We produce congruences modulo a prime p > 3 for sums ∑k (3kk) xk over ranges 0 ≤ k < q and 0 ≤ k < q/3, where q is a power of p. Here x equals either c2/(1 - c) 3, or 4s2/(27(s2 - 1)), where c and s are indeterminates. In the former case, we deal more generally with shifted binomial coefficients 3k+ek. Our method derives such congruences directly from closed forms for the corresponding series.
Mattarei, S., Tauraso, R. (2025). Congruences for partial sums of the generating series for (3kk). INTERNATIONAL JOURNAL OF NUMBER THEORY, 21(1 (February 2025)), 237-256 [10.1142/S1793042125500125].
Congruences for partial sums of the generating series for (3kk)
Mattarei S.;
2025
Abstract
We produce congruences modulo a prime p > 3 for sums ∑k (3kk) xk over ranges 0 ≤ k < q and 0 ≤ k < q/3, where q is a power of p. Here x equals either c2/(1 - c) 3, or 4s2/(27(s2 - 1)), where c and s are indeterminates. In the former case, we deal more generally with shifted binomial coefficients 3k+ek. Our method derives such congruences directly from closed forms for the corresponding series.
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