$\mathbb{Q}$-linear dependence of certain Bessel moments

Let $I_0$ and $K_0$ be modified Bessel functions of the zeroth order. We use Vanhove’s differential operators for Feynman integrals to derive upper bounds for dimensions of the $\mathbb{Q}$-vector space spanned by certain sequences of Bessel moments $$\{{\int }_0^{\mathrm{\infty }}[I_0(t){]}^a[K_0(t){]}^b{t}^{2k+1}\mathrm{d}t\mid k\in {\mathbb{Z}}_{\ge 0}\},$$where $a$ and $b$ are fixed non-negative integers. For $a\in \mathbb{Z}\cap [1,b)$, our upper bound for the $\mathbb{Q}$-linear dimension is $\lfloor (a+b-1)\mathrm{/}2\rfloor$, which improves the Borwein-Salvy bound $\lfloor (a+b+1)\mathrm{/}2\rfloor$. Our new upper bound $\lfloor (a+b-1)\mathrm{/}2\rfloor$ is not sharp for $a=2,b=6$, due to an exceptional $\mathbb{Q}$-linear relation $\int_0^\infty [I_0(t)]^2[K_0(t)]^6 t$\mathrm{d}$, t=72\int_0^\infty [I_0(t)]^2[K_0(t)]^6 t^{3}$\mathrm{d}$, t$, which is provable by integrating modular forms.