Abstract
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.
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Dedicated to the memory of Luigi Gatteschi.
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Lewanowicz, S., Woźny, P., Area, I. et al. Multivariate generalized Bernstein polynomials: identities for orthogonal polynomials of two variables. Numer Algor 49, 199–220 (2008). https://doi.org/10.1007/s11075-008-9168-9
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DOI: https://doi.org/10.1007/s11075-008-9168-9
Keywords
- Multivariate generalized Bernstein polynomials
- Multivariate q-Bernstein polynomials
- Bivariate big q-Jacobi polynomials
- Bivariate q-Hahn polynomials
- Bivariate Hahn polynomials
- Connection relations