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

We introduce polynomials $B^n_{k}(\boldmath{x};\omega|q)$ of total degree n, where $\boldmath{k} = (k_1,\ldots,k_d)\in\mathbb N_0^d, \; 0\le k_1+\ldots+k_d\le n$, and $\boldmath{x}=(x_1,x_2,\ldots,x_d)\in\mathbb R^d$, 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 $B^n_{k}(\boldmath{x};\omega|q)$ 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 and Dunkl’s (little) q-Jacobi polynomials (SIAM J Algebr Discrete Methods 1:137–151, 1980), as well as between bivariate discrete Bernstein and Hahn polynomials.