Solving recurrences for Legendre–Bernstein basis transformations

Abstract

The change of basis matrix $M$ from shifted Legendre to Bernstein polynomials and $M^{-1}$ have applications in computer graphics. Algorithms use their properties to find the matrix elements efficiently. We give new functions for the elements of $M$ as a summation, and a complete hypergeometric function. We find that Gosper’s algorithm does not produce closed-form expressions for the elements of either $M$ or $M^{-1}$. Zeilberger’s algorithm produces four second-order recurrences for the elements of the matrices that enable them to be computed in $O\left(n\right)$ time, and for the derivation of closed-form functions by row and column for the elements. Two row recurrences are special cases of those found by Woźny (2013) who used a different method. We show that the recurrences for rows of $M$ and columns of $M^{-1}$ are equivalent. The recurrences for columns of $M$ and rows of $M^{-1}$ generate functions that are the Lagrange interpolation polynomials of their elements. These polynomials are equal to hypergeometric functions, which are solutions of the recurrences.