On sums of binomial coefficients modulo p^2

@inproceedings{Sun2012OnSO,
  title={On sums of binomial coefficients modulo p^2},
  author={Zhi-Wei Sun},
  year={2012}
}
Let p be an odd prime and let a be a positive integer. In this paper we investigate the sum ∑pa−1 k=0 (hpa−1 k )(2k k ) /mk mod p2, where h and m are p-adic integers with m 6≡ 0 (mod p). For example, we show that if h 6≡ 0 (mod p) and pa > 3, then p−1 ∑ k=0 (hpa − 1 k )(2k k )( − h 2 )k ≡ ( 1− 2h pa )( 1 + h (( 4− 2 h )p−1 − 1 )) (mod p), where (−) denotes the Jacobi symbol. Here is another remarkable congruence: If pa > 3 then p−1 ∑ k=0 (pa − 1 k )(2k k ) (−1) ≡ 3p−1 ( pa 3 ) (mod p). 

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