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

Abstract

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 $$\begin{aligned} \left\{ \left. \int _0^\infty [I_0(t)]^a[K_0(t)]^b t^{2k+1}\mathrm {d}t\right| k\in {\mathbb {Z}}_{\ge 0}\right\} , \end{aligned}$$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 \(\lfloor (a+b-1)/2\rfloor \), which improves the Borwein–Salvy bound \(\lfloor (a+b+1)/2\rfloor \). Our new upper bound \(\lfloor (a+b-1)/2\rfloor \) 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.