Determinants of the Laplacians on the n-dimensional unit sphere
© Choi; licensee Springer 2013
Received: 10 April 2013
Accepted: 22 July 2013
Published: 7 August 2013
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.
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 (among others) D’Hoker and Phong [1, 2], 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. Although the first interest in the determinants of the Laplacians arose mainly for Riemann surfaces, it is also interesting and potentially useful to compute these determinants for classical Riemannian manifolds of higher dimensions, such as spheres. Here, we are particularly concerned with the evaluation of the functional determinant for the k-dimensional unit sphere () with the standard metric.
which is known to converge absolutely in a half-plane for some .
Definition 1 (cf. Osgood et al. )
which is called the functional determinant of the Laplacian Δ on M.
where denotes the greatest integer part in the order μ of the sequence .
with the same multiplicity as in (1.16).
We will exclude the zero mode, that is, start the sequence at for later use. Furthermore, with a view to emphasizing n on , we choose the notations , , , , and instead of , , , , and , respectively.
where denotes the determinants of the Laplacians on ().
Several authors (see Choi , Kumagai , Vardi , and Voros ) used the theory of multiple gamma functions (see Barnes [11–14]) to compute the determinants of the Laplacians on the n-dimensional unit sphere (). Quine and Choi  made use of zeta regularized products to compute and the determinant of the conformal Laplacian, . Choi and Srivastava [16, 17], Choi et al. , and Choi  made use of some closed-form evaluations of the series involving zeta function (see [, Chapter 3]) for the computation of the determinants of the Laplacians on (). In the sequel, here, we aim at presenting a general explicit formula for the determinants of the Laplacians on (; ) by mainly using a summation formula of the series involving zeta function.
2 The Stirling numbers of the first kind
in terms of the gamma function Γ, and .
where denotes the number of permutations of n symbols, which has exactly k cycles.
For potential use, we observe the following simple properties related to in the lemma below.
Proof It is easy to see the first expression for . For the second one, it is enough to see that the defined product is an even function of z. □
For later use, we compute the first few values of as in Lemma 2.
3 Series associated with the zeta functions
where denotes the set of nonpositive integers. It is noted that both the Riemann zeta function and the Hurwitz zeta function can be continued meromorphically to the whole complex s-plane except for a simple pole only at with their residue 1. For easy reference, we recall some properties of and as in the following lemma.
Employing the various methods and techniques used in the vast literature on the subject of the closed-form evaluations series associated with the zeta functions, Srivastava and Choi (see [, Chapter 3], [, Chapter 3], and see also the related references therein) presented a rather extensive collection of closed-form sums of series involving the zeta functions. For the use in the next section, we recall two general formulas as in the following lemma (see [, p.254]).
By setting and in (3.16) and using suitable formulas given in this section, we get a special case identity of (3.16) as in Lemma 5 below.
Further specialized formulas of the identity in (3.17), for later use, are obtained as in Lemma 6 below.
4 The determinants of the Laplacians on ()
Here, by using (1.18) and the results in the previous sections, we are ready to compute the determinants of the Laplacians on () as in the following theorem.
of which the last series is expressed in a closed-form as given in Lemma 5.
where and are given as in (4.2) and (4.3), respectively.
where is given in (2.9).
It is seen that applying (4.14) to (4.11) proves (4.1). Employing (3.11), (3.13), (3.14), and (3.15) in (4.13), we obtain the expressions (4.4), (4.5), and (4.6).
which, upon using (3.11) and (3.14) in the first finite series, yields (4.8). □
By setting in (4.1), we give as in the following corollary.
The author would like to express his gratitude for the referees’ helpful comments to have this paper in the present improved form. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (2010-0011005).
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