The gamma function, the extended gamma function, the beta function, the extended beta function, the gamma distribution, the beta distribution and the extended beta distribution have been generalized to the matrix case in vaious ways. However, generalizations of the extended gamma and extended beta functions defined by 5 and 6 , respectively, to the matrix case have not been defined and studied.
It is, therefore, of great interest to define generalizations of the extended gamma and beta functions to the matrix case, study their properties, obtain different integral representations, and establish the connection of these generalizations with other known special functions of matrix argument. This paper is divided into seven sections.
- On a Class of Incomplete Gamma Functions with Applications.
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- Generalized Extended Matrix Variate Beta and Gamma Functions and Their Applications.
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Section 2 deals with some well known definitions and results on matrix algebra, zonal polynomials and special functions of matrix argument. In Section 3, the extended matrix variate gamma function has been defined and its properties have been studied. Definition and different integral representations of the extended matrix variate beta function are given in Section 4.
Some integrals involving zonal polynomials and generalized extended matrix variate beta function are evaluated in Section 5. In Section 6, the distribution of the sum of dependent generalized inverted Wishart matrices has been derived in terms of generalized extended matrix variate beta function. We introduce the generalized extended matrix variate beta distribution in Section 7.
In this section we give several known definitions and results.
We first state the following notations and results that will be utilized in this and subsequent sections. Several generalizations of the Euler's gamma function are available in the scientific literature. The multivariate gamma function, which is frequently used in multivariate statistical analysis, is defined by. By evaluating the above integral it is easy to see that.
The generalized hypergeometric function of one matrix argument as defined by Constantine 9 and James 15 is. If any numerator parameter ai is a negative integer,. They are given by. For properties and further results on these functions the reader is referred to Herz 13 , Constantine 9 , James 15 , and Gupta and Nagar Results given in Lemma 2.
A matrix variate generalization of the generalized extended gamma function can be defined in the following way:. Two special cases of 25 are worth mentioning.
In the following theorem we establish a relationship between generalized extended gamma function of matrix argument and multivariate gamma function through an integral involving the generalized extended gamma function of matrix argument and zonal polynomials. Now, integration of Z using Lemma 2. Note that the above corollary gives an interesting relationship between the generalized extended gamma function of matrix argument and multivariate gamma function.
Incomplete gamma function
Finally, integrating Z and Y by using multivariate gamma and multivariate beta integrals and simplifying the resulting expression, we obtain the desired result. Now, applying the above inequality in 25 , and using the integral representation of the generalized extended gamma function given in 25 , we obtain the desired result. In this section, a matrix variate generalization of 6 is defined and several of its properties are studied.
From 38 , it is apparent that.
Replacing the confluent hypergeometric function by its integral repre- sentation in 38 , changing the order of integration, and integrating Z by using 22 , we obtain. This particular special case has internal closure properties of its own because it can be used to express all successive derivatives.
In such a case, one must use a limit. Some special cases of this function can be simplified. This formula can be further inflated or generalized to a huge class of Laplace transforms and Mellin transforms.
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When combined with a computer algebra system , the exploitation of special functions provides a powerful method for solving definite integrals, in particular those encountered by practical engineering applications see Symbolic integration for more details. The following indefinite integrals are readily obtained using integration by parts with the constant of integration omitted in both cases:.
The lower and the upper incomplete Gamma function are connected via the Fourier transform:. From Wikipedia, the free encyclopedia. Archived from the original PDF on Archived copy as title link Theorem 3. Retrieved from " https: Gamma and related functions Continued fractions. Archived copy as title All articles with unsourced statements Articles with unsourced statements from February Articles with inconsistent citation formats. Views Read Edit View history.
M. Aslam Chaudhry (Author of On a Class of Incomplete Gamma Functions with Applications)
It presents some basic properties of these functions, including their recurrence relations, special cases, asymptotic representations, and integral transform relationships. The authors explore applications of these generalized functions to problems in transient heat conduction, special cases of laser sources, and problems associated with heat transfer in human tissues.
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