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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≢ 0 (mod p), $P_{[p / 6]} (t) \equiv - (3 / p) \sum_{x = 0}^{p - 1} ((x ³ - 3 x + 2 t) / p) (m o d p)$ and $(\sum_{x = 0}^{p - 1} ((x ³ + m x + n) / p)) ² \equiv ((- 3 m) / p) \sum_{k = 0}^{[p / 6]} (\binom{2 k}{k}) (\binom{3 k}{k}) (\binom{6 k}{3 k}) {((4 m ³ + 27 n ²) / (12 ³ \cdot 4 m ³))}^{k} (m o d p)$ , where (a/p) is the Legendre symbol and [x] is the greatest integer function. As an application we solve some conjectures of Z. W. Sun and the author concerning $\sum_{k = 0}^{p - 1} (\binom{2 k}{k}) (\binom{3 k}{k}) (\binom{6 k}{3 k}) / m^{k} (m o d p ²)$ , where m is an integer not divisible by p.

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Interpolatorische Kubatorformeln und reelle Ideale.

Irrationality measures of $log 2$ and $π / \sqrt{3}$ .

Itération des polynômes et propriétés d'orthogonalité

Iterative square roots of Cebysev polynomials.

Jacobi type and Gegenbauer type generalization of certain polynomials

Jacobi-Sobolev orthogonal polynomials: asymptotics for $N$ -coherence of measures.

Jacobi-weighted orthogonal polynomials on triangular domains.

La fonction hypergéométrique [Book]

La multiplication d'une forme linéaire par une fraction rationnelle. Application aux formes de Laguerre-Hahn

Laguerre polynomials with four parameters

Laguerre series and the Cauchy integral representation

Laguerre-type Bell polynomials.

Laguerre-type exponentials, and the relevant $L$ -circular and $L$ -hyperbolic functions.

Landau and Kolmogoroff type polynomial inequalities.

Laplace transform pairs of N-dimensions and second order linear partial differential equations with constant coefficients.

Left-definite variations of the classical Fourier expansion theorem.

Legendre polynomials and supercongruences

Les polynômes orthogonaux auto-associés modulo deux

Linearization and Connection Coefficients of Orthogonal Polynomials.

Linearization coefficients for Sheffer polynomial sets via lowering operators.

Currently displaying 181 – 200 of 462