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Journal of Mathematical Sciences

, Volume 234, Issue 5, pp 680–696 | Cite as

An Inverse Factorial Series for a General Gamma Ratio and Related Properties of the Nørlund–Bernoulli Polynomials

  • D. B. Karp
  • E. G. Prilepkina
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The inverse factorial series expansion for the ratio of products of gamma functions whose arguments are linear functions of the variable is found. A recurrence relation for the coefficients in terms of the Nørlund–Bernoulli polynomials is provided, and the half-plane of convergence is determined. The results obtained naturally supplement a number of previous investigations of the gamma ratios, which began in the 1930-ies. The expansion obtained in this paper plays a crucial role in the study of the behavior of the delta-neutral Fox’s H-function in the neighborhood of its finite singular point. A particular case of the inverse factorial series expansion is used in deriving a possibly new identity for the Nørlund–Bernoulli polynomials.

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References

  1. 1.
    T. W. Anderson, An Introduction to Multivariate Statistical Analysis, 2nd. ed., Wiley (1984).Google Scholar
  2. 2.
    G. E. P. Box, “A general distribution theory for a class of likelihood criteria,” Biometrika, 36, 317–346 (1949).MathSciNetCrossRefGoogle Scholar
  3. 3.
    B. L. J. Braaksma, “Asymptotic expansions and analytic continuation for a class of Barnes integrals,” Comp. Math., 15, No. 3, 239–341 (1962–64).Google Scholar
  4. 4.
    A. Z. Broder, “The r-Stirling numbers,” Discrete Math., 49, 241–259 (1984).MathSciNetCrossRefGoogle Scholar
  5. 5.
    L. Carlitz, “Weighted Stirling numbers of the first and second kind. I,” Fibonacci Quart., 18, 147–162 (1980).MathSciNetzbMATHGoogle Scholar
  6. 6.
    L. Carlitz, “Weighted Stirling numbers of the first and second kind. II,” Fibonacci Quart., 18, 242–257 (1980).MathSciNetzbMATHGoogle Scholar
  7. 7.
    C. A. Charalambides, Enumerative Combinatorics, Chapman and Hall/CRC (2002).Google Scholar
  8. 8.
    O. Costin, Asymptotics and Borel summability, Chapman and Hall/CRC (2009).Google Scholar
  9. 9.
    A. B. Olde Daalhuis, “Inverse factorial-series solutions of difference equations,” Proc. Edinburgh Math. Soc., 47, 421–448 (2004).MathSciNetCrossRefGoogle Scholar
  10. 10.
    E. Delabaere and J.-M. Rasoamanana, “Sommation effective d’une somme de Borel par séries de factorielles,” Ann. Inst. Fourier, 57, No. 2, 421–456 (2007).MathSciNetCrossRefGoogle Scholar
  11. 11.
    K. Dilcher, “Asymptotic behaviour of Bernoulli, Euler, and generalized Bernoulli polynomials,” J. Appr. Theory, 49, 321–330 (1987).MathSciNetCrossRefGoogle Scholar
  12. 12.
    D. W. H. Gillam and V. P. Gurarii, “On functions uniquely determined by their asymptotic expansion,” Funct. Analysis Appl., 40, No. 4, 273–284 (2006).MathSciNetCrossRefGoogle Scholar
  13. 13.
    W. B. Ford, “The asymptotic developments of functions defined by Maclaurin series,” Univ. Michigan Stud., Sci. Ser., 11 (1936).Google Scholar
  14. 14.
    A. K. Gupta and J. Tang, “On a general distribution for a class of likelihood ratio criteria,” Austral. J. Stat., 30, No. 3, 359–366 (1988).MathSciNetCrossRefGoogle Scholar
  15. 15.
    H. K. Hughes, “The asymptotic developments of a class of entire functions,” Bull. Amer. Math. Soc., 51, 456–461 (1945).MathSciNetCrossRefGoogle Scholar
  16. 16.
    V. M. Kalinin, “Special functions and the limit properties of probability distributions. I,” Zap. Nauchn. Semin. LOMI, 13, 5–137 (1969).MathSciNetzbMATHGoogle Scholar
  17. 17.
    D. Karp and E. Prilepkina, “Completely monotonic gamma ratio and infinitely divisible H-function of Fox,” Comput. Meth. Funct. Theory, 16, 135–153 (2016).MathSciNetCrossRefGoogle Scholar
  18. 18.
    D. Karp and E. Prilepkina, “Some new facts concerning the delta neutral case of Fox’s H-function,” Comput. Meth. Funct. Theory, 17, No. 2, 343–367 (2017).MathSciNetCrossRefGoogle Scholar
  19. 19.
    A. A. Kilbas and M. Saigo, H-Transforms: Theory and Applications, Chapman & Hall/CRC (2004).Google Scholar
  20. 20.
    M. Koutras, “Non-central Stirling numbers and some applications,” Discrete Math., 42, 73–89 (1982).MathSciNetCrossRefGoogle Scholar
  21. 21.
    E. Landau, “Über die Grundlagen der Theorie der Fakultätenreihen,” Sitzsber. Akad. München, 36, 151–218 (1906).zbMATHGoogle Scholar
  22. 22.
    L. M. Milne-Thompson, The Calculus of Finite Differences, Macmillan and Co. Ltd (1933).Google Scholar
  23. 23.
    C. Mitschi and D. Sauzin, Divergent Series, Monodromy and Resurgence. I (Lect. Notes Math., 2153), Springer (2016).Google Scholar
  24. 24.
    U. S Nair, “The application of the moment function in the study of distribution laws in statistics,” Biometrika, 30, No. 3/4, 274–294 (1939).CrossRefGoogle Scholar
  25. 25.
    G. Nemes, “Generalization of Binet’s gamma function formulas,” Int. Transforms Spec. Funct., 24, No. 8, 597–606 (2013).MathSciNetCrossRefGoogle Scholar
  26. 26.
    F. Nevanlinna, “Zur Theorie der asymptotischen Potenzreihen,” Ann. Acad. Sci. Fenn., Ser. A, 12, No. 3, 1–81 (1919).Google Scholar
  27. 27.
    N. Nielsen, Die Gammafunktion, Chelsea, New York (1965); originally published by Teubner, Leipzig and Berlin (1906).Google Scholar
  28. 28.
    N. E. Norlund, “Sur les séries de facultés,” Acta Math., 37, No. 1, 327–387 (1914).MathSciNetCrossRefGoogle Scholar
  29. 29.
    N. E. Norlund, Vorlesungen über Differenzrechnung, Springer Verlag, Berlin (1924).CrossRefGoogle Scholar
  30. 30.
    N. E. Norlund, “Hypergeometric functions,” Acta Math., 94, 289–349 (1955).MathSciNetCrossRefGoogle Scholar
  31. 31.
    N. E. Norlund, “Sur les valeurs asymptotiques des nombres et des polynômes de Bernoulli,” Rend. Circ. Mat. Palermo, 10, No. 1, 27–44 (1961).MathSciNetCrossRefGoogle Scholar
  32. 32.
    F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark (Eds.), NIST Handbook of Mathematical Functions, Cambridge Univ. Press (2010).Google Scholar
  33. 33.
    R. B. Paris and D. Kaminski, Asymptotics and Mellin–Barnes Integrals, Cambridge University Press (2001).Google Scholar
  34. 34.
    F. Qi, X-T. Shi, and F.-F. Liu, “Expansions of the exponential and the logarithm of power series and applications,” Arab. J. Math., 6, 95–108 (2017).MathSciNetCrossRefGoogle Scholar
  35. 35.
    T. D. Riney, “On the coefficients in asymptotic factorial expansions,” Proc. Amer. Math. Soc., 7, 245–249 (1956).MathSciNetCrossRefGoogle Scholar
  36. 36.
    T. D. Riney, “A finite recursion formula for the coefficients in asymptotic expansions,” Trans. Amer. Math. Soc., 88, 214–226 (1958).MathSciNetCrossRefGoogle Scholar
  37. 37.
    T. D. Riney, “Coefficients in certain asymptotic factorial expansions,” Proc. Amer. Math. Soc., 10, 511–518 (1959).MathSciNetCrossRefGoogle Scholar
  38. 38.
    J. Riordan, Combinatorial Identities, John Wiley and Sons (1968).Google Scholar
  39. 39.
    A. D. Sokal, “An improvement of Watson’s theorem on Borel summability,” J. Math. Phys., 21, No. 2, 261–263 (1980).MathSciNetCrossRefGoogle Scholar
  40. 40.
    F. Tricomi and A. Erdélyi, “The asymptotic expansion of a ratio of gamma functions,” Pacific J. Math., 1, No. 1, 133–142 (1951).MathSciNetCrossRefGoogle Scholar
  41. 41.
    J. G. van der Corput, “On the coefficients in certain asymptotic factorial expansions,” Akad. Wet. (Amsterdam) Proc., Ser. A, 60, No. 4, 337–351 (1957).Google Scholar
  42. 42.
    H. van Engen, “Concerning gamma function expansions,” Tohoku Math. J., 45, 124–129 (1939).zbMATHGoogle Scholar
  43. 43.
    G. N. Watson, “A theory of asymptotic series,” Phil. Trans. Royal Soc. London (Ser. A), 211, 279–313 (1911).CrossRefGoogle Scholar
  44. 44.
    G. N. Watson, “The transformation of an asymptotic series into a convergent series of inverse factorials,” Rend. Giro. Mat. Palermo, 34, 41–88 (1912).CrossRefGoogle Scholar
  45. 45.
    W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Dover Publications, Inc. (2002).Google Scholar
  46. 46.
    E. J. Weniger, “Summation of divergent power series by means of factorial series,” Appl. Numer. Math., 60, 1429–1441 (2010).MathSciNetCrossRefGoogle Scholar
  47. 47.
    E. M. Wright, “The asymptotic expansion of the generalized hypergeometric function,” J. London Math. Soc., 10, 287–293 (1935).Google Scholar
  48. 48.
    E. M. Wright, “The asymptotic expansion of the generalized hypergeometric function,” Proc. London Math. Soc. (2), 46, 389–408 (1940).MathSciNetCrossRefGoogle Scholar
  49. 49.
    E. M. Wright, “A recursion formula for the coefficients in an asymptotic expansion,” Glasgow Math. J., 4, No. 1, 38–41 (1958).MathSciNetCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Far Eastern Federal University and Institute of Applied Mathematics of the FEBRASVladivostokRussia

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