On Sums, Derivatives, and Flips of Riordan Arrays

On Sums, Derivatives, and Flips of Riordan Arrays

Caroline Bang
Iowa State University
411 Morrill Rd
Ames, IA 50011
USA

Matias von Bell
Institute of Geometry
Graz University of Technology
Kopernikusgasse 24
Graz, A-8010
Austria

Eric Culver
Brigham Young University
Provo, UT 84602
USA

Jessica Dickson
Washington State University
PO Box 643113
Pullman, WA 99164
USA

Stoyan Dimitrov
Rutgers University
Hill Center
Piscataway, NJ 08854
USA

Rachel Perrier
Washington State University
PO Box 643113
Pullman, WA 99164
USA

Sheila Sundaram
Pierrepont School
One Sylvan Road North
Westport, CT 06880
USA

Abstract:

We study three operations on Riordan arrays. First, we investigate when the sum of Riordan arrays yields another Riordan array. We characterize the A- and Z-sequences of these sums of Riordan arrays, and also identify an analog for A-sequences when the sum of Riordan arrays does not yield a Riordan array. In addition, we define the new operations `Der' and `Flip' on Riordan arrays. We fully characterize the Riordan arrays resulting from these operations applied to the Appell and Lagrange subgroups of the Riordan group. Finally, we study the application of these operations to various known Riordan arrays, generating many combinatorial identities in the process.

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Received November 9 2022; revised versions received February 15 2023; February 27 2023. Published in Journal of Integer Sequences, February 28 2023.

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