Congruences involving S_k^(2)(2) and S_k^(2)(5/9)
Establish the mod-p^2 congruences, normalized integer relations, and p-adic integrality statements for the specializations and weighted sums specified in Conjecture 4.7.
References
Conjecture 4.7. Let p be an odd prime.
— A family of polynomials and related congruences and series
(2505.02767 - Sun, 5 May 2025) in Conjecture 4.7, Section 4
In view of T2kP and Conjecture 3.1, we are led to pose the following conjecture.
\begin{conjecture} Let $p>3$ be a prime, $n\in {\pm 156816, \pm1584, \pm 784, \pm 144, \pm 48, \pm 9}$ and $n \not\equiv 0, \pm 16 \pmod p$. Then \begin{equation*} \sum{p-1}{k=0}\binom{2k}{k}2\frac{T{2k}(n,64) }{(n+16){2k}16k} \equiv \left( \frac{n(n+16)}{p}\right)\sum_{k=0}{p-1}\binom{2k}{k}2\binom{4k}{2k}\frac{1}{n{2k} \pmod{p2}. \end{equation*} \end{conjecture}
— New Congruences Involving $p$-adic dual sequences
(2608.14453 - Otmani, 14 Aug 2026) in Section 3, conjecture immediately following the corollary containing equations (T2kP) and (T2kP1)