A novel theory of Legendre polynomials

Extreme value theory for stochastic integrals of Legendre polynomials

For t ≥ 0, let X(t) = (X0(t), . . . , Xp(t)) , where Xi(t) denotes the integral of the ith order Legendre polyonimal with respect to the same Brownian motion described by the corresponding standard deviation (0 ≤ i ≤ p). We obtain the exact tail behavior of P ( sup0≤t≤h |X(t)| > u ) as u → ∞, and the limit distribution of sup0≤t≤T |X(t)| as T → ∞. These processes naturally arise in the context ...

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