Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Calculation of the moments and the moment generating function for the reciprocal gamma distribution

Authors: Arne Fransén and Staffan Wrigge
Journal: Math. Comp. 42 (1984), 601-616
MSC: Primary 65D20; Secondary 60E10, 62E15, 65U05
MathSciNet review: 736456
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we consider the distribution $ G(x) = {F^{ - 1}}\smallint _0^x{(\Gamma (t))^{ - 1}}\;dt$. The aim of the investigation is twofold: first,to find numerical values of characteristics such as moments, variance, skewness, kurtosis,etc.; second, to study analytically and numerically the moment generating function $ \varphi (t) = \smallint _0^\infty {e^{ - tx}}/\Gamma (x)\;dx$. Furthermore, we also make a generalization of the reciprocal gamma distribution, and study some of its properties.

References [Enhancements On Off] (What's this?)

  • [1] J. J. Dorning, B. Nicolaenko & J. K. Thurber, "An integral identity due to Ramanujan which occurs in neutron transport theory," J. Math. Mech., v. 19, No. 5, 1969, pp. 429-438. MR 0254298 (40:7507)
  • [2] A. Erdélyi, W. Magnus, F. Oberhettinger & F. G. Tricomi, Higher Transcendental Functions, Vol. Ill, McGraw-Hill, New York, 1955.
  • [3] A. Fransén, Addendum and Corrigendum to "High-precision values of the gamma function and of some related coefficients," Math. Comp., v. 37, 1981, pp. 233-235. MR 616377 (82m:65002)
  • [4] A. Fransén & S. Wrigge, "High-precision values of the gamma function and of some related coefficients," Math. Comp., v. 34, 1980, pp. 553-566. MR 559204 (81f:65004)
  • [5] W. Gautschi, "Polynomials orthogonal with respect to the reciprocal gamma function," BIT, v. 22, 1982, pp. 387-389. MR 675673 (84h:65024)
  • [6] G. H. Golub & J. H. Welsch, "Calculation of Gauss quadrature rules," Math. Comp., v. 23, 1969, pp. 221-230, Microfiche supplement A1-A10. MR 0245201 (39:6513)
  • [7] S. A. Gustafson, "Rapid computation of general interpolation formulas and mechanical quadrature rules," Comm. ACM, v. 14, 1971, pp. 797-801, Algorithm 417, p. 807. MR 0311069 (46:10167a)
  • [8] G. H. Hardy, Ramanujan--Twelve Lectures on Subjects Suggested by His Life and Work, (reprinted), Chelsea, New York, 1959.
  • [9] Collected papers of G. H. Hardy, Vols. I-VII (Especially Vol. IV, pp. 544-548), Oxford at the Clarendon Press, 1969.
  • [10] W. A. Johnson, Private communication, 1982.
  • [11] A. Lindhagen, Studier öfver Gamma-Funktionen och Några Beslägtade Transcendenter (Studies of the gamma function and of some related transcendental), Doctoral Thesis, B. Almqvist & J. Wiksell's boktryckeri, Upsala, 1887.
  • [12] W. F. Lunnon, P. A. B. Pleasants and N. M. Stephens, "Arithmetic properties of Bell numbers to a composite modulus I," Acta Arith., v. 35, 1979, pp. 1-16. MR 536875 (80k:05006)
  • [13] R. E. A. C. Paley & N. Wiener, Fourier Transforms in the Complex Domain, Amer. Math. Soc., New York, 1934. MR 1451142 (98a:01023)
  • [14] J. Riordan, Combinatorial Identities, Wiley, New York, 1968. MR 0231725 (38:53)
  • [15] I. N. Sneddon, Special Functions of Mathematical Physics and Chemistry, Oliver and Boyd, Edinburgh and London, 1961. MR 0080170 (18:204a)
  • [16] S. Wrigge, "A note on the moment generating function for the reciprocal gamma distribution," Math. Comp., v. 42, 1984, pp. 617-621. MR 736457 (86f:65042b)
  • [17] M. Wyman & R. Wong, "The asymptotic behaviour of $ \mu (z,\beta ,\alpha )$," Canad. J. Math., v. 21, 1969, pp. 1013-1023. MR 0244521 (39:5835)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65D20, 60E10, 62E15, 65U05

Retrieve articles in all journals with MSC: 65D20, 60E10, 62E15, 65U05

Additional Information

Keywords: Reciprocal gamma distribution, population characteristics, generating function
Article copyright: © Copyright 1984 American Mathematical Society

American Mathematical Society