Jacobi Functions and Analysis on Noncompact Semisimple Lie Groups

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)