Stokman, J. V. The \(c\)-function expansion of a basic hypergeometric function associated to root systems. (English) Zbl 1290.33015 Ann. Math. (2) 179, No. 1, 253-299 (2014). 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) Cited in 8 Documents MSC: 33C67 Hypergeometric functions associated with root systems Keywords:basic Harish-Chandra series; basic hypergeometric functions; \(c\)-functions PDF BibTeX XML Cite \textit{J. V. Stokman}, Ann. Math. (2) 179, No. 1, 253--299 (2014; Zbl 1290.33015) Full Text: DOI arXiv OpenURL 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.