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Finite-sum Identities and Supercongruences for Cubes of Catalan Numbers

  • Yuqi Liu
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Abstract

Let Ck=binom2kk/(k+1) be the k-th Catalan number, and let p be an odd prime. We establish exact finite-sum identities that, together with known binomial supercongruences and reductions of Sun, yield ∑k=0(p−1)/2Ck3/16k≡2−24y2−5p2/2y2pmodp3 when p=x2+3y2≡1pmod3, and ∑k=0(p−1)/2Ck3/((−8)k)≡−1−48y2−33p2/16y2pmodp3 when p=x2+4y2≡1pmod4. We also evaluate the corresponding sum with denominator 16k(k+1)2, and obtain two weighted Catalan supercongruences. The finite identities follow from the ratios of consecutive summands, symmetry of double sums, and telescoping. All endpoint terms are retained over the rational numbers before any modular reduction. In each bilinear identity, one endpoint denominator equals p; its cancellation produces the necessary term of order p2. The precise arithmetic inputs are stated separately.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026finitesum,
  title = {{Finite-sum Identities and Supercongruences for Cubes of Catalan Numbers}},
  author = {Yuqi Liu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.34160},
  eprint = {2610.34160},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.34160},
}
RIS
TY  - UNPB
TI  - Finite-sum Identities and Supercongruences for Cubes of Catalan Numbers
AU  - Yuqi Liu
PY  - 2026
DA  - 2026/10/25
PB  - arXiv
DO  - 10.48550/arxiv.2610.34160
UR  - https://arxiv.org/abs/2610.34160
AB  - Let C_k=binom2kk/(k+1) be the k-th Catalan number, and let p be an odd prime. We establish exact finite-sum identities that, together with known binomial supercongruences and reductions of Sun, yield ∑_k=0^(p−1)/2C_k^3/16^k≡2−24y^2−5p^2/2y^2pmodp^3 when p=x^2+3y^2≡1pmod3, and ∑_k=0^(p−1)/2C_k^3/((−8)^k)≡−1−48y^2−33p^2/16y^2pmodp^3 when p=x^2+4y^2≡1pmod4. We also evaluate the corresponding sum with denominator 16^k(k+1)^2, and obtain two weighted Catalan supercongruences. The finite identities follow from the ratios of consecutive summands, symmetry of double sums, and telescoping. All endpoint terms are retained over the rational numbers before any modular reduction. In each bilinear identity, one endpoint denominator equals p; its cancellation produces the necessary term of order p^2. The precise arithmetic inputs are stated separately.
ER  -