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A few days ago (see this), I asked a question regarding the derivatives of the function $$P_2(x,t)=\frac{\sqrt{2}e^{-t/4}}{(4\pi t)^{3/2}}\int_x^\infty\frac{se^{-s^2/4t}ds}{\sqrt{\cosh(s)-\cosh(x)}}.$$

I was trying to check that $P_2$ satisfies the equation $$\frac{\partial^2 P_2}{\partial x^2}+\coth(x)\frac{\partial P_2}{\partial x}-\frac{\partial P_2}{\partial t}=0,$$ as asked in ''Eigenvalues in Riemannian Geometry'' by Isaac Chavel (pp. 242-246): Chavel, I. exercise. However, calculations are too heavy for Mathematica to compute them and when derivating by hand over the formulas given in answers to the previous question, I fail to get zero. Is there something else I could try to verify that $P_2$ is in fact a fundamental solution?

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You might want to check the paper Heat Kernel Bounds on Hyperbolic Space and Kleinian Groups by Davies and Mandouvalos. In odd dimensions, the heat kernel is fairly explicit (and tractable in low dimensions) and can be computed by a recurrence relation. The even dimensional kernels can be computed as suitable integral of the next highest (odd dimensional) kernel. This is Theorem 1.2 of that paper.

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