New presences of $\pi$ and $e$ in Pascal's triangle
Combinatorics
2023-02-20 v2
Abstract
The following work shows new connections between the constants and with Pascal's triangle and the Lucas triangle, established via Fibonacci polynomials and similar means. Furthermore, relations between the two famous constants and the rows of Pascal's triangle and the Lucas triangle are conjectured, together with some other important related identities.
Cite
@article{arxiv.2212.11149,
title = {New presences of $\pi$ and $e$ in Pascal's triangle},
author = {Mauricio Guevara V.},
journal= {arXiv preprint arXiv:2212.11149},
year = {2023}
}
Comments
13 pages, 13 figures. Fixed a minor mistake in the proof of theorem 4 (whose claim remains true), added a new, more general theorem relating e and the Lucas polynomials, as well as a new conjecture concerning the shallow diagonals of Pascal's triangle and pi. Also added an identity that relates e and pi to the factorials of composite numbers. Fixed some minor errata