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

TL;DR

This paper establishes improved upper bounds on the dimension of the rational vector space spanned by certain Bessel moments, using differential operators and modular form techniques, with implications for understanding their linear relations.

Contribution

The authors derive a new upper bound for the Q-linear dimension of Bessel moments, improving upon previous bounds and identifying specific relations through modular form integration.

Findings

New upper bound: loor((a+b-1)/2 for the dimension

Improved upon Borwein-Salvy bound loor((a+b+1)/2

Identified an exceptional Q-linear relation for a=2, b=6

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 \[ \left\{\left.\int_0^\infty [I_0(t)]^a[K_0(t)]^b t^{2k+1}\mathrm{d}\, t\right|k\in\mathbb Z_{\geq0}\right\},\]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.