球面上热核的渐近展开

具体计算了球面Ｓｎ（１）（ｎ２）上热核的渐近展开式中前五项的系数，而根据已有的公式只能算出前四项．最后给出了展开式一般项的递推公式，发现它与Ｂｅｒｎｏｕｌｉ数有未曾想到的联系．根据不变量理论，我们可以确定任意ｎ维紧致无边Ｒｉｅｍａｎｎ流形上热核的渐近展开式中第五项的系数．

Asymptotic Expansions of Heat Kernels of the Spheres

We calculated the first five terms of the asymptotic expansion of the heat kernels of the n-dimensional sphere S n(1), while the known results can only give the first four terms. We also obtain recursion formulas for the general terms of the asymptotic expansions of the kernel of the sphere, and find out an unexpected relation between the general terms and the Bounoulli numbers. According to the invariance theory, we can determine the fifth term of the same expansions of any compact n-dimensional Riemannian manifold.