Supercongruences of Atkin and Swinnerton-Dyer type for Legendre polynomials

Abstract

In this paper we prove some generalisations of congruences of Atkin and Swinnerton-Dyer type. This is done in the form of congruences for numbers P n(A/j']), where P,,(I) are the orthogonal polynomials of Legendre. The proofs are based on complex multiplication of elliptic functions.

Formal Groups and Invariant Differentials of Elliptic Curves

“…Proof: The invariant differential of E is defined by ω E = dw/dz 3z 2 dz, see (3). The result now follows as we have shown that…”

Supercongruences and complex multiplication

“…Theorem 7 (Coster and van Hamme, [9]). Let p be an odd prime and d a squarefree positive integer such that ( −d p ) = 1.…”

Supercongruences for sporadic sequences

“…Proceeding as in the proof of Lemma 2.3, we therefore obtain (37). The congruence (14) then follows if we can prove that, for integers k such that p ∤ k, D(mp r , k) ≡ 0 (mod p 2r ). This is an immediate consequence of the fact that D(n, k) is divisible by n k 2 .…”

“…For recent progress in this direction, see [11], [16], [22], [23], [26], [33] or [41]. Finally, Atkin-Swinnerton-Dyer supercongruences have been recently studied in [14], [15], [24] and [37]. In this paper, we consider the sequences of numbers given by…”