SpringerLink
Log in

New Approximations for the Higher Order Coefficients in an Asymptotic Expansion for the Barnes G-Function

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we provide new formulas for determining the coefficients appearing in the asymptotic expansion for the Barnes G-function as n tends to infinity for certain classes of asymptotic expansion for the Barnes G-function. We remark that our formulas can be used to approximate the coefficients appearing in an asymptotic expansion of the “random matrix factor” from the Keathing-Snaith conjecture and the coefficients appearing in an asymptotic expansion of the “Lévy-Khintchine type representation of the reciprocal of the Barnes G-function”.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adamchik, V.S.: On the Barnes function. In: Proceedings of the 2001 International Symposium on Symbolic and Algebraic Computation, (July 22–25, 2001, London, Canada),ACM, Academic Press, pp. 15–20 (2001)

  2. Symbolic and numeric computations of the Barnes function. Comput. Phys. Commun. 157, 181–190 (2004)

  3. Barnes, E.W.: The genesis of the double gamma functions. Proc. Lond. Math. Soc. 31, 358–381 (1899)

    Article  MathSciNet  Google Scholar 

  4. Barnes, E.W.: The theory of the G-function. Q. J. Pure Appl. Math. 31, 264–314 (1900)

    MATH  Google Scholar 

  5. Barnes, E.W.: The theory of the double gamma function. Philos. Trans. R. Soc. Lond. A 196, 265–388 (1901)

    Article  Google Scholar 

  6. Barnes, E.W.: On the theory of the multiple gamma function. Trans. Camb. Philos. Soc. 19, 374–425 (1904)

    Google Scholar 

  7. Bender, E.A.: An asymptotic expansion for the coefficients of some formal power series. J. Lond. Math. Soc. 9, 451–458 (1975)

    Article  MathSciNet  Google Scholar 

  8. Bender, E.A., Richmond, L.B.: An asymptotic expansion for the coefficients of some formal power series II: Lagrange Inversion. Discrete Math. 50, 135–141 (1984)

    Article  MathSciNet  Google Scholar 

  9. Berry, M.V., Howls, C.J.: Infinity interpreted. Phys. World 6, 35–39 (1993)

    Article  Google Scholar 

  10. Boyd, W.G.C.: Approximations for the late coefficients in asymptotic expansions arising in the method of steepest descents. Methods Appl. Anal. 2, 475–489 (1995)

    Article  MathSciNet  Google Scholar 

  11. Chen, C.-P.: Asymptotic expansions for Barnes \(G\)-function. J. Number Theory 135, 36–42 (2014)

    Article  MathSciNet  Google Scholar 

  12. Choi, J.: Determinant of Laplacians on \(S^3\). Math. Jpn 40, 155–166 (1994)

    Google Scholar 

  13. Comtet, L.: Advanced Combinotorics. Riedel, Dordrecht (1974)

    Book  Google Scholar 

  14. Ferreira, C., López, J. L.: An asymptotic expansion of the double gamma function. J. Approx. Theory 111, 298–314

  15. Glaisher, J.W.L.: On a numerical continued product. Messenger Math. 6, 71–76 (1877)

    Google Scholar 

  16. Issaka, A.: On Ramanujan’s Inverse digamma approximation. Ramanujan J. 39, 291–302 (2016)

    Article  MathSciNet  Google Scholar 

  17. Keating, J.P., Snait, N.C.: Random matrix theory and \(\zeta (1/2+it)\). Commun. Math. Phys. 214, 57–89 (2000)

    Article  MathSciNet  Google Scholar 

  18. Kinkelin, Ueber eine mit der gammafunction verwandte transcendente und deren anwendung auf die integralrechnung. J. Reine Angew. Math. 57, 122–158 (1860)

  19. Nemes, G.: On the coefficents of the asymptotic expansion of \(n!\). Integer Seqs. 13, 1069 (2010)

    Google Scholar 

  20. Nemes, G.: Approximations for the higher order coefficients in an asymptotic expansion for the Gamma function. J. Math. Anal. Appl. 396, 417–424 (2012)

    Article  MathSciNet  Google Scholar 

  21. Nemes, G.: New asymptotic expansion for the Gamma function. Arch. Math. 95, 161–169 (2010)

    Article  MathSciNet  Google Scholar 

  22. Nemes, G.: Error bounds and exponential improvement for the asymptotic expansion of the Barnes \(G\)-function. Proc. R. Soc. A Math. Phys. Eng. Sci. 470(2172), 14 (2014)

    MathSciNet  MATH  Google Scholar 

  23. Nikeghbali, A., Yor, M.: The Barnes G function and its relations with sums and products of generalized Gamma. Elect. Commun. Probab. 14, 396–411 (2009)

    MathSciNet  MATH  Google Scholar 

  24. Cassou-Nogués P.: Analogues p-adiques des fonctions \(\Gamma \)-multiples. In: ‘Journées Arithmétiques de Luminy, Colloq. Internat. CNRS, Centre Univ. Luminy, Luminy, 1978’ Astérisque, 61, 43–55 (1978)

  25. Odlyzko, A.M.: Asymptotic enumeration methods. In: Graham, R.L., Grẗschel, M., Lovász, L. (eds.) Handbook of Combinatorics, vol. II, pp. 1063–1229. MIT Press, Cambridge (1995)

  26. Osgood, B., Phillips, R., Sarnak, P.: Extremal of determinants of Laplacians. J. Funct. Anal. 80, 148–211 (1988)

    Article  MathSciNet  Google Scholar 

  27. Quine, J.R., Choi, Z.J.: regularized products and functional determinants on spheres. Rocky Mountain J. Math. 26, 719–729 (1996)

    Article  MathSciNet  Google Scholar 

  28. Sarnak, P.: Determinants of Laplacians. Commun. Math. Phys. 110, 113–120 (1987)

    Article  MathSciNet  Google Scholar 

  29. Todorov, P.G.: Taylor expansion of analytic functions related to \((1+z)^x-1\), J. Math. Anal. Appl. 132, 264–280 (1988)

  30. Vardi, I.: Determinants of Laplacians and multiple gamma functions. SIAM J. Math. Anal. 19, 493–507 (1988)

    Article  MathSciNet  Google Scholar 

  31. Xu, Z., Wang, W.: More asymptotic expansions for the Barnes \(G\)-function. J. Number Theory 174, 505–517 (2017)

    Article  MathSciNet  Google Scholar 

  32. Xu, A.: Asymptotic expansions related to the Glaisher–Kinkelin constant and its analogues. J. Number Theory 163 (2016)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of North Carolina at Charlotte, 9201 University City Blvd, Charlotte, 28223, NC, USA

    Aziz Issaka

Authors
  1. Aziz Issaka
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Aziz Issaka.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Issaka, A. New Approximations for the Higher Order Coefficients in an Asymptotic Expansion for the Barnes G-Function. Results Math 77, 45 (2022). https://doi.org/10.1007/s00025-021-01579-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00025-021-01579-z

Keywords

Mathematics Subject Classification

Search

Navigation