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arXiv · 2412.03169

Non-Symmetric Askey--Wilson Shift Operators

Max van Horssen·Philip Schlösser
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Abstract

We classify the shift operators for the symmetric Askey-Wilson polynomials and construct shift operators for the non-symmetric Askey-Wilson polynomials using two decompositions of non-symmetric Askey-Wilson polynomials in terms of symmetric ones. These shift operators are difference-reflection operators, and we discuss the conditions under which they restrict to shift operators for the symmetric Askey-Wilson polynomials. We use them to compute the norms of the non-symmetric Askey-Wilson polynomials and compute their specialisations for $q\to1$. These turn out to be shift operators for the non-symmetric Heckman-Opdam polynomials of type $BC_1$ that have recently been found.

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Max van Horssen, Philip Schlösser. 2024-12-04. Non-Symmetric Askey--Wilson Shift Operators. https://doi.org/10.1111/sapm.70102

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