*(English)*Zbl 0584.43010

[For the entire collection see Zbl 0543.00007.]

The Jacobi functions ${{\Phi}}_{\lambda}^{(\lambda ,\beta )}\left(t\right)$, $\alpha $ $\ne -1,-2,\xb7\xb7\xb7$, are the continuous analogue of the Jacobi polynomials. They occur for special parameters ($\alpha $,$\beta )$ as spherical functions on rank one symmetric spaces. They also occur as a special way of parametrizing the Gauss hypergeometric function ${}_{2}{F}_{1}(a,b;c;z)$, taking $a=(\alpha +\beta +1-i\lambda )$, $b=(\alpha +\beta +1+i\lambda )$, $c=\alpha +1$, $z=-{sinh}^{2}t\xb7$

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 |