On sums of binomial coefficients modulo p²

Zhi-Wei Sun. "On sums of binomial coefficients modulo p²." Colloquium Mathematicae 127.1 (2012): 39-54. <http://eudml.org/doc/284015>.

@article{Zhi2012,
abstract = {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 $∑_\{k=0\}^\{p^\{a\}-1\} (p^\{a\}-1 \atop k)(2k \atop k)(-1)^\{k\} ≡ 3^\{p-1\} (p^\{a\}/3) (mod p²)$.},
author = {Zhi-Wei Sun},
journal = {Colloquium Mathematicae},
keywords = {central binomial coefficients; congruences modulo prime powers},
language = {eng},
number = {1},
pages = {39-54},
title = {On sums of binomial coefficients modulo p²},
url = {http://eudml.org/doc/284015},
volume = {127},
year = {2012},
}

TY - JOUR
AU - Zhi-Wei Sun
TI - On sums of binomial coefficients modulo p²
JO - Colloquium Mathematicae
PY - 2012
VL - 127
IS - 1
SP - 39
EP - 54
AB - 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 $∑_{k=0}^{p^{a}-1} (p^{a}-1 \atop k)(2k \atop k)(-1)^{k} ≡ 3^{p-1} (p^{a}/3) (mod p²)$.
LA - eng
KW - central binomial coefficients; congruences modulo prime powers
UR - http://eudml.org/doc/284015
ER -