Some congruences involving binomial coefficients

Hui-Qin Cao; Zhi-Wei Sun

Hui-Qin Cao ; Zhi-Wei Sun

Colloquium Mathematicae, Tome 139 (2015), p. 127-136 / Harvested from The Polish Digital Mathematics Library

Résumé

Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let p > 3 be a prime. We show that $T_{p - 1} \equiv (p / 3) 3^{p - 1} (m o d p ²)$ , where the central trinomial coefficient Tₙ is the constant term in the expansion of $(1 + x + x^{- 1}) ⁿ$ . We also prove three congruences modulo p³ conjectured by Sun, one of which is $\sum_{k = 0}^{p - 1} (\binom{p - 1}{k}) (\binom{2 k}{k}) ({(- 1)}^{k} - {(- 3)}^{- k}) \equiv (p / 3) (3^{p - 1} - 1) (m o d p ³)$ . In addition, we get some new combinatorial identities.

Publié le : 2015-01-01

Zbl 1339.11021

EUDML-ID : urn:eudml:doc:284057

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     title = {Some congruences involving binomial coefficients},
     journal = {Colloquium Mathematicae},
     volume = {139},
     year = {2015},
     pages = {127-136},
     zbl = {1339.11021},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm139-1-8}
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Hui-Qin Cao; Zhi-Wei Sun. Some congruences involving binomial coefficients. Colloquium Mathematicae, Tome 139 (2015) pp. 127-136. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm139-1-8/