Leonard pairs having zero-diagonal TD-TD form

Fix an algebraically closed field FF and an integer n≥1n≥1. Let Matn(F)Matn(F) denote the FF-algebra consisting of the n×nn×n matrices that have all entries in FF. We consider a pair of diagonalizable matrices in Matn(F)Matn(F), each acting in an irreducible tridiagonal fashion on an eigenbasis for the other one. Such a pair is called a Leonard pair in Matn(F)Matn(F). In the present paper, we find all Leonard pairs A  , A⁎A⁎ in Matn(F)Matn(F) such that each of A   and A⁎A⁎ is irreducible tridiagonal with all diagonal entries 0. This solves a problem given by Paul Terwilliger.