Generalized Bernoulli polynomials and the Casimir effect in the Einstein universe

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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 languageEnglish (US)
Title of host publicationLie Theory and Its Applications in Physics
Subtitle of host publicationIX International Workshop
PublisherSpringer New York LLC
Pages231-238
Number of pages8
ISBN (Print)9784431542698
DOIs
StatePublished - Jan 1 2013

Publication series

NameSpringer Proceedings in Mathematics and Statistics
Volume36
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

Fingerprint

Casimir Effect
Bernoulli Polynomials
Generalized Polynomials
Albert Einstein
n-dimensional
Exact Results
Scalar Field
Closed-form
Hurwitz zeta Function
Bernoulli numbers
Regularization Technique
Minkowski Space
Representation Theory
Energy
Dirichlet Boundary Conditions
Generating Function
Regularization
Vacuum
Radius
Equivalence

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

Moylan, P. J. (2013). Generalized Bernoulli polynomials and the Casimir effect in the Einstein universe. In 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
Moylan, Patrick J. / Generalized Bernoulli polynomials and the Casimir effect in the Einstein universe. Lie Theory and Its Applications in Physics: IX International Workshop. Springer New York LLC, 2013. pp. 231-238 (Springer Proceedings in Mathematics and Statistics).
@inproceedings{9066f592a92640b7948ccd7cfc89d16d,
title = "Generalized Bernoulli polynomials and the Casimir effect in the Einstein universe",
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 {\"O}zcan (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.",
author = "Moylan, {Patrick J.}",
year = "2013",
month = "1",
day = "1",
doi = "10.1007/978-4-431-54270-4_15",
language = "English (US)",
isbn = "9784431542698",
series = "Springer Proceedings in Mathematics and Statistics",
publisher = "Springer New York LLC",
pages = "231--238",
booktitle = "Lie Theory and Its Applications in Physics",

}

Moylan, PJ 2013, Generalized Bernoulli polynomials and the Casimir effect in the Einstein universe. in 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.

Lie Theory and Its Applications in Physics: IX International Workshop. Springer New York LLC, 2013. p. 231-238 (Springer Proceedings in Mathematics and Statistics; Vol. 36).

Research output: Chapter in Book/Report/Conference proceedingConference 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 -

Moylan PJ. Generalized Bernoulli polynomials and the Casimir effect in the Einstein universe. In Lie Theory and Its Applications in Physics: IX International Workshop. Springer New York LLC. 2013. p. 231-238. (Springer Proceedings in Mathematics and Statistics). https://doi.org/10.1007/978-4-431-54270-4_15