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Special Functions: Group Theoretical Aspects and Applications pp 1–85Cite as

Jacobi Functions and Analysis on Noncompact Semisimple Lie Groups

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Part of the book series: Mathematics and Its Applications ((MAIA,volume 18))

Abstract

A Jacobi function \({\phi _\lambda }^{\left( {\alpha ,\beta } \right)}\left( {\alpha ,\beta ,\lambda \in C,\alpha \ne - 1, - 2,...} \right)\) is defined as the even C∞-function on ℝ which equals 1 at 0 and which satisfies the differential equation

$$\left( {{d^2}/d{t^2} + \left( {\left( {2\alpha + 1} \right)cotht + \left( {2\beta + 1} \right)tht} \right)d/dt + + {\lambda ^2} + {{\left( {\alpha + \beta + 1} \right)}^2}} \right){f_\lambda }^{\left( {\alpha ,\beta } \right)}\left( t \right) = 0.$$
(1.1)

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  1. Centrum voor Wiskunde en Informatica, Postbus 4079, 1009 AB, Amsterdam, Nederland

    Tom H. Koornwinder

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  1. Mathematics Department, University of Wisconsin, Madison, USA

    R. A. Askey

  2. Centre for Mathematics and Computer Science, Amsterdam, The Netherlands

    T. H. Koornwinder

  3. Department of Mathematics, University of Siegen, Germany

    W. Schempp

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Koornwinder, T.H. (1984). Jacobi Functions and Analysis on Noncompact Semisimple Lie Groups. In: Askey, R.A., Koornwinder, T.H., Schempp, W. (eds) Special Functions: Group Theoretical Aspects and Applications. Mathematics and Its Applications, vol 18. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-9787-1_1

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