Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-25T23:57:19.755Z Has data issue: false hasContentIssue false
  • English
  • Français

Combinatorics of the Heat Trace on Spheres

Published online by Cambridge University Press:  20 November 2018

Iosif Polterovich
Iosif Polterovich*
Affiliation:
Institut des Sciences Mathématiques, Université du Québec à Montréal, and Centre de Recherches Mathématiques, Université de Montréal, Montreál, Québec, email: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We present a concise explicit expression for the heat trace coefficients of spheres. Our formulas yield certain combinatorial identities which are proved following ideas of D. Zeilberger. In particular, these identities allow to recover in a surprising way some known formulas for the heat trace asymptotics. Our approach is based on a method for computation of heat invariants developed in $\left[ \text{P} \right]$.

Keywords

05A1958J35
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 54 , Issue 5 , 01 October 2002 , pp. 1086 - 1099
Copyright
Copyright © Canadian Mathematical Society 2002

References

[Be] Berger, M., Geometry of the spectrum. Proc. Symp. Pure Math. 27 (1975), 129–152.Google Scholar
[CW] Cahn, R. S. and Wolf, J. A., Zeta functions and their asymptotic expansions for compact symmetric spaces of rank one. Comment.Math. Helv. 51 (1976), 1–21.Google Scholar
[Ca] Camporesi, R., Harmonic analysis and propagators on homogeneous spaces. Phys. Rep. (1–2) 196 (1990), 1–134.Google Scholar
[ELV] Elizalde, E., Lygren, M. and Vassilevich, D. V., Antisymmetric tensor fields on spheres: functional determinants and non-local counterterms. J. Math. Phys. (7) 37 (1996), 3105–3117.Google Scholar
[Er] Erdélyi, A. et. al., Higher transcendental functions, vol. 1. McGraw-Hill, 1953.Google Scholar
[DK] Dowker, J. S. and Kirsten, K., Spinors and forms on the ball and the generalized cone. Comm. Anal. Geom. (3) 7 (1999), 641–679.Google Scholar
[Gi] Gilkey, P., The index theorem and the heat equation. Mathematics Lecture Series 4, Publish or Perish, 1974.Google Scholar
[Go] Gould, H. W., Combinatorial identities. Henry W. Gould, 1972.Google Scholar
[GR] Gradshtein, I. S. and Ryzhik, I. M., Table of integrals, series and products. Academic Press, 1980.Google Scholar
[GKP] Graham, R., Knuth, D. and Patashnik, O., Concrete Mathematics. A foundation for computer science. Addison-Wesley, 1994.Google Scholar
[MS] McKean, H. P. Jr., and Singer, I. M., Curvature and the eigenvalues of the Laplacian. J. Differential Geom. 1 (1967), 43–69.Google Scholar
[Mi] Milnor, J., Morse Theory. Princeton University Press, 1963.Google Scholar
[MP] Minakshisundaram, S. and Pleijel, A., Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds. Canad. J. Math. 1 (1949), 242–256.Google Scholar
[M¨u] M¨uller, C., Analysis of spherical symmetries in Euclidean spaces. Applied Mathematical Sciences 129, Springer-Verlag, 1998.Google Scholar
[P] Polterovich, I., Heat invariants of Riemannian manifolds. Israel J. Math. 119 (2000), 239–252.Google Scholar
[Se] Seeley, R., Complex powers of an elliptic operator. Proc. Symp. Pure Math. 10 (1967), 288–307.Google Scholar
[Wo] Wolfram, S., Mathematica: a system for doing mathematics by computer. Addison-Wesley, 1991.Google Scholar
[We] Weingart, G., A characterization of the heat kernel coefficients. math.DG/0105144.Google Scholar
[Z] Zeilberger, D., Proof of an identity conjectured by Iossif Polterovitch that came up in the Agmon-Kannai asymptotic theory of the heat kernel. http://www.math.temple.edu/.zeilberg/pj.html, 2000.Google Scholar