Orthogonal polynomials, quantum walks and the Prouhet-Tarry-Escott problem

Abstract: This paper is motivated by the following problem. Define a quantum walk on a positively weighted path (linear chain). Can the weights be tuned so that perfect state transfer occurs between the first vertex and any other position? We do not fully answer this question - in fact, we show that a particular case of this problem is equivalent to a solution of a particular case of the well known Prouhet-Tarry-Escott problem, deeming our original task certainly harder than anticipated. In our journey, we prove new results about sequences of orthogonal polynomials satisfying three-term recurrences. In particular, we provide a full characterization of when two polynomials belong to such a sequence, which (as far as we were able to ascertain) was known only for when their degrees differ by one.