Communications in Number Theory and Physics

Volume 12 (2018)

Number 2

Wrońskian factorizations and Broadhurst–Mellit determinant formulae

Pages: 355 – 407



Yajun Zhou (Program in Applied and Computational Mathematics (PACM), Princeton University, Princeton, New Jersey, U.S.A.; and Academy of Advanced Interdisciplinary Studies (AAIS), Peking University, Beijing, China)


Drawing on Vanhove’s contributions to mixed Hodge structures for Feynman integrals in two-dimensional quantum field theory, we compute two families of determinants whose entries are Bessel moments. Via explicit factorizations of certain Wrońskian determinants, we verify two recent conjectures proposed by Broadhurst and Mellit, concerning determinants of arbitrary sizes. With some extensions to our methods, we also relate two more determinants of Broadhurst–Mellit to the logarithmic Mahler measures of certain polynomials.


Bessel moments, Feynman integrals, Wrońskian determinants, Mahler measures

2010 Mathematics Subject Classification

Primary 11R06, 15A15, 33C10, 46E25. Secondary 60G50, 81Q30, 81T18, 81T40.

This research was supported in part by the Applied Mathematics Program within the Department of Energy (DOE) Office of Advanced Scientific Computing Research (ASCR) as part of the Collaboratory on Mathematics for Mesoscopic Modeling of Materials (CM4).

Received 8 November 2017

Accepted 15 February 2018

Published 21 June 2018