On p-Adic Valuation Trees Arising from Laplace Transform of Fibonacci Polynomials
Abstract
We study the p-adic valuation trees associated with the sequence en=∫0∞ e−tFn(t) dt, obtained by evaluating the Laplace transforms of the Fibonacci polynomials at $1$. We answer an open question of Givens and Moll in the negative by constructing an explicit counterexample showing that even internal branches need not be simple. We then reformulate the Givens--Moll lifting criteria for the even and odd subsequences in terms of the roots and derivatives of two associated polynomials. In particular, simple roots modulo p give rise to simple internal branches. Finally, we develop a recursive lifting procedure that determines the unique surviving residue class at each level of a branch associated with a simple root and computes the p-adic valuations of the corresponding terms.
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BibTeX
@misc{florez2026padic,
title = {{On $p$-Adic Valuation Trees Arising from Laplace Transform of Fibonacci Polynomials}},
author = {Rigoberto Flórez and Robinson Higuita-Diaz and José L. Ramírez},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37356},
eprint = {2610.37356},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37356},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On p-Adic Valuation Trees Arising from Laplace Transform of Fibonacci Polynomials
AU - Rigoberto Flórez
AU - Robinson Higuita-Diaz
AU - José L. Ramírez
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37356
UR - https://arxiv.org/abs/2610.37356
AB - We study the p-adic valuation trees associated with the sequence e_n=∫_0^∞ e^−tF_n(t) dt, obtained by evaluating the Laplace transforms of the Fibonacci polynomials at $1$. We answer an open question of Givens and Moll in the negative by constructing an explicit counterexample showing that even internal branches need not be simple. We then reformulate the Givens--Moll lifting criteria for the even and odd subsequences in terms of the roots and derivatives of two associated polynomials. In particular, simple roots modulo p give rise to simple internal branches. Finally, we develop a recursive lifting procedure that determines the unique surviving residue class at each level of a branch associated with a simple root and computes the p-adic valuations of the corresponding terms.
ER -