zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Hypergeometric functions on complex matrix space. (English) Zbl 0731.33015

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×n Hermitian matrices over 𝔽=, or . Let a 1 ,···,a p and b 1 ,···,b q be complex parameters. The hypergeometric function p F q of matrix argument is defined on S by

p F q (a 1 ,···,a p ;b 1 ,···,b q ;s)= m [a 1 ] m ···[a p ] m Z m (s) [b 1 ] m ,···,[b q ] m |m|!·

The sum is extended over all partitions m=(m 1 ,···,m n ), [a] m is the generalized truncated factorial for the matrix space, and Z m is the zonal polynomial associated with m.

In this note one considers the case 𝔽=. 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.

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.

Reviewer: J.Faraut (Paris)
33C70Other hypergeometric functions and integrals in several variables