### Abstract

We consider various regularization schemes for calculating the renormalized vacuum energy of a massless scalar field in the n-dimensional Einstein universe. We also study a related problem, namely, the Casimir energy for a massless scalar field in the n-dimensional Einstein universe subject to Dirichlet boundary conditions on a sphere of maximal radius. In a recent work the author used the representation theory of SO(2,n) to obtain exact results but not in closed form for the second problem with n arbitrary. Here we make use of generating functions for generalized Bernoulli polynomials and an extension of a result of Srivastava and Todorov about generalized Bernoulli numbers (Srivastava, Todorov, J. Math. Anal. Appl. 130:509-513, 1988) to obtain new results involving exact expressions in closed form for both problems. We also consider expansions of the generalized Bernoulli polynomials into Hurwitz zeta functions which enables us to explicitly demonstrate the equivalence of the cutoff function technique with the zeta regularization technique. Our method of approach confirms the results of Herdeiro et al. (Class. Quant. Gravit. 25:165010, 2008) and Özcan (Class. Quant. Gravit. 23:5531-5546, 2006).We conclude the paper by showing that useful information about the analogous problem in n-dimensional Minkowski space can also be obtained out of our analysis.

Original language | English (US) |
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Title of host publication | Lie Theory and Its Applications in Physics |

Subtitle of host publication | IX International Workshop |

Publisher | Springer New York LLC |

Pages | 231-238 |

Number of pages | 8 |

ISBN (Print) | 9784431542698 |

DOIs | |

State | Published - Jan 1 2013 |

### Publication series

Name | Springer Proceedings in Mathematics and Statistics |
---|---|

Volume | 36 |

ISSN (Print) | 2194-1009 |

ISSN (Electronic) | 2194-1017 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Lie Theory and Its Applications in Physics: IX International Workshop*(pp. 231-238). (Springer Proceedings in Mathematics and Statistics; Vol. 36). Springer New York LLC. https://doi.org/10.1007/978-4-431-54270-4_15

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*Lie Theory and Its Applications in Physics: IX International Workshop.*Springer Proceedings in Mathematics and Statistics, vol. 36, Springer New York LLC, pp. 231-238. https://doi.org/10.1007/978-4-431-54270-4_15

**Generalized Bernoulli polynomials and the Casimir effect in the Einstein universe.** / Moylan, Patrick J.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Generalized Bernoulli polynomials and the Casimir effect in the Einstein universe

AU - Moylan, Patrick J.

PY - 2013/1/1

Y1 - 2013/1/1

N2 - We consider various regularization schemes for calculating the renormalized vacuum energy of a massless scalar field in the n-dimensional Einstein universe. We also study a related problem, namely, the Casimir energy for a massless scalar field in the n-dimensional Einstein universe subject to Dirichlet boundary conditions on a sphere of maximal radius. In a recent work the author used the representation theory of SO(2,n) to obtain exact results but not in closed form for the second problem with n arbitrary. Here we make use of generating functions for generalized Bernoulli polynomials and an extension of a result of Srivastava and Todorov about generalized Bernoulli numbers (Srivastava, Todorov, J. Math. Anal. Appl. 130:509-513, 1988) to obtain new results involving exact expressions in closed form for both problems. We also consider expansions of the generalized Bernoulli polynomials into Hurwitz zeta functions which enables us to explicitly demonstrate the equivalence of the cutoff function technique with the zeta regularization technique. Our method of approach confirms the results of Herdeiro et al. (Class. Quant. Gravit. 25:165010, 2008) and Özcan (Class. Quant. Gravit. 23:5531-5546, 2006).We conclude the paper by showing that useful information about the analogous problem in n-dimensional Minkowski space can also be obtained out of our analysis.

AB - We consider various regularization schemes for calculating the renormalized vacuum energy of a massless scalar field in the n-dimensional Einstein universe. We also study a related problem, namely, the Casimir energy for a massless scalar field in the n-dimensional Einstein universe subject to Dirichlet boundary conditions on a sphere of maximal radius. In a recent work the author used the representation theory of SO(2,n) to obtain exact results but not in closed form for the second problem with n arbitrary. Here we make use of generating functions for generalized Bernoulli polynomials and an extension of a result of Srivastava and Todorov about generalized Bernoulli numbers (Srivastava, Todorov, J. Math. Anal. Appl. 130:509-513, 1988) to obtain new results involving exact expressions in closed form for both problems. We also consider expansions of the generalized Bernoulli polynomials into Hurwitz zeta functions which enables us to explicitly demonstrate the equivalence of the cutoff function technique with the zeta regularization technique. Our method of approach confirms the results of Herdeiro et al. (Class. Quant. Gravit. 25:165010, 2008) and Özcan (Class. Quant. Gravit. 23:5531-5546, 2006).We conclude the paper by showing that useful information about the analogous problem in n-dimensional Minkowski space can also be obtained out of our analysis.

UR - http://www.scopus.com/inward/record.url?scp=84883148053&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84883148053&partnerID=8YFLogxK

U2 - 10.1007/978-4-431-54270-4_15

DO - 10.1007/978-4-431-54270-4_15

M3 - Conference contribution

SN - 9784431542698

T3 - Springer Proceedings in Mathematics and Statistics

SP - 231

EP - 238

BT - Lie Theory and Its Applications in Physics

PB - Springer New York LLC

ER -