Series involving the Zeta function and multiple Gamma functions

Applied Mathematics and Computation

Volumn 159, Issue 2, 2004, Pages 509-537

  Choi, Junesang ^a   Cho, Young Joon ^b   Srivastava, H M ^c  

^a Dongguk University   (South Korea)

^b Pusan National University   (South Korea)

^c UNIVERSITY OF VICTORIA   (Canada)

Author keywords

Determinants of the Laplacians; Glaisher Kinkelin constant; Hurwitz (or generalized) Zeta function; Meromorphic continuation; Multiple Gamma functions; Regularization procedure; Riemann's function; Series involving the Zeta function

Indexed keywords

COMPUTATIONAL METHODS; MATHEMATICAL MODELS; MATHEMATICAL OPERATORS; STATISTICAL METHODS; THEOREM PROVING;

FINITE PART PRESCRIPTION; MULTIPLE GAMMA FUNCTIONS; RIEMANN METHOD; WEIERSTRASS CANONICAL PRODUCTS;

FUNCTIONS;

EID: 5644235276     PISSN: 00963003     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.amc.2003.08.134     Document Type: Article

References (46)

1. Adamchik V.S. Polygamma functions of negative order. J. Comput. Appl. Math. 100:1998;191-199.
2. V.S. Adamchik, Integral representations for the Barnes function (Preprint, 2001).
3. Adamchik V.S. On the Barnes function. Mourrain B. Proceedings of the 2001 International Symposium on Symbolic and Algebraic Computation. London, Ontario, 22-25 July 2001 :2001;15-20 ACM Press, New York.
4. Adamchik V.S., Srivastava H.M. Some series of the Zeta and related functions. Analysis. 18:1998;131-144.
5. Alexeiewsky W.P. Über eine Classe von Funktionen, die der Gammafunktion analog sind. Leipzig Weidmannsche Buchhandlung. 46:1894;268-275.
6. Apostol T.M. Some series involving the Riemann Zeta function. Proc. Amer. Math. Soc. 5:1954;239-243.
7. Barnes E.W. The theory of the. G -function Quart. J. Math. 31:1899;264-314.
8. Barnes E.W. On the theory of the multiple Gamma function. Trans. Cambridge Philos. Soc. 19:1904;374-439.
9. Bendersky L. Sur la fonction Gamma généralisée. Acta Math. 61:1933;263-322.
10. P. Cassou-Noguès, Analogues p-adiques des fonctions Γ -multiples, in: Journées Arithmétiques de Luminy (Colloq. Internat. CNRS, Centre Univ. Luminy, Luminy, 1978), Astérisque, 61, Soc. Math. France, Paris, 1979, pp. 43-55.
11. 3 Math. Japon. 40:1994;155-166.
12. Choi J., Quine J.R. E.W. Barnes' approach of the multiple Gamma functions. J. Korean Math. Soc. 29:1992;127-140.
13. Choi J., Seo T.Y. The double Gamma function. East Asian Math. J. 13:1997;159-174.
14. Choi J., Srivastava H.M. Sums associated with the Zeta function. J. Math. Anal. Appl. 206:1997;103-120.
15. Choi J., Srivastava H.M. Certain classes of series involving the Zeta function. J. Math. Anal. Appl. 231:1999;91-117.
16. Choi J., Srivastava H.M. An application of the theory of the double Gamma function. Kyushu J. Math. 53:1999;209-222.
17. Choi J., Srivastava H.M. Certain classes of series associated with the Zeta function and multiple Gamma functions. J. Comput. Appl. Math. 118:2000;87-109.
18. Choi J., Srivastava H.M. A certain class of series associated with the Zeta function. Integral Transform. Spec. Funct. 12:2001;237-250.
19. Choi J., Srivastava H.M. A certain family of series associated with the Zeta and related functions. Hiroshima Math. J. 32:2002;417-429.
20. Choi J., Srivastava H.M., Adamchik V.S. Multiple Gamma and related functions. Appl. Math. Comput. 134:2003;515-533.
21. Choi J., Srivastava H.M., Quine J.R. Some series involving the Zeta function. Bull. Austral. Math. Soc. 51:1995;383-393.
22. D'Hoker E., Phong D.H. On determinant of Laplacians on Riemann surface. Comm. Math. Phys. 104:1986;537-545.
23. Dufresnoy J., Pisot Ch. Sur la relation fonctionnelle. f(x+1)-f(x)=φ(x) Bull. Soc. Math. Belg. 15:1963;259-270.
24. n Messenger Math. 7:1877;43-47.
25. Gradshteyn I.S., Ryzhik I.M. Table of Integrals, Series, and Products. Sixth ed. 2000;Academic Press, New York. (Corrected and Enlarged Ed. prepared by A. Jeffrey, D. Zwillinger).
26. Hölder O. Über eine von Abel untersuchte Transzendente und eine merkwürdige Funktionalbeziehung. Beriche über die Verhandlungen der Saechsischen Akademie der Wissenschaften zu Leipzig. Mathematisch-Physische Klasse. 80:1928;312-325.
27. Hölder O. Über eine Transcendente Funktion. 1886;Dieterichsche Verlags-Buchhandlung, Göttingen. pp. 514-522.
28. Kanemitsu S., Katsurada M., Yoshimoto M. On the Hurwitz-Lerch Zeta-function. Aequationes Math. 59:2000;1-19.
29. Kanemitsu S., Kumagai H., Yoshimoto M. Sums involving the Hurwitz Zeta function. Ramanujan J. 5:2001;5-19.
30. 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.
31. Kumagai H. The determinant of the Laplacian on the. n -sphere Acta Arith. 91:1999;199-208.
32. Matsumoto K. Asymptotic series for double Zeta, double Gamma, and Hecke. L -functions Math. Proc. Cambridge Philos. Soc. 123:1998;385-405 Corrigendum Addendum 132 (2002) 377-384.
33. Matsumoto K. Asymptotic series for double Zeta, double Gamma, and Hecke. L -functions Math. Proc. Cambridge Philos. Soc. 123:1998;385-405 Corrigendum Addendum 132 (2002) 377-384.
34. Osgood B., Phillips R., Sarnak P. Extremals of determinants of Laplacians. J. Funct. Anal. 80:1988;148-211.
35. Quine J.R., Choi J. Zeta regularized products and functional determinants on spheres. Rocky Mountain J. Math. 26:1996;719-729.
36. Sarnak P. Determinants of Laplacians. Comm. Math. Phys. 110:1987;113-120.
37. Shallit J.D., Zikan K. A theorem of Goldbach. Amer. Math. Monthly. 93:1986;402-403.
38. Shintani T. A proof of the classical Kronecker limit formula. Tokyo J. Math. 3:1980;191-199.
39. Srivastava H.M. A unified presentation of certain classes of series of the Riemann Zeta function. Riv. Mat. Univ. Parma (Ser. 4). 14:1988;1-23.
40. Srivastava H.M. Sums of certain series of the Riemann Zeta function. J. Math. Anal. Appl. 134:1988;129-140.
41. Srivastava H.M., Choi J. Series Associated with the Zeta and Related Functions. 2001;Kluwer Academic Publishers, Dordrecht, Boston, and London.
42. A. Terras, Harmonic Analysis on Symmetric Spaces and Applications, Vol. I, Springer-Verlag, New York, 1985.
43. Vardi I. Determinants of Laplacians and multiple Gamma functions. SIAM J. Math. Anal. 19:1988;493-507.
44. M.-F. Vignéras. L'équation fonctionnelle de la fonction Zêta de Selberg du groupe moudulaire PSL(2,Z), in: Journées Arithmétiques de Luminy (Colloq. Internat. CNRS, Centre Univ. Luminy, Luminy, 1978), Astérisque, 61, Soc. Math. France, Paris, 1979, pp. 235-249.
45. Voros A. Special functions, spectral functions and the Selberg Zeta function. Commun. Math. Phys. 110:1987;439-465.
46. Whittaker E.T., Watson G.N. A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; With an Account of the Principal Transcendental Functions. Fourth Ed. 1963;Cambridge University Press, Cambridge, London, and New York.