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The ***k*-Binomial Transforms and the Hankel Transform

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Michael Z. Spivey

Department of Mathematics and Computer Science

University of Puget Sound

Tacoma, Washington 98416-1043

USA

Laura L. Steil

Department of Mathematics and Computer Science

Samford University

Birmingham, Alabama 35229

USA

**Abstract:**
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.

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(Concerned with sequences
A000012
A000032
A000045
A000079
A000108
A000142
A000165
A000166
A000354
A000364
A000609
A000984
A001003
A001006
A001653
A001907
A002078
A002315
A002426
A002801
A003645
A005799
A007052
A007070
A007680
A007696
A010844
A010845
A014445
A014448
A032031
A047053
A052562
A055209
A056545
A056546
A059231
A059304
A075271
A075272
A082032
A084770
A084771
A097814
A097815 and
A097816
.)

Received June 24 2005;
revised version received November 1 2005.
Published in *Journal of Integer Sequences* November 15 2005.

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