_{n\geq 0}\left(\frac{(2n-1)!!}{(2n)!!}\right)^3 (2kk')^{2n} \tag{7}$$ that holds for any $0<k<\frac{1}{\sqrt{2}}$ due to the (AGM-like) invariance properties of the complete elliptic integral of the first kind.