On Minakshisundaram-Pleijel zeta functions of spheres

Abstract:

The aim of this paper is to show that the Minakshisundaram-Pleijel zeta function ${Z_k}(s)$ of k-dimensional sphere ${\mathbb {S}^k},k \geq 2$ (defined in $\Re e(s) > \frac {k}{2}$ by \[ {Z_k}(s) = \sum \limits _{n = 1}^\infty {\frac {{{P_k}(n)}}{{{{[n(n + k - 1)]}^s}}}} \] with $(k - 1)!{P_k}(n) = \mathcal {R}(n + 1,k - 2)(2n + k - 1)$ where the "rising factorial" $\mathcal {R}(x,n) = x(x + 1) \cdots (x + n - 1)$ is defined for real number x and n nonnegative integer) can be put in the form \[ (k - 1)!{Z_k}(s) = {\sum \limits _{l = 0}^\infty {{{( - 1)}^l}\left ( {\frac {{k - 1}}{2}} \right )} ^{2l}}\left ( {\begin {array}{*{20}{c}} { - s} \\ l \\ \end {array} } \right )\sum \limits _{j = 0}^{k - 1} {{B_{k,}}_j\zeta (2s + 2l - j,\frac {{k + 1}}{2})} \] where ${B_{k,j}}$ are explicitly computed. The above formula allows us to find explicitly the residue of ${Z_k}(s)$ at the pole $s = \frac {k}{2} - n,n \in \mathbb {N}$, \[ \frac {1}{{(k - 1)!}}\sum \limits _{h = 0}^{\frac {k}{2} - 1} {{{\sum \limits _{\begin {array}{*{20}{c}} {l + h = n} \\ {l \geq 0} \\ \end {array} } {{{( - 1)}^l}\left ( {\frac {{k - 1}}{2}} \right )} }^{2l}}\left ( {\begin {array}{*{20}{c}} {n - \frac {k}{2}} \\ l \\ \end {array} } \right )} {B_{k,k - 2h - 1}}.\] In passing, we also obtain apparently new relations among the Stirling numbers.