Accurate Evaluation of Polynomials in Legendre Basis

Some Experiments with Evaluation of Legendre Polynomials

Common practice is to recommend evaluation of polynomials by Horner’s rule. Here’s an example where it is fast but doesn’t work nearly as accurately as another fairly easy method. Can a method for Legendre polynomials be both fast and accurate? 1 1 Legendre Polynomials A substantial literature has grown up around the uses for orthonormal polynomials. Here we look at the example of Legendre poly...

On Polar Legendre Polynomials

We introduce a new class of polynomials {Pn}, that we call polar Legendre polynomials, they appear as solutions of an inverse Gauss problem of equilibrium position of a field of forces with n + 1 unit masses. We study algebraic, differential and asymptotic properties of this class of polynomials, that are simultaneously orthogonal with respect to a differential operator and a discrete-continuou...

Congruences concerning Legendre Polynomials

Let p be an odd prime. In the paper, by using the properties of Legendre polynomials we prove some congruences for È p−1 2 k=0 2k k ¡ 2 m −k (mod p 2). In particular, we confirm several conjectures of Z.W. Sun. We also pose 13 conjectures on supercongruences.

Legendre polynomials and supercongruences

Let p > 3 be a prime, and let Rp be the set of rational numbers whose denominator is not divisible by p. Let {Pn(x)} be the Legendre polynomials. In this paper we mainly show that for m,n, t ∈ Rp with m 6≡ 0 (mod p), P[ p 6 ](t) ≡ − (3 p ) p−1 ∑ x=0 (x3 − 3x + 2t p ) (mod p)

Fractional differential equations with Legendre polynomials