Abstract
Hypergeometric functions have been increasingly present in several disciplines including Statistics, but there is much confusion on their proper uses, as well as on their existence and domain of definition. In this article, we try to clarify several points and give a general overview of the topic, going from the univariate case to the matrix case, in one and then in several arguments. We also survey some results in fields close to Statistics, where hypergeometric functions are actively used, studied and developed.
Highlights
Hypergeometric functions in one or several variables, introduced first in Mathematics, have been used in Physics and Applied Mathematics for some time
It is understandable that the huge volume of relations between hypergeometric functions of all types presented in the literature, and new ones regularly introduced in journals, raise various pertinent questions: Are they correct? How can we recognize a series as being of hypergeometric type? Can some of them be merely modified versions of existing ones? What are the mechanisms to derive new results from existing fundamental ones? Can we identify those which are really basic?
It is not surprising that hypergeometric functions are used in Economics and related fields, where advanced mathematics are often used for modeling and computation
Summary
Hypergeometric functions in one or several variables, introduced first in Mathematics, have been used in Physics and Applied Mathematics for some time. There are, at present, three survey articles on hypergeometric functions in the literature: one is from the Encyclopedia of Statistical sciences [2], one by Schlosser [3], and the third one by Abadir [4] Each of these surveys has its own merits, but the first one is limited to one variable and does not cover several topics related to mathematics. NOTE: In this survey we will limit our consideration to the real case, for scalar, vector and matrix variables, since the complex case is seldom encountered in Statistics, and its inclusion would considerably lengthen the article Classical treatises on this topic are Erdelyi et al [15], Slater [16] [17], Bailey [18]. Our view is only one among so many others, that could differ sharply from ours
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