Reverse Legendre polynomials.

For a nonnegative integer n, let P n be the vector space of polynomials of degree at most n, equipped with the inner product ⟨ f (x) , g (x) ⟩ = ∫ - 1 1 f (x) g (x) d x . Let B = { x n , x n - 1 , ... , 1 } , an ordered basis of P . Let { P ← n n (x) , P ← n - 1 n (x) , ⋯ , P ← 0 n (x) } be the orthogonal basis of P n obtained by applying the Gram–Schmidt procedure to B , with the resulting polynomials normalized by the condition that P ← k n (1) = 1 . We call these polynomials reverse Legendre polynomials. In this paper, we give explicit formulas for the reverse Legendre polynomials { P ← k n (x) } and determine some of their properties. [ABSTRACT FROM AUTHOR]