Browsing by Author "Simeonov, Plamen"

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    A Polynomial Blossom for the Askey–Wilson Operator
    (Springer, 2018) Simeonov, Plamen; Goldman, Ron
    We introduce a blossoming procedure for polynomials related to the Askey–Wilson operator. This new blossom is symmetric, multiaffine, and reduces to the complex representation of the polynomial on a certain diagonal. This Askey–Wilson blossom can be used to find the Askey–Wilson derivative of a polynomial of any order. We also introduce a corresponding Askey–Wilson Bernstein basis for which this new blossom provides the dual functionals. We derive a partition of unity property and a Marsden identity for this Askey–Wilson Bernstein basis, which turn out to be the terminating versions of Rogers’ 6ϕ5 summation formula and a very-well-poised 8ϕ7 summation formula. Recurrence and symmetry relations and differentiation and degree elevation formulas for the Askey–Wilson Bernstein bases, as well as degree elevation formulas for Askey–Wilson Bézier curves, are also given.
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    q-Blossoming for analytic functions
    (Springer, 2018) Goldman, Ron; Simeonov, Plamen
    We construct aﾠq-analog of the blossom for analytic functions, the analyticﾠq-blossom. Thisﾠq-analog also extends the notion ofﾠq-blossoming from polynomials to analytic functions. We then apply this analyticﾠq-blossom to derive identities for analytic functions represented in terms of theﾠq-Poisson basis, includingﾠq-versions of the Marsden identity and the de Boor-Fix formula for analytic functions.