Algebraic representation for Bernstein polynomials on the symmetric interval and combinatorial relations

We pose the question of explicit algebraic representation for Bernstein polynomials. The general statement of the problem on an arbitrary interval $[a,b]$ is briefly discussed. For completeness, we recall Wigert formulas for the polynomials coefficients on the standard interval $[0,1]$. However, the focus of the paper is the case of the symmetric interval $[-1,1]$, which is of fundamental interest for approximation theory. The exact expressions for the coefficients of Bernstein polynomials on $[-1,1]$ are found. For the interpretation of the results we introduce a number of new numerical objects named Pascal trapeziums. They are constructed by analogy with a classical triangle, but with their own “initial” and “boundary” conditions. The elements of Pascal trapeziums satisfy various relations which remind customary combinatorial identities. A systematic research on such properties is fulfilled, and summaries of formulas are given. The obtained results are applicable for the study of the behavior of the coefficients in Bernstein polynomials on $[-1,1]$. For example, it appears that there exists a universal connection between two coefficients $a_{2m,m}(f)$ and $a_{m,m}(f)$, and this is true for all $m\in\mathbb N$ and for all functions $f\in C[-1,1]$. Thus, it is set up that the case of symmetric interval $[-1,1]$ is essentially different from the standard case of $[0,1]$. Perspective topics for future research are proposed. A number of this topics is already being studied.