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Root systems and hypergeometric functions. II. (English) Zbl 0656.17007
Keeping the notation used in part I (see the preceding review Zbl 0656.17006), the author continues studying a class of solutions \(\phi\) of the equation \(L\phi =(\lambda -\rho,\lambda +\rho)\phi.\) The main concern of the paper is the analyticity of \(\phi\). The author proves, among other things, that a certain linear combination of \(\phi\) could be extended to a Weyl group invariant analytic function on all of \(\exp (E^*)\). As an application of the main result he establishes an orthogonality relation on the Jacobi polynomials associated to a root system R.
For parts III, IV, see ibid. 67, No.1, 21-49 (1988); No.2, 191-209 (1988).
Reviewer: Tu Guizhang

17B20 Simple, semisimple, reductive (super)algebras
35C10 Series solutions to PDEs
34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
33C05 Classical hypergeometric functions, \({}_2F_1\)
13N05 Modules of differentials
Full Text: Numdam EuDML
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