Some congruences involving binomial coefficients

• Autorzy: Hui-Qin Cao , Zhi-Wei Sun
• Języki publikacji: EN

Abstrakty

EN

Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let p > 3 be a prime. We show that
$T_{p-1} ≡ (p/3) 3^{p-1} (mod p²)$,
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
$∑_{k=0}^{p-1} \binom{p-1}{k}\binom{2k}{k} ((-1)^k - (-3)^{-k}) ≡ (p/3)(3^{p-1} - 1) (mod p³)$.
In addition, we get some new combinatorial identities.

Kategorie tematyczne

• 05A19: Combinatorial identities, bijective combinatorics
• 11B39: Fibonacci and Lucas numbers and polynomials and generalizations
• 11B65: Binomial coefficients; factorials; q -identities
• 05A10: Factorials, binomial coefficients, combinatorial functions
• 11A07: Congruences; primitive roots; residue systems

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2015
• Tom: 139
• Numer: 1
• Strony: 127-136

Daty

wydano

2015

Twórcy

autor

Hui-Qin Cao

• Department of Applied Mathematics, Nanjing Audit University, Nanjing 211815, People's Republic of China

autor

Zhi-Wei Sun

• Department of Mathematics, Nanjing University, Nanjing 210093, People's Republic of China

Identyfikatory

DOI

10.4064/cm139-1-8