## Abstract and Applied Analysis

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

Junesang Choi

#### Abstract

We introduce a set of mathematical constants which is involved naturally in the theory of multiple Gamma functions. Then we present general asymptotic inequalities for these constants whose special cases are seen to contain all results very recently given in Chen 2011.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 121795, 11 pages.

Dates
First available in Project Euclid: 28 March 2013

https://projecteuclid.org/euclid.aaa/1364475998

Digital Object Identifier
doi:10.1155/2012/121795

Mathematical Reviews number (MathSciNet)
MR3004899

Zentralblatt MATH identifier
1256.33002

#### Citation

Choi, Junesang. A Set of Mathematical Constants Arising Naturally in the Theory of the Multiple Gamma Functions. Abstr. Appl. Anal. 2012 (2012), Article ID 121795, 11 pages. doi:10.1155/2012/121795. https://projecteuclid.org/euclid.aaa/1364475998

#### References

• E. W. Barnes, “The theory of the G-function,” Quarterly Journal of Mathematics, vol. 31, pp. 264–314, 1899.
• E. W. Barnes, “The genesis of the double gamma functions,” Proceedings of the London Mathematical Society, vol. 31, no. 1, pp. 358–381, 1899.
• E. W. Barnes, “The theory of the double Gamma function,” Philosophical Transactions of the Royal Society A, vol. 196, pp. 265–388, 1901.
• E. W. Barnes, “On the theory of the multiple Gamma functions,” Transactions of the Cambridge Philosophical Society, vol. 19, pp. 374–439, 1904.
• O. Hölder, Uber Eine Transcendente Funktion, vol. 1886, Dieterichsche, Göttingen, Germany, 1886.
• W. P. Alexeiewsky, Uber Eine Classe von Funktionen, die der Gammafunktion Analog Sind, vol. 46, Leipzig Weidmannsche Buchhandlung, 1894.
• 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.
• J. Choi, “Determinant of Laplacian on ${S}^{3}$,” Mathematica Japonica, vol. 40, no. 1, pp. 155–166, 1994.
• H. Kumagai, “The determinant of the Laplacian on the $n$-sphere,” Acta Arithmetica, vol. 91, no. 3, pp. 199–208, 1999.
• B. Osgood, R. Phillips, and P. Sarnak, “Extremals of determinants of Laplacians,” Journal of Functional Analysis, vol. 80, no. 1, pp. 148–211, 1988.
• 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.
• I. Vardi, “Determinants of Laplacians and multiple gamma functions,” SIAM Journal on Mathematical Analysis, vol. 19, no. 2, pp. 493–507, 1988.
• A. Voros, “Spectral functions, special functions and the Selberg zeta function,” Communications in Mathematical Physics, vol. 110, no. 3, pp. 439–465, 1987.
• J. Choi, “Some mathematical constants,” Applied Mathematics and Computation, vol. 187, no. 1, pp. 122–140, 2007.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001.
• J. W. L. Glaisher, “On the product ${1}^{1}$${2}^{2}\cdots {n}^{n},” Messenger of Mathematics, vol. 7, pp. 43–47, 1877. • J. W. L. Glaisher, “On the constant which occurs in the formula for {1}^{1}$${2}^{2}\cdots {n}^{n}$,” Messenger of Mathematics, vol. 24, pp. 1–16, 1894.
• http://mathworld.wolfram.com/Glaisher-KinkelinConstant.html.
• C.-P. Chen, “Glaisher-Kinkelin constant,” Integral Transforms and Special Functions, IFirst, pp. 1–8, 2011.
• H. M. Srivastava and J. Choi, Zeta and Q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, The Netherlands, 2012.
• V. S. Adamchik, “Polygamma functions of negative order,” Journal of Computational and Applied Mathematics, vol. 100, no. 2, pp. 191–199, 1998.
• L. Bendersky, “Sur la fonction gamma généralisée,” Acta Mathematica, vol. 61, no. 1, pp. 263–322, 1933.
• G. H. Hardy, Divergent Series, Clarendon Press, Oxford University Press, Oxford, UK, 1949.
• G. H. Hardy, Divergent Series, Chelsea PublishingčommentComment on ref. [24b?]: We split this reference to [24a,24b?]. Please check.Company, New York, NY, USA, 2nd edition, 1991.
• J. Edwards, A Treatise on the Integral Calculus with Applications: Examples and Problems, vol. 1-2, Chelsea Publishing Company, New York, NY, USA, 1954.
• 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.
• 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.