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Zbl 0641.33003
Vardi, Ilan
Determinants of Laplacians and multiple gamma functions. (English)
[J] SIAM J. Math. Anal. 19, No.2, 493-507 (1988). ISSN 0036-1410; ISSN 1095-7154/e

The author reinterpretes the classical formula $\Gamma(.)=\sqrt{\pi}$ in the form $$ \Gamma(.)=2\sp{-1/2}(\det \Delta\sb 1)\sp{1/4},$$ where $\Delta\sb 1=-d\sp 2/dx\sp 2$ denotes the Laplacian on $S\sp 1$. He then introduces so-called multiple Gamma functions $\Gamma\sb n$ for all $n\ge 0$ and then his main result states that $\Gamma\sb n(.)$ can be evaluated in terms of det $\Delta\sb m$ $(m=1,...,n)$, where $\Delta\sb m$ is the Laplacian on the m-sphere $S\sp m$. The proof splits into two parts: First, $\Gamma\sb n(.)$ is expressed in terms of the numbers $\zeta'(-m)$ $(m=0,1,...,n-1)$, where $\zeta$ denotes the Riemann zeta function. Second, det $\Delta\sb n$ is also expressed in terms of $\zeta'(-m)$ $(m=0,1,...,n-1)$. As a by-product, the author establishes the formula $\log A=(1/12)-\zeta'(-1)$ for the Kinkelin constant A. \par The paper under review is closely related with work of {\it A. Voros} [Commun. Math. Phys. 110, 439-465 (1987; Zbl 0631.10025)] and {\it P. Sarnak} [Commun. Math. Phys. 110, 113-120 (1987; Zbl 0618.10023)]. In particular, Voros points out that A already was computed in the literature.
[J.Elstrodt]

MSC 2000:: *33B15 Gamma-functions, etc.
58J50 Spectral problems; spectral geometry; scattering theory

Keywords: determinant of the Laplacian; Barnes' double Gamma function; Gamma functions

Citations: Zbl 0631.10025; Zbl 0618.10023

Cited in: Zbl 1068.33004

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