### Some Mathematical Properties for Marginal Model of Poisson-Gamma Distribution

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**Authors:**M. G. BIN-SAAD, J. A. YOUNIS AND A. HASANOV

**DOI:**10.46793/KgJMat2301.105B

**Abstract:**

Recently, Casadei [?] provided an explicit formula for statistical marginal model in terms of Poisson-Gamma mixture. This model involving certain polynomials which play the key role in reference analysis of the signal and background model in counting experiments. The principal object of this paper is to present a natural further step toward the mathematical properties concerning this polynomials. We ﬁrst obtain explicit representations for these polynomials in form of the Laguerre polynomials and the conﬂuent hyper-geometric function and then based on these representations we derive a number of useful properties including generating functions, recurrence relations, diﬀerential equation, Rodrigueś formula, ﬁnite sums and integral transforms.

**Keywords:**

Poisson-Gamma distribution, marginal models, Laguerre polynomials, hyper-geometric functions.

**References:**

[1] L. C. Andrews, Special Functions for Engineers and Applied Mathematician, The Macmillan Company, New York, 1985.

[2] J. M. Bernardo, Modern Bayesian Inference: Foundations and Objective Methods, Elsevier, Amsterdam, 2009.

[3] J. M. Bernardo and A. F. M. Smith, Bayesian Theory, John Wiley, New York, 1994.

[4] D. Casadei, Reference analysis of the signal + background model in counting experiments, J. Instrum. 7(1) (2011), 1–34.

[5] D. Casadei, Statistical methods used in ATLAS for exclusion and discovery, in: Proceedings of PHYSTAT2011, CERN, 2011, 17–20.

[6] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, McGraw-Hill Book Company, New York, Toronto, London, 1953.

[7] P. J. Heagerty and S. L. Zeger, Marginalized multilevel models and likelihood inference, Statist. Sci. 15(1) (2000), 1–26.

[8] N. N. Lebedev, Special Functions and Their Applications, Prentice-Hall, Inc., Englewood Cliﬀs, New Jersey, 1965.

[9] Y. L. Luke, The Special Functions and Their Approximations, Academic press, New York, London, 1969.

[10] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Diﬀerential Equations, John Wiley and Sons, Inc., New York, 1993.

[11] J. W. Pearson, S. Olver and M. A. Porter, Numerical methods for the computation of the conﬂuent and Gauss hypergeometric functions, Numer. Algor. 74(3) (2017), 821–866, DOI org2/10.1007/2Fs11075-016-0173-0.

[12] A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and Series, Gordon and Breach, New York, 1990.

[13] E. D. Rainville, Special Functions, Macmillan Company, New York, 1960; Reprinted by Chelsea Publishing Company, Bronx, New York, 1971.

[14] R. Reynolds, Numerical Evaluation Of The Contour Integral Representation For The 1-D and 3-D Coulomb Wave Functions, M. S. Thesis, York University, Toronto, Ontario, 2010, hosted at [http://www.math.yorku.ca/Who/Faculty/Stauﬀer/thesis.pdf].

[15] J. B. Seaborn, Hypergeoemtric Functions and Their Applications, Springer, New York, 1991.

[16] D. Sun and J. O. Berger, Reference priors with partial information, Biometrika 85(1) (1998), 55–71, DOI 10.1093/biomet/85.1.55.