A Study of Integer Sequences, Riordan Arrays, Pascal-like Arrays and Hankel Transforms

We study integer sequences and transforms that operate on them. Many of these transforms are defined by triangular arrays of integers, with a particular focus on Riordan arrays and Pascal-like arrays. In order to explore the structure of these transforms, use is made of methods coming from the theory of continued fractions, hypergeometric functions, orthogonal polynomials and most importantly from the Riordan groups of matrices. We apply the Riordan array concept to the study of sequences related to graphs and codes. In particular, we study sequences derived from the cyclic groups that provide an infinite family of colourings of Pascal’s triangle. We also relate a particular family of Riordan arrays to the weight distribution of MDS error-correcting codes. The Krawtchouk polynomials are shown to give rise to many different families of Riordan arrays. We define and investigate Catalannumber-based transformations of integer sequences, as well as transformations based on Laguerre and related polynomials. We develop two new constructions of families of Pascallike number triangles, based respectively on the ordinary Riordan group and the exponential Riordan group, and we study the properties of sequences arising from these constructions, most notably the central coefficients and the generalized Catalan numbers associated to the triangles. New exponential-factorial constructions are developed to further extend this theory. The study of orthogonal polynomials such as those of Chebyshev, Hermite, Laguerre and Charlier are placed in the context of Riordan arrays, and new results are found. We also extend results on the Stirling numbers of the first and second kind, using exponential Riordan arrays. We study the integer Hankel transform of many families of integer sequences, exploring links to related orthogonal polynomials and their coefficient arrays. Two particular cases of power series inversion are studied extensively, leading to results concerning the Narayana triangles.