## The $$c$$-function expansion of a basic hypergeometric function associated to root systems.(English)Zbl 1290.33015

An explicit expansion in symmetric Macdonald polynomials of a basic hypergeometric function associated to a root system was obtained by Cherednik in the case the associated twisted affine root system is reduced. An extension of Cherednik’s construction in the form of a meromorphic Weyl group invariant solution of the spectral problem of the Macdonald $$q$$-difference operators was obtained by the author of the present paper. Now, the author derives an explicit $$c$$-function expansion of a basic hypergeometric function $$\mathcal{E}_+$$ associated to root systems. The obtained explicit $$c$$-function expansion is an expansion in terms of the basis of the space of meromorphic solutions of the spectral problem with coefficients expressed in terms of a $$q$$-analog of the Harish-Chandra $$c$$-function. The basic Harish-Chandra series $$\hat \Phi _\eta (\cdot, \gamma )$$ with base point given by a torus element $$\eta$$ is a meromorphic common eigenfunction of the Macdonald $$q$$-difference operators. The obtained $$c$$-function expansion of a basic hypergeometric function $$\mathcal{E}_+$$ has the form $$\mathcal{E}_+(t,\gamma )=\hat c_\eta (\gamma _0)^{-1}\sum_{w\in W_0}\hat c_\eta (w\gamma )\hat \Phi _\eta (t,w\gamma )$$, where $$W_0$$ is the Weyl group of the underlying finite root system.

### MSC:

 33C67 Hypergeometric functions associated with root systems
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### References:

 [1] R. Askey and J. Wilson, ”Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials,” Mem. Amer. Math. Soc., vol. 54, iss. 319, p. iv, 1985. · Zbl 0572.33012 [2] J. V. Stokman, ”Connection coefficients for basic Harish-Chandra series,” SIGMA Symmetry Integrability Geom. Methods Appl., vol. 8, iss. 3, p. 039, 2012. · Zbl 0849.17025
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