Study of $p^{k}$-Eulerian polynomials and $p^{k}$-Fibonacci numbers for every odd prime $p$ and $k\geq0$

M. Parvathi; A. Tamilselvi; D. Hepsi

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arXiv · 2506.09941

Study of $p^{k}$-Eulerian polynomials and $p^{k}$-Fibonacci numbers for every odd prime $p$ and $k\geq0$

M. Parvathi·A. Tamilselvi·D. Hepsi

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Abstract

In this paper, we define the notion of descent for the paths in the $p$-Bratteli diagram. This leads to the definition of $p^{k}$-Eulerian polynomials, whose coefficients count the number of paths with a given number of descents. We provide a method for constructing the $p^{k}$-Eulerian polynomials at each vertex. Furthermore, we compute the total number of descents of all paths ending at a given vertex as the corresponding $p^{k}$-Fibonacci numbers. We show that the derivative of the $p^{k}$-Eulerian polynomial evaluated at 1 for a fixed vertex equals the corresponding $p^{k}$-Fibonacci number. Finally, we discuss the generating function for the sequence of $p^{k}$-Fibonacci numbers and the recurrence relations they satisfy.

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M. Parvathi, A. Tamilselvi, D. Hepsi. 2025-06-11. Study of $p^{k}$-Eulerian polynomials and $p^{k}$-Fibonacci numbers for every odd prime $p$ and $k\geq0$. https://arxiv.org/abs/2506.09941

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