Riordan-Bernstein Polynomials, Hankel Transforms and Somos Sequences

Using the language of Riordan arrays, we define a notion of generalized Bernstein polynomials which are defined as elements of certain Riordan arrays. We characterize the general elements of these arrays, and examine the Hankel transform of the row sums and the first columns of these arrays. We propose conditions under which these Hankel transforms possess the Somos-$4$ property. We use the generalized Bernstein polynomials to define generalized B\'ezier curves which can provide a visualization of the effect of the defining Riordan array.