J. Number Theory 133(2013), 1950-1976. CONGRUENCES CONCERNING LEGENDRE POLYNOMIALS II
نویسنده
چکیده
Abstract. Let p > 3 be a prime, and let m be an integer with p ∤ m. In the paper we solve some conjectures of Z.W. Sun concerning Pp−1 k=0 2k k 3 /mk (mod p2), Pp−1 k=0 2k k 4k 2k /mk (mod p) and Pp−1 k=0 2k k 2 4k 2k /mk (mod p2). In particular, we show that P p−1 2 k=0 2k k 3 ≡ 0 (mod p2) for p ≡ 3, 5, 6 (mod 7). Let {Pn(x)} be the Legendre polynomials. In the paper we also show that P[ p 4 ](t) ≡ −( 6 p ) Pp−1 x=0( x− 3 2 (3t+5)x+9t+7 p ) (mod p), where t is a rational p−adic integer, [x] is the greatest integer not exceeding x and ( a p ) is the Legendre symbol. As
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تاریخ انتشار 2013