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A Family of Riordan Group Automorphisms
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Ângela Mestre and José Agapito

Centro de Análise Funcional, Estruturas Lineares e Aplicações

Grupo de Estruturas Algébricas, Lineares e Combinatórias

Departamento de Matemática

Faculdade de Cîencias, Universidade de Lisboa

1749-016 Lisboa

Portugal

**Abstract:**

In 2006, Bacher introduced a family of Riordan group automorphisms
parametrized by three complex numbers. Bacher's family is a subgroup
of the group of automorphisms of the Riordan group and so is the
subfamily parametrized only by two real numbers. Here, we study some
of the algebraic properties of this subfamily and use the elements to
point out isomorphisms between Riordan subgroups. In this context,
we prove that the set of Riordan arrays whose row sum sequence is a
sequence of partial sums, forms a Riordan subgroup. Moreover, we show
that the well-known recursive matrices may be constructed from sequences
of images of a Riordan array under automorphisms. Our construction
also discloses a correspondence between the recursive matrices and a
pair of well-defined Riordan arrays.

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(Concerned with sequences
A000012
A000027
A000045
A000096
A000108
A001478
A005586
A007318
A014137
A014138
A090826
A110555
A115140.)

Received July 10 2019; revised versions received December 23 2019; December 24 2019.
Published in *Journal of Integer Sequences*,
December 26 2019.

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