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DETERMINANTS OF THE LAPLACIANS ON THE n-DIMENSIONAL UNIT SPHERE Sn (n = 8, 9)
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  • Journal title : Honam Mathematical Journal
  • Volume 33, Issue 3,  2011, pp.321-333
  • Publisher : The Honam Mathematical Society
  • DOI : 10.5831/HMJ.2011.33.3.321
 Title & Authors
DETERMINANTS OF THE LAPLACIANS ON THE n-DIMENSIONAL UNIT SPHERE Sn (n = 8, 9)
Choi, June-Sang;
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 Abstract
During the last three decades, the problem of evaluation of the determinants of the Laplacians on Riemann manifolds has received considerable attention by many authors. The functional determinant for the n-dimensional sphere with the standard metric has been computed in several ways. Here we aim at computing the determinants of the Laplacians on (n = 8, 9) by mainly using ceratin known closed-form evaluations of series involving Zeta function.
 Keywords
Gamma function;Psi-(or Digamma) function;Riemann Zeta function;Hurwitz Zeta function;Selberg Zeta function;Zeta regularized product;Determinants of Laplacians;Series associated with the Zeta functions;
 Language
English
 Cited by
1.
Determinants of the Laplacians on the n-dimensional unit sphere Sn, Advances in Difference Equations, 2013, 2013, 1, 236  crossref(new windwow)
 References
1.
E. W. Barnes, The theory of the G-function, Quart. J. Math. 31 (1899), 264-314.

2.
E. W. Barnes, Genesis of the double Gamma function, Proc. London Math. Soc. (Ser. 1) 31 (1900), 358-381.

3.
E. W. Barnes, The theory of the double Gamma function, Philos. Trans. Roy. Soc. London Ser. A 196 (1901), 265-388. crossref(new window)

4.
E. W. Barnes, On the theory of the multiple Gamma functions, Trans. Cambridge Philos. Soc. 19 (1904), 374-439.

5.
J. Choi, Determinant of Laplacian on $S^3$, Math. Japon. 40 (1994), 155-166.

6.
J. Choi, Y. J. Cho and H. M. Srivastava, Series involving the Zeta function and multiple Gamma functions, Appl. Math. Comput. 159 (2004), 509-537. crossref(new window)

7.
J. Choi and H. M. Srivastava, An application of the theory of the double Gamma function Kyushu J. Math. 53 (1999), 209-222. crossref(new window)

8.
J. Choi and H. M. Srivastava, Certain classes of series associated with the Zeta function and multiple Gamma functions J. Comput. Appl. Math. 118 (2000), 87-109. crossref(new window)

9.
E. D'Hoker and D. H. Phong, On determinant of Laplacians on Riemann surface, Comm. Math. Phys. 104 (1986), 537-545. crossref(new window)

10.
E. D'Hoker and D. H. Phong, Multiloop amplitudes for the bosonic polyakov string, Nucl. Phys. B 269 (1986), 204-234.

11.
H. Kumagai, The determinant of the Laplacian on the n-sphere, Acta Arith. 91 (1999), 199-208.

12.
B. Osgood, R. Phillips and P. Sarnak, Extremals of determinants of Laplacians, J. Funct. Anal. 80 (1988), 148-211. crossref(new window)

13.
J. R. Quine and J. Choi, Zeta regularized products and functional determinants on spheres, Rocky Mountain J. Math. 26 (1996), 719-729. crossref(new window)

14.
P. Sarnak, Determinants of Laplacians, Comm. Math. Phys. 110 (1987), 113-120. crossref(new window)

15.
J. D. Shallit and K. Zikan, A theorem of Goldbach, Amer. Math. Monthly 93 (1986), 402-403. crossref(new window)

16.
H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, Boston and London, 2001.

17.
H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier, 2011, in press.

18.
A. Terras, Harmonic Analysis on Symmetric Spaces and Applications, Vol. I, Springer-Verlag, New York, 1985.

19.
I. Vardi, Determinants of Laplacians and multiple Gamma functions, SIAM J. Math. Anal. 19 (1988), 493-507. crossref(new window)

20.
A. Voros, Special functions, spectral functions and the Selberg Zeta function Comm. Math. Phys. 110 (1987), 439-465. crossref(new window)