Special values of multiple gamma functions

Duke, William; Imamoḡlu, Özlem

Journal de théorie des nombres de Bordeaux, Volume 18 (2006) no. 1, pp. 113-123.

We give a Chowla-Selberg type formula that connects a generalization of the eta-function to $GL\left(n\right)$ with multiple gamma functions. We also present some simple infinite product identities for certain special values of the multiple gamma function.

Nous donnons une formule de type Chowla-Selberg qui relie une généralisation de la fonction éta à $GL\left(n\right)$ avec les fonctions gamma multiples. Nous présentons également quelques identités de produit infinis pour certaines valeurs spéciales de la fonction gamma multiple.

[1] V. Adamchik, Multiple Gamma Function and Its Application to Computation of Series. Ramanujan Journal 9 (2005), 271–288. | MR | Zbl

[2] E. W. Barnes, On the theory of the multiple Gamma function. Cambr. Trans. 19 (1904), 374–425.

[3] J. Elstrodt, F. Grunewald, J. Mennicke, Groups acting on hyperbolic space. Harmonic analysis and number theory. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. | MR | Zbl

[4] P. Epstein, Zur Theorie allgemeiner Zetafunktionen. Math. Ann. 56 (1903), 615–644. | MR

[5] E. Hecke, Analytische Arithmetik der positive quadratischen Formen, (1940) # 41 in Mathematische Werke. | MR | Zbl

[6] M.J. Liouville, Journal de Mathématiques Pure et Appliquées.

[7] S. Minakshisundaram, Å. Pleijel, Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds. Canadian J. Math. 1 (1949), 242–256. | MR | Zbl

[8] J. Dufresnoy, Ch. Pisot, Sur la relation fonctionnelle $f(x+1)-f\left(x\right)=\varphi \left(x\right)$. Bull. Soc. Math. Belg. 15 (1963), 259–270. | MR | Zbl

[9] P. Sarnak, Determinants of Laplacians; heights and finiteness. Analysis, et cetera, 601–622, Academic Press, Boston, MA, 1990. | MR | Zbl

[10] S. Selberg, S. Chowla, On Epstein’s zeta-function. J. Reine Angew. Math. 227 (1967), 86–110. | MR | Zbl

[11] T. Shintani, On special values of zeta functions of totally real algebraic number fields. Proceedings of the International Congress of Mathematicians (Helsinki, 1978), pp. 591–597, Acad. Sci. Fennica, Helsinki, 1980. | MR | Zbl

[12] C. L. Siegel, Über die analytische Theorie der quadratischen Formen. Ann. of Math. (2) 36 (1935), no. 3, 527–606. [in Gesammelte Abhandlungen] | MR | Zbl

[13] C. L. Siegel, Lectures on advanced analytic number theory. Notes by S. Raghavan. Tata Institute of Fundamental Research Lectures on Mathematics, No. 23 Tata Institute of Fundamental Research, Bombay 1965. | MR | Zbl

[14] H. M. Srivastava, J. Choi, Series associated with the zeta and related functions. Kluwer Academic Publishers, Dordrecht, 2001. | MR | Zbl

[15] A. Terras, Bessel series expansions of the Epstein zeta function and the functional equation. Trans. Amer. Math. Soc. 183 (1973), 477–486. | MR | Zbl

[16] I. Vardi, Determinants of Laplacians and multiple gamma functions. SIAM J. Math. Anal. 19 (1988), no. 2, 493–507. | MR | Zbl

[17] B. A. Venkov, Elementary number theory. Translated from the Russian and edited by Helen Alderson Wolters-Noordhoff Publishing, Groningen 1970. | MR | Zbl

[18] M.-F. Vignéras, L’équation fonctionnelle de la fonction zêta de Selberg du groupe modulaire $PSL(2,\mathbb{Z})$. Journées Arithmétiques de Luminy , 235–249, Astérisque 61 , Soc. Math. France, Paris, 1979. | Numdam | MR | Zbl