## Elliptic integral evaluations of Bessel moments

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Venue: | Jan 2008, Journal of Physics A: Mathematical and General |

Citations: | 3 - 3 self |

### BibTeX

@ARTICLE{Bailey_ellipticintegral,

author = {David H. Bailey and Jonathan M. Borwein and David Broadhurst and M. L. Glasser},

title = {Elliptic integral evaluations of Bessel moments},

journal = {Jan 2008, Journal of Physics A: Mathematical and General},

year = {},

volume = {41},

pages = {205203}

}

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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 cn,k � (t)dt with integers n = 1,2,3,4 and k ≥ 0, obtaining new results for the even 0 tkKn 0 moments c3,2k and c4,2k. We also derive new closed forms for the odd moments sn,2k+1 � 0 t2k+1I0 (t)K n−1 0 (t)dt with n = 3,4 and for tn,2k+1: = � ∞ 0 t2k+1I2 0 (t)Kn−2 0 (t)dt with n = 5, relating the latter to Green functions on hexagonal, diamond and cubic lattices. We conjecture the values of s5,2k+1, make substantial progress on the evaluation of c5,2k+1, s6,2k+1 and t6,2k+1 and report more limited progress regarding c5,2k, c6,2k+1 and c6,2k. In the process, we obtain 8 conjectural evaluations, each of which has been checked