Reverse Legendre polynomials (Q2140598)

Let \(\mathcal{P}\) be the vector space of all polynomials, equipped with the inner product \(\langle f(x), g(x)\rangle=\int_{-1}^{1} f(x) g(x) d x\). The Legendre polynomials \(P_{0}(x), P_{1}(x), \ldots\) are the polynomials obtained by applying the Gram-Schmidt procedure to the ordered basis \(\mathcal{B}=\left\{1, x, x^{2}, \ldots\right\}\) of \(\mathcal{P}\), except that the resulting polynomials are normalized by the condition that \(P_{n}(1)=1\) for every \(n\). This paper poses the following question: what happens if the Gram-Schmidt procedure is applied ``from the top down''? For any nonnegative integer \(n\), let \(\mathcal{P}_{n}\) be the vector space of polynomials of degree at most \(n\), equipped with the same inner product, and apply the Gram-Schmidt procedure to the ordered basis \(\mathcal{B}=\left\{x^{n}, x^{n-1}, \ldots, 1\right\}\) of \(\mathcal{P}_{n}\) to obtain polynomials \(\left\{\overleftarrow{P}_{n}^{n}(x), \overleftarrow{P}_{n-1}^{n}(x), \ldots, \overleftarrow{P}_{0}^{n}(x)\right\}\) with the analogous normalization \(\overleftarrow{P}_{k}^{n}(1)=1\) for every \(n, k\). Since these polynomials are obtained by reversing the order of the basis elements, the authors of this paper call these reverse Legendre polynomials. The objective of this paper is to explicitly determine the polynomials \(\left\{\overleftarrow{P}_{k}^{n}(x)\right\}\) and to describe some of their properties.