Certain classes of series associated with the Zeta function and multiple gamma functions

Journal of Computational and Applied Mathematics

Volumn 118, Issue 1-2, 2000, Pages 87-109

  Choi, Junesang ^a   Srivastava, H M ^b  

^a Dongguk University   (South Korea)

^b UNIVERSITY OF VICTORIA   (Canada)

Author keywords

11M06; 11M99; 33B99; Alexeweisky's theorem; Determinants of Laplacians; Euler Maclaurin summation formula; Hecke's L functions; Multiple gamma functions; Riemann's function

Indexed keywords

EID: 0039602863     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0377-0427(00)00311-3     Document Type: Article

References (30)

1. Alexeiewsky W. Über eine Classe von Funktionen, die der Gammafunktion analog sind. Leipzig Weidmannsche Buchhandluns. 46:1894;268-275.
2. Apostol T.M. Introduction to Analytic Number Theory. 1976;Springer, New York.
3. Barnes E.W. The theory of the. G -function Quart. J. Math. 31:1899;264-314.
4. Barnes E.W. Genesis of the double Gamma function. Proc. London Math. Soc. 31:1900;358-381.
5. Barnes E.W. The theory of the double Gamma function. Philos. Trans. Roy. Soc. London Ser. A. 196:1901;265-388.
6. Barnes E.W. On the theory of the multiple Gamma function. Philos. Trans. Roy. Soc. London Ser. A. 199:1904;374-439.
7. Cassou-Noguès P. Analogues. p -adiques des fonctions Γ -multiples Journées Arithmétiques de Luminy (Colloq. Internat. CNRS, Centre Univ. Luminy, Luminy, 1978) Astérisque. 61:1979;43-55.
8. Chen M.-P., Srivastava H.M. Some families of series representations for the Riemann. ζ(3) Resultate Math. 33:1998;179-197.
9. 3 Math. Japon. 40:1994;155-166.
10. Choi J., Srivastava H.M. Sums associated with the Zeta function. J. Math. Anal. Appl. 206:1997;103-120.
11. Choi J., Srivastava H.M. Certain classes of series involving the Zeta function. J. Math. Anal. Appl. 231:1999;91-117.
12. Choi J., Srivastava H.M. An application of the theory of the double Gamma function. Kyushu J. Math. 53:1999;209-222.
13. Choi J., Srivastava H.M., Quine J.R. Some series involving the Zeta function. Bull. Austral. Math. Soc. 51:1995;383-393.
14. Davis H.T. The Summation of Series. 1962;The Principia Press of Trinity University, San Antonio, TX.
15. Edwards J. A Trestise on the Integral Calculus with Applications, Examples and Problems, Vol. 2. 1954;Chelsea, New York.
16. Gradshteyn I.S., Ryzhik I.M. Table of Integrals, Series, and Products (Corrected and Enlarged Ed. prepared by A. Jeffrey). 1980;Academic Press, New York.
17. Hardy G.H. Divergent Series. 1949;Clarendon (Oxford University) Press, Oxford.
18. D'Hoker E., Phong D.H. On determinants of Laplacians on Riemann surfaces. Comm. Math. Phys. 104:1986;537-545.
19. V.O. Hölder, Über eine Transcendente Funktion, Dieterichsche Verlags-Buchhandlung, Göttingen, 1886, pp. 514-522.
20. Kinkelin V.H. Über eine mit der Gamma Funktion verwandte Transcendente und deren Anwendung auf die Integralrechnung. J. Reine Angew. Math. 57:1860;122-158.
21. Matsumoto K. Asymptotic series for double Zeta, double Gamma, and Hecke. L -functions Math. Proc. Cambridge Philos. Soc. 123:1998;385-405.
22. Osgood B., Phillips R., Sarnak P. Extremals of determinants of Laplacians. J. Funct. Anal. 80:1988;148-211.
23. Quine J.R., Choi J. Zeta regularized products and functional determinants on spheres. Rocky Mountain J. Math. 26:1996;719-729.
24. Shintani T. A proof of the classical Kronecker limit formula. Tokyo J. Math. 3:1980;191-199.
25. Srivastava H.M. A unified presentation of certain classes of series of the Riemann Zeta function. Riv. Mat. Univ. Parma (4). 14:1988;1-23.
26. Srivastava H.M. Some rapidly converging series for. ζ(2n+1) Proc. Amer. Math. Soc. 127:1999;385-396.
27. Vardi I. Determinants of Laplacians and multiple Gamma functions. SIAM J. Math. Anal. 19:1988;493-507.
28. Vignéras M.-F. L'équation fonctionnelle de la fonction Zêta de Selberg du groupe moudulaire. PSL(2,Z) Journées Arithmétiques de Luminy (Colloq. Internat. CNRS, Centre Univ. Luminy, Luminy, 1978), Astérisque. 61:1979;235-249.
29. Voros A. Special functions, spectral functions and the Selberg Zeta function. Comm. Math. Phys. 110:1987;439-465.
30. Whittaker E.T., Watson G.N. A Course of Modern Analysis. Fourth Edition:1963;Cambridge University Press, Cambridge.