Legendre polynomials and supercongruences

Zhi-Hong Sun

Zhi-Hong Sun

Acta Arithmetica, Tome 161 (2013), p. 169-200 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Publié le : 2013-01-01

Zbl 1287.11004

EUDML-ID : urn:eudml:doc:286585

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     author = {Zhi-Hong Sun},
     title = {Legendre polynomials and supercongruences},
     journal = {Acta Arithmetica},
     volume = {161},
     year = {2013},
     pages = {169-200},
     zbl = {1287.11004},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa159-2-6}
}

Zhi-Hong Sun. Legendre polynomials and supercongruences. Acta Arithmetica, Tome 161 (2013) pp. 169-200. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa159-2-6/