Congruences concerning Legendre Polynomials
نویسنده
چکیده
Let p be an odd prime. In the paper, by using the properties of Legendre polynomials we prove some congruences for È p−1 2 k=0 2k k ¡ 2 m −k (mod p 2). In particular, we confirm several conjectures of Z.W. Sun. We also pose 13 conjectures on supercongruences.
منابع مشابه
Congruences concerning Legendre polynomials III
Suppose that p is an odd prime and d is a positive integer. Let x and y be integers given by p = x2 + dy2 or 4p = x2 + dy2. In this paper we determine x (mod p) for many values of d. For example,
متن کاملJ. Number Theory 143(2014), no.3, 293-319. GENERALIZED LEGENDRE POLYNOMIALS AND RELATED SUPERCONGRUENCES
Abstract. For any positive integer n and variables a and x we define the generalized Legendre polynomial Pn(a, x) by Pn(a, x) = Pn k=0 a k −1−a k ( 1−x 2 ). Let p be an odd prime. In this paper we prove many congruences modulo p related to Pp−1(a, x). For example, we show that Pp−1(a, x) ≡ (−1)〈a〉p Pp−1(a,−x) (mod p), where a is a rational p− adic integer and 〈a〉p is the least nonnegative resid...
متن کاملJ. Number Theory 133(2013), 1950-1976. CONGRUENCES CONCERNING LEGENDRE POLYNOMIALS II
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 ]...
متن کاملSchur congruences, Carlitz sequences of polynomials and automaticity
We rst generalize the Schur congruence for Legendre polynomials to sequences of polyno-mials that we call \d-Carlitz". This notion is more general than a similar notion introduced by Carlitz. Then, we study automaticity properties of double sequences generated by these sequences of polynomials, thus generalizing previous results on double sequences produced by one-dimensional linear cellular au...
متن کاملOn Modified Dickson Polynomials
were defined and studied by P. Fllipponi In the case a-\ in [1], where several identities and congruences were established. In this note we generalize some of those theorems and present some new properties of these polynomials. One basic result is Proposition 2 in [1] which states that if/? is an odd prime and k is an integer, then Zp(k,l) = (k\p) (mod/0, (1-2) where (k \p) is the Legendre symb...
متن کامل