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arXiv · 2608.29354

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 $\binom{m+\frac12}{k+1}$ in terms of Catalan numbers and products of odd linear factors. Combining this representation with Pascal-type identities and two identities of Bat{\i}r 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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Kunle Adegoke, Robert Frontczak, Taras Goy. 2026-08-29. Harmonic Catalan Number Series via Half-Integer Binomial Coefficients. https://arxiv.org/abs/2608.29354

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