Abstract
We settle the dual addition formula for continuous q-ultraspherical polynomials as an expansion in terms of special q-Racah polynomials for which the constant term is given by the linearization formula for the continuous q-ultraspherical polynomials. In a second proof we derive the dual addition formula from the Rahman–Verma addition formula for these polynomials by using the self-duality of the polynomials. We also consider the limit case of continuous q-Hermite polynomials.
Highlights
In this paper, as a natural continuation of our recent derivation [18] of the dual addition formula for ultraspherical polynomials, we derive the dual addition formula for continuous q-ultraspherical polynomials
The second proof exploits the self-duality of the continuous q-ultraspherical polynomials
The dual addition formula follows from the known addition formula [22] for these polynomials
Summary
As a natural continuation of our recent derivation [18] of the dual addition formula for ultraspherical polynomials, we derive the dual addition formula for continuous q-ultraspherical polynomials. While (1.1), (1.2), (1.3), and their generalizations to ultraspherical polynomials, are formulas established long ago and staying within the realm of classical orthogonal polynomials, it is remarkable that the dual addition formula steps out from this and needs Racah polynomials, which live high up in the Askey scheme. Dual addition formulas: the case of continuous. Polynomials have the property of self-duality, which is lost in the limit to q = 1 This notion means that, for a suitable function σ , an orthogonal polynomial pn(x) has the property that pn(σ (m)) = pm(σ (n)) It contains the two proofs of the dual addition formula for continuous q-ultraspherical polynomials. For formulas on orthogonal polynomials in the (q-)Askey scheme we will often refer to Chapters 9 and 14 in [16]
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