$a$-potent Schwarz matrices and Bessel-like Jacobi polynomials

We consider the problem of the reconstruction of a Schwarz matrix from exactly one given eigenvalue. This inverse eigenvalue problem leads to the Jacobi orthogonal polynomials~$\{P_k^{(-n,n)}{\}}_{k=0}^{n-1}$ that can be treated as a discrete finite analogue of Bessel polynomials.