### On sums of binomial coefficients modulo p^2On sums of binomial coefficients modulo p^2

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 Author Sun, Zhi-Wei Source arXiv.org Content type Text File Format PDF Date of Submission 2009-10-29 Language English
 Subject Domain (in DDC) Natural sciences & mathematics ♦ Mathematics Subject Keyword Mathematics - Number Theory ♦ Mathematics - Combinatorics ♦ 11B65 ♦ 05A10 ♦ 11A07 ♦ 11S99 ♦ math Abstract Let p be an odd prime and let a be a positive integer. In this paper we investigate the sum $\sum_{k=0}^{p^a-1}\binom{hp^a-1}{k}\binom{2k}{k}/m^k$ mod p^2, where h,m are p-adic integers with m\not=0 (mod p). For example, we show that if h\not=0 (mod p) and p^a>3 then $sum_{k=0}^{p^a-1}\binom{hp^a-1}{k}\binom{2k}{k}(-h/2)^k =(\frac{1-2h}{p^a})(1+h((4h-2)^{p-1}/h^{p-1}-1)) (mod p^2),$ where (-) denotes the Jacobi symbol. Here is another remarkable congruence: If p>3 then $\sum_{k=0}^{p^a-1}\binom{p^a-1}{k}\binom{2k}{k}(-1)^k =3^{p-1}(\frac{p^a}3) (mod p^2).$ Educational Use Research Learning Resource Type Article Page Count 13