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arXiv · 2602.15208

Self-Convolutions of Generalized Narayana Numbers

Greg Dresden·Yuechen Xiao·Guanzhang Zhou
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Abstract

For the Fibonacci numbers $F_n$, we have the self-convolution formula $5 \sum_{i=0}^n F_i F_{n-i} = (2n)F_{n+1} - (n+1)F_n$. We find the corresponding self-convolution formula for the Narayana numbers $R_n$ which satisfy $R_n = R_{n-1} + R_{n-3}$, and then generalize it to the $k$-step Narayana numbers $\mathcal{R}_n$ with order-$k$ recurrence formula $\mathcal{R}_n = \mathcal{R}_{n-1} + \mathcal{R}_{n-k}$.

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Greg Dresden, Yuechen Xiao, Guanzhang Zhou. 2026-05-20. Self-Convolutions of Generalized Narayana Numbers. https://arxiv.org/abs/2602.15208

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