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

(v1) i+21 pages. Simplification and extension of some results in Section 5 of arXiv:1706.08308; (v2) i+29 pages; (v3) 20 pages, accepted version

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Cited by in corpus (3)

• Feynman Integrals in Dimensional Regularization and Extensions of Calabi-Yau Motives
• Quadratic relations between Bessel moments
• On the Periods of Twisted Moments of the Kloosterman Connection