On sums of binomial coefficients modulo p²

• Autorzy: Zhi-Wei Sun
• Języki publikacji: EN

Abstrakty

EN

Let p be an odd prime and let a be a positive integer. In this paper we investigate the sum $∑_{k=0}^{p^{a}-1} (hp^{a}-1 \atop k) (2k \atop k)/m^{k} (mod p²)$, where h and m are p-adic integers with m ≢ 0 (mod p). For example, we show that if h ≢ 0 (mod p) and $p^{a} > 3$, then
$∑_{k=0}^{p^{a}-1} (hp^{a}-1 \atop k)(2k \atop k)(-h/2)^{k} ≡ ((1-2h)/(p^{a}))(1 + h((4-2/h)^{p-1} - 1)) (mod p²)$,
where (·/·) denotes the Jacobi symbol. Here is another remarkable congruence: If $p^{a} > 3$ then
$∑_{k=0}^{p^{a}-1} (p^{a}-1 \atop k)(2k \atop k)(-1)^{k} ≡ 3^{p-1} (p^{a}/3) (mod p²)$.

Kategorie tematyczne

• 11B39: Fibonacci and Lucas numbers and polynomials and generalizations
• 11S99: None of the above, but in this section
• 11B65: Binomial coefficients; factorials; q -identities
• 11E25: Sums of squares and representations by other particular quadratic forms
• 11A07: Congruences; primitive roots; residue systems
• 05A10: Factorials, binomial coefficients, combinatorial functions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2012
• Tom: 127
• Numer: 1
• Strony: 39-54

Daty

wydano

2012

Twórcy

autor

Zhi-Wei Sun

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

Identyfikatory

DOI

10.4064/cm127-1-3