Generalized Legendre polynomials and related supercongruences

q-Analogues of two “divergent” Ramanujan-type supercongruences

“…Although supercongruences have been widely studied by many mathematicians including Beukers [3], Long and Ramakrishna [15], Rodriguez-Villegas [21], Z.-H. Sun [24], Z.-W. Sun [25], van Hamme [31], and Zudilin [34], etc., there are still many problems on q-congruences which are worthwhile to investigate. In fact, it is not always easy to give q-analogues of known congruences, and many congruences might have no q-analogues.…”

Congruences for Truncated Hypergeometric Series

“…where • p denotes the Legendre symbol. For more proofs of Theorem 1.1, see [3,14,18,19]. Some extensions of the congruences in Theorem 1.1 to modulus p 3 were obtained in [15,16].…”

Super congruences for two Apéry-like sequences

“…Suppose that p is a prime with p > 3. Then Now applying[S8, Theorem 3.4] yields the result.Theorem 5.3. Let p be an odd prime, m ∈ Z p and (m + 2)(m − 2) ≡ 0 (mod p).…”