Congruences involving S_k^(2)(−1/3)

Prove the mod-p^2 congruences and p-adic integrality statement for the sums involving S_k^{(2)}(−1/3) specified in Conjecture 4.2.

Background

This conjecture gives supercongruences for a specialization of S_k{(2)} at −1/3 and relates the resulting values to representations of primes by x2+ y2, together with a companion weighted congruence.

References

Conjecture 4.2. Let p > 3 be a prime.

— A family of polynomials and related congruences and series  (2505.02767 - Sun, 5 May 2025) in Conjecture 4.2, Section 4

Conjecture 4.2 Let p>3 be a prime. Then $$\begin{align*}&\sum_{n=0}{p-1}(2n+1)3V_n\Big( 12\Big)2 6p4-{143}3p6\mod {p7}, &\sum_{n=0}{p-1}(2n+1)3V_n\Big(- 12\Big)2 - p2\mod {p4}, &\sum_{n=0}{p-1}(2n+1)3V_n\Big(- 13\Big)2 - 59\Ls p3p2\mod {p4}, &\sum_{n=0}{p-1}(2n+1)3V_n\Big(- 14\Big)2 - 58(-1){p-1}2}p2\mod {p4}, &\sum_{n=0}{p-1}(2n+1)3V_n\Big(- 16\Big)2 - {13}{18}\Ls p3p2\mod {p4}.\end{align*}$$

— Generalizations of the Christoffel-Darboux formula and congruences involving Apéry-like numbers  (2608.13192 - Sun, 13 Aug 2026) in Conjecture 4.2, Section 4