Asymptotics of the Euler transform of Fibonacci numbers

TL;DR

This paper studies the asymptotic behavior of a generalized generating function involving Fibonacci numbers, extending known results for specific sequences in the OEIS through analysis of a broader class of functions.

Contribution

It introduces a generalized form of the Euler transform of Fibonacci numbers and derives its asymptotic properties, expanding understanding of related OEIS sequences.

Findings

Derived asymptotic formulas for the generalized generating function

Extended results to multiple OEIS sequences (A166861, A200544, A260787)

Provided insights into the growth behavior of Fibonacci-based products

Abstract

The generating function for the sequence A166861 in the OEIS (Euler transform of Fibonacci numbers) is Product_{k>0} 1/(1-x^k)^F(k), where F(k) are the Fibonacci numbers. This paper analyzes the more general generating function U(x) = Product_{k>0} 1/(1-x^k)^F(k+z), where z is a nonnegative integer, which provides asymptotics for the sequences A166861 (z=0), A200544 (z=1) and A260787 (z=2) in the OEIS.