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Erdélyi-Kober Fractional Integrals Involving Many Real Matrices

  • A. M. Mathai
  • H. J. Haubold
Chapter
Part of the SpringerBriefs in Mathematical Physics book series (BRIEFSMAPHY, volume 31)

Abstract

All the matrices appearing in this chapter are p × p real positive definite unless stated otherwise. In order to avoid too many symbols we will use \(u_1=\frac {x_2}{x_1}\) for the ratio of x2 to x1 in the real scalar variable case, \(U_1=X_2^{\frac {1}{2}}X_1^{-1}X_2^{\frac {1}{2}}\), symmetric ratio, in the real p × p matrix-variate case. The corresponding density of u1 and U1 will be indicated by g1; we will use u2 = x1x2 for the product in the real scalar variable case and \(U_2=X_2^{\frac {1}{2}}X_1X_2^{\frac {1}{2}}\), the symmetric product, in the real p × p matrix-variate case. The corresponding density of u2 or U2 will be indicated by g2. If x1 and x2 are statistically independently distributed real scalar random variables, and X1 and X2 are statistically independently distributed real matrix-variate random variables, then g2(u2) or g2(U2) and g1(u1) or g1(U1) will denote product and ratio distributions or M-convolutions of product and ratio whatever be the set of variables. In all the preceding chapters the basic claim is that fractional integrals are of two kinds, the first kind or left-sided and the second kind or right-sided. The first kind of fractional integrals belong to the class of Mellin convolution of a ratio and the second kind of fractional integrals belong to the class of Mellin convolution of a product. In the matrix-variate case these will be M-convolutions of ratio and product respectively. If the variables are random variables then g1 and g2 correspond to the densities of ratio and product respectively. We will give the following formal definition of fractional integral operators of the first kind and second kind. For the sake of completeness we will recall the general definitions from Chaps.  3 and  4.

References

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Copyright information

© The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  • A. M. Mathai
    • 1
  • H. J. Haubold
    • 2
  1. 1.Department of Mathematics and StatisticsMcGill UniversityMontrealCanada
  2. 2.Office for Outer Space Affairs United NationsViennaAustria

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