Journal of Integer Sequences, Vol. 26 (2023), Article 23.2.7

On Sums, Derivatives, and Flips of Riordan Arrays

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

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

Eric Culver
Brigham Young University
Provo, UT 84602

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

Stoyan Dimitrov
Rutgers University
Hill Center
Piscataway, NJ 08854

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

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


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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(Concerned with sequences A000007 A000012 A000027 A000032 A000045 A000108 A000225 A000302 A001006 A001045 A001070 A001350 A001519 A001629 A001700 A001906 A002054 A004146 A005043 A005408 A007598 A007852 A008549 A026671 A026737 A027941 A029907 A033453 A040000 A045925 A052952 A067324 A086615 A129869 A144109 A152163.)

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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