Fibonacci polynomials, generalized Stirling numbers, and Bernoulli, Genocchi and tangent numbers

TL;DR

This paper explores the relationships between Fibonacci and Lucas polynomials, generalized Stirling numbers, Bernoulli, Genocchi, and tangent numbers, revealing new identities and connections through matrix transformations.

Contribution

It introduces matrices that connect Fibonacci and Lucas polynomials of different indices and links these to generalized Stirling numbers and special number sequences, uncovering novel identities.

Findings

Matrices relate Fibonacci and Lucas polynomials of different indices.

New identities between Bernoulli, Genocchi, tangent, and Stirling numbers.

Connections with the Akiyama-Tanigawa algorithm.

Abstract

We study matrices which transform the sequence of Fibonacci or Lucas polynomials with even index to those with odd index and vice versa. They turn out to be intimately related to generalized Stirling numbers and to Bernoulli, Genocchi and tangent numbers and give rise to various identities between these numbers. There is also a close connection with the Akiyama-Tanigawa algorithm.