Self-Convolutions of Generalized Narayana Numbers

TL;DR

This work proves a unified self-convolution identity for the $k$-step Narayana numbers $\mathcal{R}_n$, extending known results for Fibonacci ($k=2$) and Narayana families ($k=3,4$) to general order-$k$ recurrences. The authors employ generating-function techniques, expressing $\mathcal{R}_n$ via $\sum_{n\ge0} \mathcal{R}_n x^n = \frac{x}{1-x-x^k}$ and establishing an equality $A(x)=B(x)-C(x)$ for three carefully defined generating-functions components, using two auxiliary lemmas. The resulting identity is

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 $\mathcal{R}_n = \mathcal{R}_{n-1} + \mathcal{R}_{n-k}$.

Theorems & Definitions (3)

• Theorem 4.1
• Lemma 5.1
• Lemma 5.2