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Symmetry of terminating basic hypergeometric series representations of the Askey-Wilson polynomials. (English) Zbl 1520.33011

Summary: We explore the symmetric nature of the terminating basic hypergeometric series representations of the Askey-Wilson polynomials and the corresponding terminating basic hypergeometric transformations that these polynomials satisfy. In particular we identify and classify the set of 4 and 7 equivalence classes of terminating balanced \({}_4\phi_3\) and terminating very-well-poised \({}_8 W_7\) basic hypergeometric series which are connected with the Askey-Wilson polynomials. We study the inversion properties of these equivalence classes and also identify the connection of both sets of equivalence classes with the symmetric group \(S_6\), the symmetry group of the terminating balanced \({}_4\phi_3\). We then use terminating balanced \({}_4\phi_3\) and terminating very-well poised \({}_8 W_7\) transformations to give a broader interpretation of Watson’s \(q\)-analog of Whipple’s theorem and its converse.

MSC:

33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)

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References:

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