  In [J. Phys. A: Math. Theor. 45 (2012)], while looking for spin chains that
admit perfect state transfer, Vinet and Zhedanov found an apparently new
sequence of orthogonal polynomials, that they called para-Krawtchouk
polynomials, defined on a bilinear lattice. In this note we present necessary
and sufficient conditions for the regularity of solutions of the corresponding
functional equation. Moreover, the functional Rodrigues formula and a closed
formula for the recurrence coefficients are presented. As a consequence, we
characterize all solutions of the functional equation, including as very
particular cases the Meixner, Charlier, Krawtchouk, Hahn, and para-Krawtchouk
polynomials.
