On the Hurwitz Transform of Sequences

Given a sequence an, we denote by hn the general term of the sequence with hn = |ai+j|0≤i,j≤n. The sequence hn is called the Hankel transform of an [19, 20, 21]. This sequence of Hankel determinants has attracted much attention of late amongst those working in the area of integer and polynomial sequences in particular [7, 18, 24, 32]. In this note we shall introduce the notion of a related Hurwitz transform, and we shall study some of its properties. As with the Hankel transform, this transform is based on classical results which have a rich literature. Part of this literature is captured in the review article by Holtz and Tyaglov [17], which forms a good background to this note. Our Hurwitz transform will give rise to a sequence of determinant values, which can be related to the Hankel transform. In the sequel, we shall be mainly concerned with integer sequences. Known integer sequences are often referred to by their OEIS number [26, 27]. For instance, the sequence of Catalan numbers Cn = 1 n+1 ( 2n n ) is A000108. Its generating function, defined by ∞ n=0 Cnx , is equal to c(x) = 1− √ 1−4x 2x . Its first elements are