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

DOI:
https://doi.org/10.1090/S0025-5718-1984-0736456-3

MathSciNet review:
736456

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Abstract: In this paper we consider the distribution . 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 . Furthermore, we also make a generalization of the reciprocal gamma distribution, and study some of its properties.

**[1]**J. J. Dorning, B. Nicolaenko, and J. K. Thurber,*An integral identity due to Ramanujan which occurs in neutron transport theory*, J. Math. Mech.**19**(1969/1970), 429–438. MR**0254298**, https://doi.org/10.1512/iumj.1970.19.19040**[2]**A. Erdélyi, W. Magnus, F. Oberhettinger & F. G. Tricomi,*Higher Transcendental Functions*, Vol. Ill, McGraw-Hill, New York, 1955.**[3]**Arne Fransén,*Addendum and corrigendum to: “High-precision values of the gamma function and of some related coefficients” [Math. Comp. 34 (1980), no. 150, 553–566; MR 81f:65004] by Fransén and S. Wrigge*, Math. Comp.**37**(1981), no. 155, 233–235. MR**616377**, https://doi.org/10.1090/S0025-5718-1981-0616377-4**[4]**Arne Fransén and Staffan Wrigge,*High-precision values of the gamma function and of some related coefficients*, Math. Comp.**34**(1980), no. 150, 553–566. MR**559204**, https://doi.org/10.1090/S0025-5718-1980-0559204-5**[5]**Walter Gautschi,*Polynomials orthogonal with respect to the reciprocal gamma function*, BIT**22**(1982), no. 3, 387–389. MR**675673**, https://doi.org/10.1007/BF01934452**[6]**Gene H. Golub and John H. Welsch,*Calculation of Gauss quadrature rules*, Math. Comp. 23 (1969), 221-230; addendum, ibid.**23**(1969), no. 106, loose microfiche suppl, A1–A10. MR**0245201**, https://doi.org/10.1090/S0025-5718-69-99647-1**[7]**Sven-Ȧke Gustafson,*Rapid computation of general interpolation formulas and mechanical quadrature rules*, Comm. ACM**14**(1971), 797–801. MR**0311069**, https://doi.org/10.1145/362919.362941**[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.**35**(1979), no. 1, 1–16. MR**536875**, https://doi.org/10.4064/aa-35-1-1-16**[13]**Raymond E. A. C. Paley and Norbert Wiener,*Fourier transforms in the complex domain*, American Mathematical Society Colloquium Publications, vol. 19, American Mathematical Society, Providence, RI, 1987. Reprint of the 1934 original. MR**1451142****[14]**John Riordan,*Combinatorial identities*, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR**0231725****[15]**Ian N. Sneddon,*Special functions of mathematical physics and chemistry*, Oliver and Boyd, Edinburgh and London; Intersciences Publishers, Inc., New York, 1956. MR**0080170****[16]**Arne Fransén and Staffan Wrigge,*Calculation of the moments and the moment generating function for the reciprocal gamma distribution*, Math. Comp.**42**(1984), no. 166, 601–616. MR**736456**, https://doi.org/10.1090/S0025-5718-1984-0736456-3**[17]**M. Wyman and R. Wong,*The asymptotic behaviour of 𝜇(𝑧,𝛽,𝛼)*, Canadian J. Math.**21**(1969), 1013–1023. MR**244521**, https://doi.org/10.4153/CJM-1969-112-4

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1984-0736456-3

Keywords:
Reciprocal gamma distribution,
population characteristics,
generating function

Article copyright:
© Copyright 1984
American Mathematical Society