Harmonic Analysis and Special Functions

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ABSTRACT Richards Richards will continue his program of research on hypergeometric functions defined on matrix spaces and on symmetric cones. Special attention will be paid to the theory of operator-valued hypergeometric functions on the symmetric cones. In the second part of the project the principal investigator will continue his research on applications of these special functions, and general techniques in harmonic analysis and representation theory, to the areas of multivariate statistics, total positivity, probability inequalities, and related combinatorial problems. Richards is interested in research problems Modern Analysis, a subfield of mathematics, for two reasons. First, Modern Analysis is an important branch of mathematics with fundamental links to astronomy, chemistry, physics, statistics, as well as to other areas of mathematics. Richards primary expertise in Modern Analysis is in the study of new classes of hypergeometric functions which generalize the classical theory developed by Euler, Gauss, Bessel, Ramanujan and many other famous scientists. Second, the principal investigator is interested in these new types of hypergeometric functions because of their fundamental importance in multivariate statistics. The research program will study these new hypergeometric functions and their properties with the aim of finding solutions to unsolved statistical problems which are of importance to applied researchers in medical and environmental fields.