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Elliptic integral evaluations of Bessel moments

Bailey, David H. and Borwein, Jonathan M. and Broadhurst, David J. and Glasser, M.L. (2008) Elliptic integral evaluations of Bessel moments. J. Phys. A: Math. Theory, 41 . pp. 5203-5231.

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    Abstract

    We record 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. More generally, 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 $c_{n,k}:=\int_0^\infty t^k K_0^n(t)\,{\rm d}t$ with integers $n=1,2,3,4$ and $k\ge0$, obtaining new results for the even moments $c_{3,2k}$ and $c_{4,2k}$. We also derive new closed forms for the odd moments $s_{n,2k+1}:=\int_0^\infty t^{2k+1}I_0^{}(t)\,K_0^{n-1}(t)\,{\rm d}t$ with $n=3,4$ and for $t_{n,2k+1}:=\int_0^\infty t^{2k+1}I_0^2(t)\,K_0^{n-2}(t)\,{\rm d}t$ with $n=5$, relating the latter to Green functions on hexagonal, diamond and cubic lattices. We conjecture the values of $s_{5,2k+1}$, make substantial progress on the evaluation of $c_{5,2k+1}$, $s_{6,2k+1}$ and $t_{6,2k+1}$ and report more limited progress regarding $c_{5,2k}$, $c_{6,2k+1}$ and $c_{6,2k}$. In the process, we obtain 8 conjectural evaluations, each of which has been checked to 1200 decimal places. One of these lies deep in 4-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.

    Item Type: Article
    Additional Information: pubdom FALSE
    Subjects: 81-xx Quantum Theory
    Faculty: UNSPECIFIED
    Depositing User: lingyun ye
    Date Deposited: 09 Jan 2008
    Last Modified: 11 Jan 2015 17:50
    URI: https://docserver.carma.newcastle.edu.au/id/eprint/380

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