New presences of $\pi$ and $e$ in Pascal's triangle

TL;DR

This paper explores novel mathematical connections between the constants pi and e with Pascal's and Lucas triangles, using Fibonacci polynomials, and proposes conjectures about their relationships.

Contribution

It introduces new links between pi, e, Pascal's triangle, and Lucas triangle through Fibonacci polynomials and formulates related conjectures.

Findings

New identities connecting pi and e with Pascal's and Lucas triangles.

Conjectures proposing further relationships between these constants and the triangle rows.

Identification of identities involving Fibonacci polynomials and these constants.

Abstract

The following work shows new connections between the constants $\pi$ and $e$ 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.