A Polynomial Blossom for the Askey–Wilson Operator
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.