Multivariate generalized Bernstein polynomials: identities for orthogonal polynomials of two variables

摘要

We introduce polynomials Bnk (x; ω|q) of total degree n, wherek = (k1, . . . , kd) ∈ Nd0 , 0 ≤ k1 + . . . + kd ≤ n, and x = (x1, x2, . . . , xd) ∈ Rd, depending on two parameters q and ω, which generalize the multivariate classical and discrete Bernstein polynomials. For ω = 0, we obtain an extension of univariate q-Bernstein polynomials, introduced by Phillips (Ann Numer Math 4:511–518, 1997). Basic properties of the new polynomials are given, including recurrence relations, q-differentiation rules and de Casteljau algorithm. For the case d = 2, connections between Bnk (x; ω|q) and bivariate orthogonal big q-Jacobi polynomials—introduced recently by the first two authors—are given, with the connection coefficients being expressed in terms of bivariate q-Hahn polynomials. As limiting forms of these relations, we give connections between bivariate q-Bernstein andDunkl’s (little) q-Jacobi polynomials (SIAM J Algebr Discrete Methods 1:137–151, 1980), as well as between bivariate discrete Bernstein and Hahn polynomials.