Convoluted convolved Fibonacci numbers

TL;DR

This paper explores properties of convolved Fibonacci numbers, their relation to number theory constants, and studies a specific transform involving formal series and the Möbius function.

Contribution

It introduces related numbers expressed via convolved Fibonacci numbers and analyzes a transform involving formal series and the Möbius function.

Findings

Related numbers expressed in terms of convolved Fibonacci numbers

Connection to numerical evaluation of number theoretical constants

Analysis of a transform involving formal series and Möbius function

Abstract

The convolved Fibonacci numbers F_j^(r) are defined by (1-z-z^2)^{-r}=\sum_{j>=0}F_{j+1}^(r)z^j. In this note some related numbers that can be expressed in terms of convolved Fibonacci numbers are considered. These numbers appear in the numerical evaluation of a certain number theoretical constant. This note is a case study of the transform {1/n}\sum_{d|n}mu(d)f(z^d)^{n/d}, with f any formal series and mu the Moebius function), which is studied in a companion paper entitled `The formal series Witt transform'.