Congruences concerning Legendre polynomials II

By Zhi-Hong Sun. 2010

77 citations recorded by the source corpus.

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 ∑k=0p−1(2kk)3/mk(modp2), ∑k=0p−1(2kk)(4k2k)/mk(modp) and ∑k=0p−1(2kk)2(4k2k)/mk(modp2). In particular, we show that ∑k=0p−12(2kk)3≡0(modp2) for p≡3,5,6(mod7). Let {Pn(x)} be the Legendre polynomials. In the paper we also show that P[p4](t)≡−(6p)∑x=0p−1(x3−32(3t+5)x+9t+7p)(modp), where t is a rational p-adic integer, [x] is the greatest integer not exceeding x and (ap) is the Legendre symbol. As consequences we determine P[p4](t)(modp) in the cases t=−53,−79,−6563 and confirm many conjectures of Z.W. Sun.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7be74163e5104a3437976f37f160eb7623c60fcc

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Citations in the corpus

On sums of Ap\'ery polynomials and related congruences (2011, background): “Sun [S3] in the case p ≡ 2 (mod 3), and the mod p version in the case p ≡ 1 (mod 3) follows from (4)-(5) in Ahlgren [A, Theorem 5].”
Conjectures and results on $x^2$ mod $p^2$ with $4p=x^2+dy^2$ (2011, background): “5) CONJECTURES AND RESULTS ON x2 mod p2 WITH 4p = x2 + dy2 7 Remark 1.1. The congruence modulo p can be easily deduced from (6) in Ahlgren [A, Theorem 5]. Recently the author’s twin brother Z. H. Sun [S2] conﬁrmed the conjecture in the case (p 7) = −1 via Legendre polynomials. The imaginary quadrati”
On a q-deformation of modular forms (2018, background): “On a q-deformation of modular forms 3 (see [25, 30]), which link corresponding partial sums with the Fourier coefficients of (2.”