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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


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