A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths

We study integer sequences using methods from the theory of continued fractions, or- thogonal polynomials and most importantly from the Riordan groups of matrices, the ordinary Riordan group and the exponential Riordan group. Firstly, we will intro- duce the Riordan group and their links through orthogonal polynomials to the Stieltjes matrix. Through the context of Riordan arrays we study the classical orthogonal polynomials, the Chebyshev polynomials. We use Riordan arrays to calculate determi- nants of Hankel and Toeplitz-plus-Hankel matrices, extending known results relating to the Chebyshev polynomials of the third kind to the other members of the family of Chebyshev polynomials. We then define the form of the Stieltjes matrices of important subgroups of the Riordan group. In the following few chapters, we develop the well es- tablished links between orthogonal polynomials, continued fractions and Motzkin paths through the medium of the Riordan group. Inspired by these links, we extend results to the Lukasiewicz paths, and establish relationships between Motzkin, Schr¨oder and certain Lukasiewicz paths. We concern ourselves with the Binomial transform of inte- ger sequences that arise from the study of Lukasiewicz and Motzkin paths and we also study the effects of this transform on lattice paths. In the latter chapters, we apply the Riordan array concept to the study of sequences related to MIMO communica- tions through integer arrays relating to the Narayana numbers. In the final chapter, we use the exponential Riordan group to study the historical Euler-Seidel matrix. We calculate the Hankel transform of many families of sequences encountered throughout.