An Extension of the Kratzel Function and Associated Inverse Gaussian Probability Distribution Occurring in Reliability Theory
In view of their importance and usefulness in reliability theory and probability distributions, several generalizations of the inverse Gaussian distribution and the Krtzel function are investigated in recent years. This has motivated the authors to introduce and study a new generalization of the inverse Gaussian distribution and the Krtzel function associated with a product of a Bessel function of the third kind )(zKQ and a Z - Fox-Wright generalized hyper geometric function introduced in this paper. The introduced function turns out to be a unified gamma-type function. Its incomplete forms are also discussed. Several properties of this gamma-type function are obtained. By means of this generalized function, we introduce a generalization of inverse Gaussian distribution, which is useful in reliability analysis, diffusion processes, and radio techniques etc. The inverse Gaussian distribution thus introduced also provides a generalization of the Krtzel function. Some basic statistical functions associated with this probability density function, such as moments, the Mellin transform, the moment generating function, the hazard rate function, and the mean residue life function are also obtained.KeywordsFox-Wright function, Inverse Gaussian distribution, Krtzel function & Bessel function of the third kind.
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 M. Abramowitz, and I. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972.
 I. Ali, S. L. Kalla and H. G. Khajah, A generalized inverse Gaussian distribution with W confluent hypergeometric function, Integral Transforms Spec. Funct. 12, No.2, pp. 101-114, 2001.
 B. Al-Saqabi, S. L. Kalla and R. Scherer, On a generalized inverse Gaussian distribution, International Journal of Applied mathematics,20, No.1,pp. 11-27, 2007.
 M. A. Chaudhry, S. M. Zubair, Generalized incomplete gamma functions with Applications, J. Comput. Appl. Math., 55, pp. 99-124, 1996.
 M. A. Chaudhary, S. M. Zubair, On the extension of generalized incomplete gamma function with applications, J. Aust. Math. Soc. Ser,B, 37, pp. 392-405, 1996.
 R. S. Chinkara, and J. Leroy, Folks, The inverse Gaussian distribution: Theory, Methodology and Applications, Marcel Dekker, New York, 1989.
 M. R. Dotsenko, On some applications of Wrights hypergeometric function, C. R. Acad. Bulgare Sci. 44, pp. 13-16, 199.
 M. R. Dotsenko, On some applications of Wrights hypergeometric function, Mat. Fiz. Nelinein Mekh, No. 18 (52), pp. 47-52, 1993.
 A. Erdlyi, W. Magnus, F. Oberhettinger, and F. G.Tricomi, Higher Transcendental Functions, Vol. II, McGraw- Hill, New York- Toronto- London, 1953; Reprinted : Krieger, Melbourne, Florida, 1953, and 1981.
 A. Erdlyi, W. Magnus, F.Oberhettinger, and F. G. Tricomi, Tables of Integral Transforms, Vol. II, McGraw- Hill, New York- Toronto- London, 1954; Reprinted : Krieger, Melbourne, Florida, 1954, and 1981.
 J. Good, The population frequencies of species and the estimation of population parameters, Biometrika, 40, pp. 237-260, 1953.
 H. J. Haubold, and A. M. Mathai, An integral arising frequently in astronomy and physics, SIAM Rev.40 (4), pp. 995-997, 1998.
 J. N. Hoem, The statistical theory of demographic rates, Scand. J. Statist., 3, pp. 169-185, 1976.
 B. Jorgensen, Statistical Properties of Generalized Inverse Gaussian Distributions, Lecture Notes in Statistics, 9, Springer, New York, 1982.
 S. L. Kalla, B. N. Al-Saqabi, H. G. Khajah, A unified form of gamma-type distribution, Appl. Math. Comput. 118, No. 2-3, pp. 175-187, 2001.
 A. A. Kilbas and M. Saigo, H-Transforms Theory and Applications,Chapman and Hall/ CRC, Roca Rotan, FL, New York, 2004.
 A. A. Kilbas, R. K. Saxena, and Juan J. Trujillo, Krtzel function as a function of .hypergeometric type, Frac. Calc. Appl. Anal. 9, pp. 109-131, 2006.
 A. A. Kilbas, R. K. Saxena, Megumi Saigo, and Juan J. Trujillo, Generalized Wright function as the H- function In Analytic Methods of Analysis and Differential Equations, AMADE 2003, Cambridge Scientific Publishers, pp. 117-134, 2006.
 E. Krtzel, Eine Verallgemeineirungder, Laplace- und Meijer Transformation, Wiss. Z.Friedrich- Shiller Univ. Mathnaturwiss.Reihe 14, No.5, pp. 369-381, 1965.
 E. Krtzel, Integral transformation of Bessel type, In Generalized Functions and Operational Calculus (Proc. Conf. Varna, 1975. Bulg. Acad. Sci., Sofia, 1979), pp. 148-155, 1979.
 A. M. Mathai, A Handbook of Special functions for Startistical and Physical Sciences, Clarendon Press, Oxford, 1993.
 A. M. Mathai, and R. K. Saxena, The H-function with Applications in Statistics and Other Disciplines, Wilery, New York, 1978.
 S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives Theory and Applications, Gordon and Breach, Reading, 1993.
 R. K. Saxena, On a unified inverse Gaussian distribution, to appear in the Proceedings of the 8th conference of Society for Special Functions and Their Applications held at Palai,Kerala, India, 2007.
 R. K. Saxena and S. L. Kalla, On a generalization of Kratzel function and associated inverse Gaussian probability distributions, Algebras, Groups, and Geometries 24, pp. 303-324, 2007.
 R. K. Saxena, A. M. Mathai and H. J. Haubold, Astrophysical thermonuclear functions for Boltzmann- Gibbs statistics and Tsallis statistics, Physica A 344, pp. 649-656, 2004.
 N. Virchenko, On some generalizations of the functions of hypergeometric type, Fract. Calc. Appl. Anal.2, No.3, pp. 233-244, 1999.
 N. Virchenko, S. L. Kalla, and A. Al- Zamel, Some results on a generalized hypergeometric function, Integral Transform Spec. Funct. 12, No.1, pp. 89- 100, 2001.
 E. M. Wright, The asymptotic expansion of the generalized hypergeometric function, J. London Math. Soc.10, pp. 286-293, 1935.
 E. M. Wright, The asymptotic expansion of the generalized hypergeometric function, Proc. London Math. Soc. (Ser.2), 46, pp. 389-408, 1940.