Let \(S^d\) be a sphere with the standard Riemannian metric of curvature \(+1\). The Laplace–Beltrami operator \(\Delta\) on \(S^d\) has eigenvalues \(\lambda_{k,d}=k(k+d-1)\) with respective multiplicity \[ \mu_{k,d}=\frac {(2k+d-1)\,(k+d-2)!} {k!\,(d-1)!},\quad k\geq1 \] and \(\mu_{0,d}=1\).
The subject of the paper is the aymptotic expansion for the trace of the heat operator \(e^{-t\Delta}\) as \(t\to0^+\), \[ \sum _{\lambda} ^{}e^{-t\lambda}=\sum _{k=0} ^{\infty}\mu_{k,d} e^{-t\lambda_{k,d}}\sim \sum _{n=0} ^{\infty}a_{n,d}t^{n-\frac {d} {2}}, \] and more precisely the calculation of the so-called heat trace coefficients \(a_{n,d}\). Earlier work [see for example R. S. Cahn and J. A. Wolf, Comment. Math. Helv. 51, 1–21 (1976; Zbl 0327.43013)] had used Lie theoretic methods to calculate these coefficients. The author proposes a new method, which is based on his earlier paper [Isr. J. Math. 119, 239–252 (2000; Zbl 0996.59019)], and which makes use of a mixture of tools from hypergeometric series theory and from difference operator calculus in an essential way. (As the author acknowledges, the latter ideas are due to D. Zeilberger.) Thus, he arrives at concise expressions for the heat trace coefficients, which, in particular, allow him to recover the earlier results.