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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.

Leonard pairs from the equitable basis of $s l_{2}$ .

Alnajjar, Hasan, Curtin, Brian (2010)

ELA. The Electronic Journal of Linear Algebra [electronic only]

Les polynômes orthogonaux auto-associés modulo deux

Maroni, P. (1983/1984)

Portugaliae mathematica

Lettres sur la théorie des fonctions elliptiques

C. G. J. Jacobi (1869)

Annales scientifiques de l'École Normale Supérieure

Level sets of polynomials in several real variables.

Heiberg, C.H. (1983)

Publications de l'Institut Mathématique. Nouvelle Série

L'identité de Barnes pour les corps finis

Anna Helversen-Pasotto (1977/1978)

Séminaire Delange-Pisot-Poitou. Théorie des nombres

Lie Derivatives, Killing΄s Equations GegenBauer Polynomials: A Combined Use for the Evaluation of some Useful Integrals

N. Spyrou (1978)

Δελτίο της Ελληνικής Μαθηματικής Εταιρίας

Lie Theoretic Generating Functions

H. L. Manocha (1976)

Publications de l'Institut Mathématique

Lie theory and certain identities satisfied by $_{p} F_{p}$

Wong, C.F., Narain Kesarwani, R. (1975)

Portugaliae mathematica

Linear differential equations and multiple zeta values. I. Zeta(2)

Michał Zakrzewski, Henryk Żołądek (2010)

Fundamenta Mathematicae

Certain generating fuctions for multiple zeta values are expressed as values at some point of solutions of linear meromorphic differential equations. We apply asymptotic expansion methods (like the WKB method and the Stokes operators) to solutions of these equations. In this way we give a new proof of the Euler formula ζ(2) = π²/6. In further papers we plan to apply this method to study some third order hypergeometric equation related to ζ(3).

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