Legendre polynomials in irrationality proofs

Little q - Legendre polynomials and irrationality of

Little q-Legendre polynomials and irrationality of certain Lambert series

Certain q-analogs hp(1) of the harmonic series, with p = 1/q an integer greater than one, were shown to be irrational by Erdős [9]. In 1991–1992 Peter Borwein [4] [5] used Padé approximation and complex analysis to prove the irrationality of these q-harmonic series and of q-analogs lnp(2) of the natural logarithm of 2. Recently Amdeberhan and Zeilberger [1] used the qEKHAD symbolic package to f...

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)