On recursive sequences with periodic coefficients and their associated Hankel matrices

In this paper, we study the positivity of the infinite Hankel matrix associated with a recursive sequence defined by periodic coefficients of period p≥2. As a preliminary step, we consider a family of constant-coefficient subsequences derived from the original sequence, and we establish a positivity criterion for their corresponding infinite Hankel matrices using analytic formulas. We then address the case of the full sequence with periodic coefficients by transforming its infinite Hankel matrix into an equivalent block matrix and applying properties of the Kronecker product. In particular, we thoroughly investigate the special case r=p=2, in which the connection between the positivity of the infinite Hankel matrix of the original sequence and those of the associated subsequences is fully characterized.