The k-Binomial Transforms and the Hankel Transform

Document Type


Publication Date


Publication Title

Journal of Integer Sequences


Mathematics and Computer Science


We give a new proof of the inv ariance of the Hankel transform under the binomial transform of a sequence. Our method of proof leads to three v ariations of the binomial transform; we call these the k-binomial transforms. We give a simple means of constructing these transforms via a triangle of numbers. We show how the exponential generating function of a sequence changes after our transforms are applied, and we use this to prove that several sequences in the On-Line Encyclopedia of Integer Sequences are related via our transforms. In the process, we prove three conjectures in the OEIS. Addressing a question of Layman, we then show that the Hankel transform of a sequence is inv ariant under one of our transforms, and we show how the Hankel transform changes after the other two transforms are applied. Finally , we use these results to determine the Hankel transforms of several integer sequences.