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PreprintCC BY 4.0arXiv

Harmonic Catalan Number Series via Half-Integer Binomial Coefficients

  • Kunle Adegoke
  • Robert Frontczak
  • Taras Goy
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Abstract

We develop a method for evaluating finite and infinite series involving Catalan numbers by specializing alternating binomial identities at half-integer parameters. The key ingredient is an explicit representation of the generalized binomial coefficients binomm+1/2k+1 in terms of Catalan numbers and products of odd linear factors. Combining this representation with Pascal-type identities and two identities of Batır and Sofo, we derive several families of closed-form summation formulas involving Catalan numbers, harmonic numbers, and odd harmonic numbers. As special cases, we obtain evaluations of finite sums together with their corresponding infinite-series analogues. The approach provides a systematic mechanism for transforming identities for generalized binomial coefficients into identities for Catalan-number series.

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Cite

BibTeXRIS
BibTeX
@misc{adegoke2026harmonic,
  title = {{Harmonic Catalan Number Series via Half-Integer Binomial Coefficients}},
  author = {Kunle Adegoke and Robert Frontczak and Taras Goy},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.29354},
  eprint = {2608.29354},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.29354},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Harmonic Catalan Number Series via Half-Integer Binomial Coefficients
AU  - Kunle Adegoke
AU  - Robert Frontczak
AU  - Taras Goy
PY  - 2026
DA  - 2026/08/29
PB  - arXiv
DO  - 10.48550/arxiv.2608.29354
UR  - https://arxiv.org/abs/2608.29354
AB  - We develop a method for evaluating finite and infinite series involving Catalan numbers by specializing alternating binomial identities at half-integer parameters. The key ingredient is an explicit representation of the generalized binomial coefficients binomm+1/2k+1 in terms of Catalan numbers and products of odd linear factors. Combining this representation with Pascal-type identities and two identities of Batır and Sofo, we derive several families of closed-form summation formulas involving Catalan numbers, harmonic numbers, and odd harmonic numbers. As special cases, we obtain evaluations of finite sums together with their corresponding infinite-series analogues. The approach provides a systematic mechanism for transforming identities for generalized binomial coefficients into identities for Catalan-number series.
ER  -