Determinants of the Laplacians on the n-dimensional unit sphere S n

Junesang Choi

doi:10.1186/1687-1847-2013-236

Abstract

During the last three decades, the problem of evaluation of the determinants of the Laplacians on Riemann manifolds has received considerable attention from many authors. The functional determinant for the k-dimensional unit sphere with the standard metric has been computed in several ways. Here we aim at computing the determinants of the Laplacians on ( ) by mainly using certain closed-form evaluations of the series involving zeta function. MSC:11M35, 11M36, 11M06, 33B15.

Highlights

1 Introduction and preliminaries During the last three decades, the problem of evaluation of the determinants of the Laplacians on Riemann manifolds has received considerable attention from many authors including D’Hoker and Phong [, ], Sarnak [ ], and Voros [ ], who computed the determinants of the Laplacians on compact Riemann surfaces of constant curvature in terms of special values of the Selberg zeta function
The first interest in the determinants of the Laplacians arose mainly for Riemann surfaces, it is interesting and potentially useful to compute these determinants for classical Riemannian manifolds of higher dimensions, such as spheres
We are concerned with the evaluation of the functional determinant for the k-dimensional unit sphere Sk (k = n + ) with the standard metric

Summary

It is not difficult to see also that

⎨ (n = ), s(n, ) = ⎩ (n ∈ N), s(n, n) = , s(n, k) = (k > n), s(n, ) = (– )n+ (n – )!, n s(n, n – ) = – and n s(n, k) = n ∈ N \ { } ; k= n s(n + , j + )nj–k = s(n, k). n ∈ N ; z ∈ C \ Z– , in terms of the gamma function , and Z– := { , – , – , . . .}. From the definition ( . ) of s(n, k), the Pochhammer symbol in ( . ) can be written in the form n (z)n = z(z + ) · · · (z + n – ) = (– )n+ks(n, k)zk, k= where (– )n+ks(n, k) denotes the number of permutations of n symbols, which has exactly k cycles. For potential use, we observe the following simple properties related to s(n, k) in the lemma below. (– )n+js(n, j)zj := C (n)z .

Then we have

Now it is easy to see that