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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 121795, 11 pages
http://dx.doi.org/10.1155/2012/121795
Research Article

A Set of Mathematical Constants Arising Naturally in the Theory of the Multiple Gamma Functions

Department of Mathematics, Dongguk University, Gyeongju 780-714, Republic of Korea

Received 7 August 2012; Accepted 10 September 2012

Academic Editor: Sung G. Kim

Copyright © 2012 Junesang Choi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. E. W. Barnes, “The theory of the G-function,” Quarterly Journal of Mathematics, vol. 31, pp. 264–314, 1899.
  2. E. W. Barnes, “The genesis of the double gamma functions,” Proceedings of the London Mathematical Society, vol. 31, no. 1, pp. 358–381, 1899. View at Publisher · View at Google Scholar
  3. E. W. Barnes, “The theory of the double Gamma function,” Philosophical Transactions of the Royal Society A, vol. 196, pp. 265–388, 1901.
  4. E. W. Barnes, “On the theory of the multiple Gamma functions,” Transactions of the Cambridge Philosophical Society, vol. 19, pp. 374–439, 1904.
  5. O. Hölder, Uber Eine Transcendente Funktion, vol. 1886, Dieterichsche, Göttingen, Germany, 1886.
  6. W. P. Alexeiewsky, Uber Eine Classe von Funktionen, die der Gammafunktion Analog Sind, vol. 46, Leipzig Weidmannsche Buchhandlung, 1894.
  7. V. H. Kinkelin, “Uber eine mit der Gamma Funktion verwandte transcendente und deren Anwendung auf die integralrechnung,” Journal für Die Reine und Angewandte Mathematik, vol. 57, pp. 122–158, 1860.
  8. J. Choi, “Determinant of Laplacian on S3,” Mathematica Japonica, vol. 40, no. 1, pp. 155–166, 1994. View at Zentralblatt MATH
  9. H. Kumagai, “The determinant of the Laplacian on the n-sphere,” Acta Arithmetica, vol. 91, no. 3, pp. 199–208, 1999.
  10. B. Osgood, R. Phillips, and P. Sarnak, “Extremals of determinants of Laplacians,” Journal of Functional Analysis, vol. 80, no. 1, pp. 148–211, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. J. R. Quine and J. Choi, “Zeta regularized products and functional determinants on spheres,” The Rocky Mountain Journal of Mathematics, vol. 26, no. 2, pp. 719–729, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. I. Vardi, “Determinants of Laplacians and multiple gamma functions,” SIAM Journal on Mathematical Analysis, vol. 19, no. 2, pp. 493–507, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. A. Voros, “Spectral functions, special functions and the Selberg zeta function,” Communications in Mathematical Physics, vol. 110, no. 3, pp. 439–465, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. J. Choi, “Some mathematical constants,” Applied Mathematics and Computation, vol. 187, no. 1, pp. 122–140, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. J. Choi, Y. J. Cho, and H. M. Srivastava, “Series involving the zeta function and multiple gamma functions,” Applied Mathematics and Computation, vol. 159, no. 2, pp. 509–537, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. J. Choi and H. M. Srivastava, “Certain classes of series involving the zeta function,” Journal of Mathematical Analysis and Applications, vol. 231, no. 1, pp. 91–117, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. J. Choi and H. M. Srivastava, “An application of the theory of the double gamma function,” Kyushu Journal of Mathematics, vol. 53, no. 1, pp. 209–222, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. J. Choi and H. M. Srivastava, “Certain classes of series associated with the zeta function and multiple gamma functions,” Journal of Computational and Applied Mathematics, vol. 118, no. 1-2, pp. 87–109, 2000, Higher transcendental functions and their applications. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. J. Choi, H. M. Srivastava, and V. S. Adamchik, “Multiple gamma and related functions,” Applied Mathematics and Computation, vol. 134, no. 2-3, pp. 515–533, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. J. Choi, H. M. Srivastava, and J. R. Quine, “Some series involving the zeta function,” Bulletin of the Australian Mathematical Society, vol. 51, no. 3, pp. 383–393, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001.
  22. J. W. L. Glaisher, “On the product 1122nn,” Messenger of Mathematics, vol. 7, pp. 43–47, 1877.
  23. J. W. L. Glaisher, “On the constant which occurs in the formula for 1122nn,” Messenger of Mathematics, vol. 24, pp. 1–16, 1894.
  24. http://mathworld.wolfram.com/Glaisher-KinkelinConstant.html.
  25. C.-P. Chen, “Glaisher-Kinkelin constant,” Integral Transforms and Special Functions, IFirst, pp. 1–8, 2011.
  26. H. M. Srivastava and J. Choi, Zeta and Q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, The Netherlands, 2012.
  27. V. S. Adamchik, “Polygamma functions of negative order,” Journal of Computational and Applied Mathematics, vol. 100, no. 2, pp. 191–199, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. L. Bendersky, “Sur la fonction gamma généralisée,” Acta Mathematica, vol. 61, no. 1, pp. 263–322, 1933. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  29. G. H. Hardy, Divergent Series, Clarendon Press , Oxford University Press, Oxford, UK, 1949.
  30. G. H. Hardy, Divergent Series, Chelsea PublishingCompany, New York, NY, USA, 2nd edition, 1991.
  31. J. Edwards, A Treatise on the Integral Calculus with Applications: Examples and Problems, vol. 1-2, Chelsea Publishing Company, New York, NY, USA, 1954.
  32. R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley Publishing Company, Reading, Mass, USA, 2nd edition, 1994.
  33. Y.-H. Zhu and B.-C. Yang, “Accurate inequalities for partial sums of a type of divergent series,” Acta Scientiarum Naturalium Universitatis Sunyatseni, vol. 37, no. 4, pp. 33–37, 1998.