A gamma type distribution involving a confluent hypergeometric function of the second kind* 

Johan Fereira¹, Susana Salinas² 

¹Departamento de Matemáticas.

²CIMA. Facultad de Ingeniería, Universidad del Zulia. Maracaibo, Venezuela.
jfereira13@hotmail.com ; susanaderomero@hotmail.com  

Abstract 

In the present work a new gamma type distribution is obtained which involves a confluent hypergeometric function of the second kind. A generalized form of the incomplete gamma function and its complementary are introduced to obtain some statistical functions. The associated statistical functions with the probability density function are deduced such as the k-moment, expect value, risk function, half-life function and other special cases. 

Key words:  Generalized distribution, incomplete gamma type, hypergeometric confluent function of the second kind. 

Una distribución generalizada tipo gamma que involucra la función hipergeométrica confluente de segunda clase 

Resumen 

En el presente trabajo se obtiene una nueva distribución tipo gamma que involucra a la función hipergeométrica confluente de segunda clase. Una forma generalizada de la función gamma incompleta y su forma complementaria son introducidas para obtener algunas funciones estadísticas. Se deducen las funciones estadísticas asociadas con la función de densidad de probabilidad, tales como el k-ésimo momento, el valor esperado, la función de riesgo y la función de vida media, y otros casos especiales. 

Palabras clave:  Distribución generalizada, tipo gamma incompleta, función hipergeométrica confluente de segunda clase. 

Recibido el 30 de Julio de 2009 

En forma revisada el 3 de Mayo de 2010 

* 2010: Mathematics Subject Classification: MSC 2000: 33C15, 62E15.

Introduction 

A new class of functions were introduced and developed by Virchenko [1, 2], which may be called t-hypergeometric and t-confluent hypergeometric functions. Those functions are natural generalizations of classical hypergeometric functions. Agarwal and Kalla [3] have defined a generalized gamma function distribution derived from a generalized Kobayashi gamma function [4], which is a confluent hypergeometric function of the second kind. 

A unified form of gamma type distribution, was given by Kalla and others [5] based on the generalized gamma function defined by Al-Musallan and Kalla [6, 7]. 

Recently A. Al-Zamel [8] has introduced a new gamma type distribution involving the t-confluent hypergeometric function and has discussed some basic functions associated with the distribution. 

In this paper, we present first some properties of t-hypergeometric function and the hypergeometric confluent function of the second kind and we also define a generalized form of the incomplete gamma function and its complementary. Moreover a density function asociated with the hypergeometric function of second kind is defined. The gamma, generalized gamma, Weibull and another type of gamma incomplete distribution are obtained as particular cases of the density generalized function. Some properties associated with the density function and other frequently used functions such as the k-th moment, risk function, life time are derived. 

The gamma function is defined [9], as follows: 

Generalized gamma function 

By using the confluent hypergeometric function of the second kind given in (6), we define the following incomplete generalized gamma function as: 

Probability density function 

In this case, we use the confluent hypergeometric function of the second kind to define the probability density function as follows (see equation (19) below) (Figure 1), where are constants such as ,and

Statistical functions 

In this section we present the basic statistical functions associated with the density function defined in (19). 

Risk function 

The mean life function 

Acknowledgment 

The author(s) wish to thank the Universidad del Zulia for the support which allowed them to carry out this work at the CIMA (Research Center for Applied Mathematics). 

References 

1.  Virchenko, N. On some generalizations of the functions of hypergeometric type. Fractional Calculus and Applied Analysis, Vol. 2, (1999), 233-244. 

2.  Virchenko, N., Kalla, S.L. and Al-Zamel, A. Some results on a generalized hypergeometric function. Integral Transforms and Special Functions. Vol. 12 (1), (2001), 89-100. 

3.  Agarwal, S.K. and Kalla, S.L. A generalized gamma distribution and its applications in relativity. Comm. in Statist. Theory and Methods, Vol. 25, (1996), 201-210. 

4.  Kobayashi, K. On generalized gamma function ocurring in differaction theory. Journal of Physical Society of Japan, Vol. 60, (1991), 1501-1512. 

5.  Kalla, S.L., Al-Saqabi, B.N. and Khajah, H. A unified form of gamma type distributions. Applied Mathematics and Computation, Vol. 118, (2-3), (2001), 175-187. 

6.  Al-Musallam, F. and Kalla, S.L. Asymptotic expansions for generalized gamma and incomplete gamma functions. Applied Analysis, Vol. 66, (1997), 173-187. 

7.  Al-Musallam, F. and Kalla, S.L. Further results on a generalized gamma function ocurring in diffraction theory. Integral Transforms and Special Function, Vol. 7, (1998), 175-190. 

8.  Al-Zamel, A. On generalized gamma-type distribution t-confluent hypergeometric function. Kuwait J. Sci. Engrg., Vol. 28, (2001), 25-36. 

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10.  Galué, L. Differintegrals of Wright’s generalized Hypergeometric Function. International Journal of Applied Mathematics, Vol. 30 (3), (2002), 255-267. 

11.  Walpole, R. Probabilidad y Estadística para Ingenieros. Prentice-Hall, México, (1999).          [ Links ]