Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0731.33015
Gross, Kenneth I.; Richards, Donald St.P.
Hypergeometric functions on complex matrix space. (English)
[J] Bull. Am. Math. Soc., New Ser. 24, No.2, 349-355 (1991). ISSN 0273-0979; ISSN 1088-9485/e

In this note new results about hypergeometric functions of matrix argument are presented without proof. The proofs together with a detailed study will appear in a forthcoming paper. Denote by S the space of $n\times n$ Hermitian matrices over ${\bbfF}={\bbfR}$, ${\bbfC}$ or ${\bbfH}$. Let $a\sb 1,...,a\sb p$ and $b\sb 1,...,b\sb q$ be complex parameters. The hypergeometric function ${}\sb pF\sb q$ of matrix argument is defined on S by $$\sb pF\sb q(a\sb 1,...,a\sb p;b\sb 1,...,b\sb q;s)=\sum\sb{m}\frac{[a\sb 1]\sb m...[a\sb p]\sb mZ\sb m(s)}{[b\sb 1]\sb m,...,[b\sb q]\sb m\vert m\vert !}. $$ The sum is extended over all partitions $m=(m\sb 1,...,m\sb n)$, $[a]\sb m$ is the generalized truncated factorial for the matrix space, and $Z\sb m$ is the zonal polynomial associated with m. \par In this note one considers the case ${\bbfF}={\bbfC}$. Then the hypergeometric function can be written as a determinant whose entries are classical hypergeometric functions. From this expansion one deduces an asymptotic formula and a system of partial differential equations of which hypergeometric functions are solutions. \par Further one defines the operator-valued hypergeometric function inductively by using the Laplace transform associated with the cone P of positive definite Hermitian matrices. As a special case one obtains the operator-valued Bessel function studied in the 70th by Gross and Kunze.
[J.Faraut (Paris)]

MSC 2000:: *33C70 Other hypergeometric functions and integrals in several variables

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]