Some conjectures on the ratio of Hankel transforms for sequences and series reversion

The Hankel transform for sequences (defined below) has attracted an increasing amount of attention in recent years. The paper [4] situated its study within the mainstream of research into integer sequences, while papers such as [2] hinted at how the study of certain Hankel transforms can lead to results concerning classical sequences. That paper exploited a link between continued fractions and the Hankel transform, as explained by Krattenhaler [3]. The best known example of a Hankel transform for sequences is that of the Catalan numbers. One of the earlier contributors to our stock of knowledge about the Hankel transform, Christian Radoux, had published several proofs of this result, along with other interesting examples [7], [8],[9],[10],[11]. One should also note the interesting umbral interpretation of the Hankel transform given in [13]. In this paper we indicate that the term-wise ratio of Hankel transforms of shifted sequences are noteworthy objects of study, giving us more insight into the processes involved in the Hankel transform.