Construction Of Pseudo-Involutions In The Riordan Group
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Date
2017
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Department
Mathematics
Program
Doctor of Philosophy
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This item is made available by Morgan State University for personal, educational, and research purposes in accordance with Title 17 of the U.S. Copyright Law. Other uses may require permission from the copyright owner.
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Abstract
We investigate pseudo-involutions in the Riordan group. The focus of this dissertation is to analyze the relationship between the generating functions g(z) and f(z) of a pseudo-involution in the Riordan group. We use the relationship to construct pseudo-involutions in the Riordan group. In doing this, we answer outstanding questions in the literature. We introduce the concept of a bi-invertible generating g(z) and we prove constructively that given a bi-invertible function g(z), there is a unique generating function f(z) that makes the Riordan matrix denoted by the pair of generating functions (g(z),f(z)) a pseudo-involution. Also, given an f(z) whose additive inverse has compositional order 2, we find the set of generating functions g(z) such that (g(z),f(z)) is a pseudo-involution. We then prove that this set of g(z)'s is an infinite group that exhibits a number of cyclic subgroups of the group. We also characterize certain subgroups of the Riordan group. As a consequence of the pseudo-involutions, we introduce some new subgroups of the Riordan group and study some of their related properties.