In mathematics, the Romanovski polynomials are one of three finite subsets of real orthogonal polynomials discovered by Vsevolod Romanovsky (Romanovski in French transcription) within the context of probability distribution functions in statistics. They form an orthogonal subset of a more general family of little-known Routh polynomials introduced by Edward John Routh in 1884. The term Romanovski polynomials was put forward by Raposo, with reference to the so-called 'pseudo-Jacobi polynomials in Lesky's classification scheme. It seems more consistent to refer to them as Romanovski–Routh polynomials, by analogy with the terms Romanovski–Bessel and Romanovski–Jacobi used by Lesky for two other sets of orthogonal polynomials.
In some contrast to the standard classical orthogonal polynomials, the polynomials under consideration differ, in so far as for arbitrary parameters only a finite number of them are orthogonal, as discussed in more detail below.