The Ramanujan Journal

, Volume 48, Issue 1, pp 159–172 | Cite as

Hilbert transforms and sum rules of Bessel moments

  • Yajun ZhouEmail author


Using Hilbert transforms, we establish two families of sum rules involving Bessel moments, which are integrals associated with Feynman diagrams in two-dimensional quantum field theory. With these linear relations among Bessel moments, we verify and generalize two conjectures by Bailey–Borwein–Broadhurst–Glasser and Broadhurst–Mellit.


Hilbert transforms Bessel functions Feynman integrals 

Mathematics Subject Classification

44A15 33C10 33C20 (Primary)  81T18 81T40 81Q30 (Secondary) 



In early 2017, I wrote up this paper in Beijing, mostly drawing on my research notes prepared at Princeton during 2012 and 2013. I thank Prof. Weinan E for arranging my stays in Princeton and Beijing, as well as organizing a seminar on constructive quantum field theory at Princeton. After completion of the initial draft of this article, I received from Dr. David Broadhurst his slides for recent talks [6, 7, 8] on Bessel moments, which set his conjectures in a wider context. I thank Dr. Broadhurst for his constant encouragements and incisive comments on this project. I am indebted to an anonymous referee for thoughtful suggestions on improving the presentation of this paper. In January 2013, I benefited from fruitful discussions with Prof. Jon Borwein on his previous contributions to Bessel moments and elliptic integrals; I was equally grateful to his friendly communications on my then-unpublished work related to Hilbert transforms. I dedicate this work to his memory.


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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Program in Applied and Computational Mathematics (PACM)Princeton UniversityPrincetonUSA
  2. 2.Academy of Advanced Interdisciplinary Sciences (AAIS)Peking UniversityBeijingPeople’s Republic of China

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