A New Characterization of Catalan Numbers Related to Hankel Transforms and Fibonacci Numbers

Abstract

The Hankel determinant of order n ≥ 1 of a sequence A = {ao, a1, a2,...} of real numbers, denoted by hn, is the determinant given by h = det[ai+jlosij≤n-1. Many authors have studied Hankel determinants containing combinatorial numbers (See [2][4][7][9][10][15]). Of special in- terest are Hankel determinants consisting of the sequence of Catalan numbers defined by Cn (3) 2n n+1 (2n)! n!(n + 1)!' n≥ 0. Cvetković, Rajković, and Ivković proved that the Hankel deter- minant of the sequence of sums of two successive Catalan num- bers is the sequence of Fibonacci numbers with odd indices. Later on, T. Benjamin, Cameron, Quinn, and Yerger extended this result and proved that if we remove by one term this se- quence of sums then the Hankel transform is the sequence of Fibonacci numbers with even indices. It is naturel, then, to ask the following question: Is the Catalan number sequence the unique nonnegative integer sequence sat- isfying this property? The aim of this project is to answer this question. The first chapter contains notations and preliminary results on the theory of orthogonal polynomials that we will use in the sequel such as calculus on the forms, orthogonality of polynomials, regularity of forms, classical polynomials, etc. In the second chapter, we present some combinatoric interpre- tations of Catalan numbers. Then, we prove that the Catalan number sequence is the unique nonnegative integer sequence fulfilling the mentioned property.