Bessel矩的椭圆积分计算
Elliptic integral evaluations of Bessel moments
摘要：
我们记录什么是已知的封闭形式的各种贝塞尔函数的时刻所产生的量子场论，凝聚态理论和其他部分的数学物理。更一般地，我们开发的积分公式的产品的六个或更少的贝塞尔函数。因此，我们能够发现和证明$c_{n,k}:=\int_0^\infty t^k K_0^n(t) {\rm d}t$与整数$n=1,2,3,4$和$k\ge0$的封闭形式，获得新的结果，甚至时刻$c_{3,2k}$和$c_{4,2k}$。我们还推导出新的封闭形式的奇时刻$s_{n,2k+1}:=\int_0^\infty t^{2k+1}I_0^{}(t) K_0^{n-1}(t) {\rm d}t$与$n=3,4$和$t_{n,2k+1}:=\int_0^\infty t^{2k+1}I_0^2(t) K_0^{n-2}(t) {\rm d}t$与$n=5$，涉及后者的六方晶系，钻石和立方晶格上的格林函数。我们推测了$s_{5,2k+1}$的值，在评价$c_{5,2k+1}$、$s_{6,2k+1}$和$t_{6,2k+1}$方面取得了实质性进展，并报告了关于$c_{5,2k}$、$c_{6,2k+1}$和$c_{6,2k}$的更有限的进展。在这个过程中，我们得到了8个数学评价，每个评价都被检查到小数点后1200位。其中之一深埋于四维量子场论中，而另外两个可能可以用精细的组合数学来证明。对于数学家和物理学家来说，仍然有五个定理的核心，它们的证明将是最有指导意义的。
英文摘要：
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.
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