Congruences involving S_k^(2)(2)

Establish the mod-p^2 congruences, integer-valued normalized sums, and p-adic integrality assertions involving S_k^{(2)}(2) stated in Conjecture 4.3.

Background

Conjecture 4.3 extends the preceding pattern to the specialization S_k{(2)}(2), including a weighted congruence and normalized congruences for arbitrary positive powers of p.

References

Conjecture 4.3. Let p be an odd prime.

— A family of polynomials and related congruences and series  (2505.02767 - Sun, 5 May 2025) in Conjecture 4.3, Section 4

Conjecture 4.3 Let p>7 be a prime. Then $$\begin{align*}&\sum_{n=0}{p-1}(2n+1)5V_n\Big( 12\Big)2 {p}{64}\mod {p4}, &\sum_{n=0}{p-1}(2n+1)5V_n\Big(- 12\Big)2 {19}{32}p\mod {p4}, &\sum_{n=0}{p-1}(2n+1)5V_n\Big(- 13\Big)2 {709}{945}\Ls p3p2\mod {p4}, &\sum_{n=0}{p-1}(2n+1)5V_n\Big(- 14\Big)2 {1841}{1920}(-1){p-1}2}p2\mod {p4}, &\sum_{n=0}{p-1}(2n+1)5V_n\Big(- 16\Big)2 {8813}{6912}\Ls p3p2\mod {p4}.\end{align*}$$

— Generalizations of the Christoffel-Darboux formula and congruences involving Apéry-like numbers  (2608.13192 - Sun, 13 Aug 2026) in Conjecture 4.3, Section 4