Congruences concerning Legendre polynomials
By Zhi-Hong Sun. 2010
105 citations recorded by the source corpus.
Let p be an odd prime. In this paper, by using the properties of Legendre polynomials we prove some congruences for Σ p―1 2 k=0 ( 2k k m ―k (mod p 2 ). In particular, we confirm several conjectures of Z.W. Sun. We also pose 13 conjectures on supercongruences.
Subfields: Algebra and Number Theory, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/2947187025b82e05c77d6f44308e2b5e55fa5a04
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In Algebra and Number Theory
In Statistical and Nonlinear Physics
Citations in the corpus
- On sums involving products of three binomial coefficients (2010, unknown): “er related results.) In 2009, the author (see [Su2]) conjectured that (p−X1)/2 k=0 2k k 2 8k ≡ (p−X1)/2 k=0 2k k 2 (−16)k ≡ (−1)(p−1)/4 2x− p 2x (mod p2), (3.3) and this was confirmed by Z.-H. Sun [S1] via (3.2) and the Legendre polynomials. ProofofTheorem1.2(i). By (1.2), S(p−1)/2 ≡ (p−X1)/2 k=0 2”
- Open Conjectures on Congruences (2009, methodology): “−k k xk(1+ x)k. Z. H. Sun [Su2] proved that Pp−1 k=0 2k k 3 /256k ≡ 0 (mod p2) for any prime p ≡ 5 (mod 6), and Pp−1 k=0 2k k 3 /(−512)k ≡ 0 (mod p2) for any prime p ≡ 3 (mod 4). See also Z. H. Sun [Su1] for his conjectures on Pp−1 k=0 4k k,k,k,k /mk mod p2 with m = −144, 648, −3969 motivated by th”
- Conjectures and results on $x^2$ mod $p^2$ with $4p=x^2+dy^2$ (2011, unknown): “xn](x2+ 3x+2)n (mod p), the author obtained in 2009 that 2 p pX−1 k=0 2k k 2 8k ≡ pX−1 k=0 2k k 2 32k ≡ ˆ 2x (mod p) if p = x2 +y2 (4 | x− 1, 2 | y), 0 (mod p) if ≡ 3 (mod 4). Later, Zhi-Hong Sun [S1] proved the author’s following conjecture: If p ≡ 1 (mod 4) and p = x2 + y2 with x ≡ 1 (mod 4) and”
- Super congruences and Euler numbers (2011, background): “formula. See also K. Ono’s book [O] and the papers [AO], [K], [MO], [M03b, M05, M08], [Mc] and [OS] for such advanced approach to super congruences. Quite recently the author’s twin brother Z. H. Sun [S3], as well as R. Tauraso [T3], gave a simple proof of the last super congruence. In [S10b] the au”
- On sums of Ap\'ery polynomials and related congruences (2011, unknown): “ p2). Remark4.1. For those x = −4,9,−48,81,−324,2401,9801,−25920,−777924,96059601, 81 256 , the author (cf. [Su2]) had conjectures on Pp−1 k=0 4 k k,k,k,k /(256x) mod p2. Motivated by this, Z. H. Sun [S2] guessed Pp−1 k=0 4k k,k,k,k /(256x)k mod p2 for x = −9/16, 81/32, −3969/256 in a similar way. I”
- Super congruences and Euler numbers (2010, background): “formula. See also K. Ono’s book [O] and the papers [AO], [K], [MO], [M03b, M05, M08], [Mc] and [OS] for such advanced approach to super congruences. Quite recently the author’s twin brother Z. H. Sun [S3], as well as R. Tauraso [T3], gave a simple proof of the last super congruence. In [S10b] the au”