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**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.

Reviewer: Nicolae Cotfas (Bucureşti)

### MSC:

33C67 | Hypergeometric functions associated with root systems |

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\textit{J. V. Stokman}, Ann. Math. (2) 179, No. 1, 253--299 (2014; Zbl 1290.33015)

### 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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