# zbMATH — the first resource for mathematics

##### Examples
 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.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 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)
Jacobi functions and analysis on noncompact semisimple Lie groups. (English) Zbl 0584.43010
Special functions: Group theoretical aspects and applications, Math. Appl., D. Reidel Publ. Co. 18, 1-85 (1984).

[For the entire collection see Zbl 0543.00007.]

The Jacobi functions ${{\Phi }}_{\lambda }^{\left(\lambda ,\beta \right)}\left(t\right)$, $\alpha$ $\ne -1,-2,···$, are the continuous analogue of the Jacobi polynomials. They occur for special parameters ($\alpha$,$\beta \right)$ as spherical functions on rank one symmetric spaces. They also occur as a special way of parametrizing the Gauss hypergeometric function ${}_{2}{F}_{1}\left(a,b;c;z\right)$, taking $a=\left(\alpha +\beta +1-i\lambda \right)$, $b=\left(\alpha +\beta +1+i\lambda \right)$, $c=\alpha +1$, $z=-{sinh}^{2}t·$

The paper under review is a well written and most valuable survey of the theory developed around Jacobi functions and the interplay between these special functions and Lie group theory. The emphasis is on harmonic analysis. There is a large and very usefull bibliography of the subject.

The content is in short described by: The Jacobi transform and its inverse. A short introduction to group theoretic tools needed. Examples of the Jacobi transform from group theory. The Abel transform. The Paley- Wiener and Plancherel theorems. Convolution structures and addition formulas. Further results and references.

##### MSC:
 43A85 Analysis on homogeneous spaces 22E46 Semisimple Lie groups and their representations 33C80 Connections of hypergeometric functions with groups and algebras 43-02 Research monographs (abstract harmonic analysis) 33C45 Orthogonal polynomials and functions of hypergeometric type