Conjectures and results on some generalized Rueppel sequences

TL;DR

This paper explores generalized Rueppel sequences using continued fractions, Riordan arrays, and Hankel transforms, proposing conjectures and new polynomial definitions related to these sequences.

Contribution

It introduces parameterized polynomial sequences with Riordan array structures and formulates conjectures on Hankel transforms of related sequences.

Findings

Representation of generating functions via Stieltjes continued fractions.

Definition of generalized Rueppel sequences with polynomial forms.

Conjecture on the product of Hankel transforms for Rueppel-related sequences.

Abstract

In this note we use the analogy between the Catalan sequence and the Rueppel sequence to derive a variety of conjectures surrounding the Hankel transforms of a number of sequences closely related to the Rueppel sequence. Use is made of the representation of suitable generating functions by Stieltjes continued fractions. We define polynomial sequences by introducing parameters that define generalized Rueppel sequences, and we show that such polynomials have coefficient arrays that are Riordan arrays. Finally we conjecture the form of a product of Hankel transforms arising from the Rueppel sequence.