A new formula for hyper-Fibonacci numbers, and the number of occurrences

Authors: TAKAO KOMATSU, LASZLO SZALAY

Abstract: In this paper, we develop a new formula for hyper-Fibonacci numbers $F_n^{[k]}$, wherein the coefficients (related to Stirling numbers of the first kind) of the polynomial ingredient $p_k(n)$ are determined. As an application we investigate the number of occurrences of positive integers among $F_n^{[k]}$ and determine all the solutions in nonnegative integers $x$ and $y$ to the Diophantine equation $F_x^{[k]}=F_y^{[\ell]}$, where $0\le k<\ell\le 70$. Moreover, we prove that if $\ell$ is fixed, then $F_x^{[k]}=F_y^{[\ell]}$ has finitely many effectively computable solutions in the nonnegative integers $x$, $y$, and $k\le\ell$.

Keywords: Hyper-Fibonacci numbers, Stirling numbers of the first kind, Diophantine equation, number of occurrences

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