Heat kernel coefficients of the Laplace operator on the D-dimensional ball

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Abstract
We present a very quick and powerful method for the calculation of heat kernel coefficients. It makes use of rather common ideas, as integral representations of the spectral sum, Mellin transforms, non‐trivial commutation of series and integrals and skillful analytic continuation of zeta functions on the complex plane. We apply our method to the case of the heat kernel expansion of the Laplace operator on a D‐dimensional ball with either Dirichlet, Neumann or, in general, Robin boundary conditions. The final formulas are quite simple. Using this case as an example, we illustrate in detail our scheme —which serves for the calculation of an (in principle) arbitrary number of heat kernel coefficients in any situation when the basis functions are known. We provide a complete list of new results for the coefficients B3,..., B10, corresponding to the D‐dimensional ball with all the mentioned boundary conditions and D=3,4,5.