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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

On Minakshisundaram-Pleijel zeta functions of spheres


Authors: E. Carletti and G. Monti Bragadin
Journal: Proc. Amer. Math. Soc. 122 (1994), 993-1001
MSC: Primary 58G26; Secondary 11M36
MathSciNet review: 1249872
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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

$\displaystyle {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

$\displaystyle (k - 1)!{Z_k}(s) = {\sum\limits_{l = 0}^\infty {{{( - 1)}^l}\left... ...)\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}$,

$\displaystyle \frac{1}{{(k - 1)!}}\sum\limits_{h = 0}^{\frac{k}{2} - 1} {{{\sum... ...{*{20}{c}} {n - \frac{k}{2}} \\ l \\ \end{array} } \right)} {B_{k,k - 2h - 1}}.$

In passing, we also obtain apparently new relations among the Stirling numbers.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1249872-9
PII: S 0002-9939(1994)1249872-9
Article copyright: © Copyright 1994 American Mathematical Society