Group Representations and Special Functions of a Matrix Argument

1992 ◽  
pp. 251-360
Author(s):  
N. Ja. Vilenkin ◽  
A. U. Klimyk
Keyword(s):  
Special Functions ◽  
Group Representations ◽  
Matrix Argument

Boundary Value Problems for Harmonic Functions on the Heisenberg Group

1986 ◽  
Vol 38 (2) ◽  
pp. 478-512 ◽  
Author(s):  
Charles F. Dunkl
Keyword(s):  
Heisenberg Group ◽  
Harmonic Functions ◽  
Poisson Kernel ◽  
Differential Operators ◽  
Special Functions ◽  
Continuous Functions ◽  
Group Representations ◽  
Hypoelliptic Operators ◽  
Analysis Group ◽  
Partial Differential Operators

Analysis on the Heisenberg group has become an important area with strong connections to Fourier analysis, group representations, and partial differential operators. We propose to show in this work that special functions methods can also play a significant part in this theory. There is a one-parameter family of second-order hypoelliptic operators Lγ, (γ ∊ C), associated to the Laplacian L0 (also called the subelliptic or Kohn Laplacian). These operators are closely related to the unit ball for reasons of homogeneity and unitary group invariance. The associated Dirichlet problem is to find functions with specified boundary values and annihilated by Lγ inside the ball (that is, Lγ-harmonic). This is the topic of this paper.Gaveau [9] proved the first positive result, showing that continuous functions on the boundary can be extended to L0-harmonic functions in the ball, by use of diffusion-theoretic methods. Jerison [15] later gave another proof of the L0-result. Hueber [14] has recently obtained some results dealing with special values of the Poisson kernel for L0.

scholarly journals Extended k-Gamma and k-Beta Functions of Matrix Arguments

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1715
Author(s):  
Ghazi S. Khammash ◽  
Praveen Agarwal ◽  
Junesang Choi
Keyword(s):  
Gamma Function ◽  
Hypergeometric Functions ◽  
Special Functions ◽  
Integral Formula ◽  
Beta Function ◽  
Integral Representations ◽  
Matrix Argument ◽  
Functional Relations

Various k-special functions such as k-gamma function, k-beta function and k-hypergeometric functions have been introduced and investigated. Recently, the k-gamma function of a matrix argument and k-beta function of matrix arguments have been presented and studied. In this paper, we aim to introduce an extended k-gamma function of a matrix argument and an extended k-beta function of matrix arguments and investigate some of their properties such as functional relations, inequality, integral formula, and integral representations. Also an application of the extended k-beta function of matrix arguments to statistics is considered.

Group Representations and Special Functions (By Antoni Wawrzynczyk)

SIAM Review ◽  
1987 ◽  
Vol 29 (2) ◽  
pp. 336-337
Author(s):  
V. S. Varadarajan
Keyword(s):  
Special Functions ◽  
Group Representations

scholarly journals A Brief History of the Centre and Prof. Dr. A. M. Mathai’s Research and Education Programs at the Occasion of His 85th Anniversary

2020 ◽  
Author(s):  
Anggi Cecilia Safaningrum
Keyword(s):  
Special Functions ◽  
Solar Neutrinos ◽  
Rate Theory ◽  
Education Programs ◽  
Matrix Argument ◽  
Normal Probability ◽  
History Of ◽  
Pathway Models ◽  
Research And Education

A brief history of the Centre for Mathematical and Statistical Sciences, Kerala, India, is given and an overview of Mathai’s research and education programs in the following topics is outlined: Fractional Calculus; Functions of Matrix Argument—M-transforms, M-convolutions; Krätzel integrals; Pathway Models; Geometrical Probabilities; Astrophysics—reaction rate theory, solar neutrinos; Special Functions—G and H-functions; Multivariate Analysis; Algorithms for Non-linear Least Squares; Characterizations—characterizations of densities, information measure, axiomatic definitions, pseudo analytic functions of matrix argument and characterization of the normal probability law; Mathai’s Entropy—entropy optimization; Analysis of Variance; Population Problems and Social Sciences; Quadratic and Bilinear Forms; Linear Algebra; Probability and Statistics.

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