2008/01/31 by David H. Bailey, Jonathan M. Borwein, David Broadhurst +1 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #hep-ph #hep-th #math-ph #math.MP
paper · pdf · doi:10.1088/1751-8113/41/20/205203
published as J. Phys. A: Math. Theor. 41 (2008) 205203 · 51 pages, 1 Postscript figure, uses amsmath.sty, added references
arxiv created 2008/02/08 · openalex publication_date 2008/04/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/02
We record and substantially extend what is known about the closed forms for various Bessel function moments arising in quantum field theory, condensed matter theory and other parts of mathematical physics. In particular, we develop formulae for integrals of products of six or fewer Bessel functions. In consequence, we are able to discover and prove closed forms for c n , k := ∫ ∞ 0 t k K n 0 ( t ) d t with integers n = 1, 2, 3, 4 and k ⩾ 0, obtaining new results for the even moments c 3,2 k and c 4,2 k . We also derive new closed forms for the odd moments s n ,2 k +1 := ∫ ∞ 0 t 2 k +1 I 0 ( t ) K n −1 0 ( t ) d t with n = 3, 4 and for t n ,2 k +1 := ∫ ∞ 0 t 2 k +1 I 2 0 ( t ) K n −2 0 ( t ) d t with n = 5, relating the latter to Green functions on hexagonal, diamond and cubic lattices. We conjecture the values of s 5,2 k +1 , make substantial progress on the evaluation of c 5,2 k +1 , s 6,2 k +1 and t 6,2 k +1 and report more limited progress regarding c 5,2 k , c 6,2 k +1 and c 6,2 k . In the process, we obtain eight conjectural evaluations, each of which has been checked to 1200 decimal places. One of these lies deep in four-dimensional quantum field theory and two are probably provable by delicate combinatorics. There remains a hard core of five conjectures whose proofs would be most instructive, to mathematicians and physicists alike.