The Euler-Seidel Matrix, Hankel Matrices and Moment Sequences

The purpose of this note is to investigate the close relationship that exists between the Euler-Seidel matrix [3, 4, 5, 6, 10] of an integer sequence, and the Hankel matrix [9] of that sequence. We do so in the context of sequences that have integral moment representations, though many of the results are valid in a more general context. While partly expository in nature, the note assumes a certain familiarity with generating functions, both ordinary and exponential, orthogonal polynomials [2, 8, 16] and Riordan arrays [12, 15] (again, both ordinary, where we use the notation (g, f), and exponential, where we use the notation [g, f ]). Many interesting examples of sequences and Riordan arrays can be found in Neil Sloane’s