Geodesic
The $k$-binomial transforms and the Hankel transform
Journal of integer sequences, Tome 9 (2006) no. 1.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: We give a new proof of the invariance of the Hankel transform under the binomial transform of a sequence. Our method of proof leads to three variations 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 invariant 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.
Classification : 11B65, 11B75
Keywords: binomial transform, Hankel transform 
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     title = {The $k$-binomial transforms and the {Hankel} transform},
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Spivey, Michael Z.; Steil, Laura L. The $k$-binomial transforms and the Hankel transform. Journal of integer sequences, Tome 9 (2006) no. 1. http://geodesic.mathdoc.fr/item/JIS_2006__9_1_a2/