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Concept
Romanovski polynomials
Summary
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.
The Romanovski polynomials solve the following version of the hypergeometric differential equation
Curiously, they have been omitted from the standard textbooks on special functions in mathematical physics and in mathematics and have only a relatively scarce presence elsewhere in the mathematical literature.
The weight functions are
they solve Pearson's differential equation
that assures the self-adjointness of the differential operator of the hypergeometric
ordinary differential equation.
For α = 0 and β < 0, the weight function of the Romanovski polynomials takes the shape of the Cauchy distribution, whence the associated polynomials are also denoted as Cauchy polynomials in their applications in random matrix theory.
The Rodrigues formula specifies the polynomial R(x) as
where Nn is a normalization constant. This constant is related to the coefficient cn of the term of degree n in the polynomial R(x) by the expression
which holds for n ≥ 1.
As shown by Askey this finite sequence of real orthogonal polynomials can be expressed in terms of Jacobi polynomials of imaginary argument and thereby is frequently referred to as complexified Jacobi polynomials.
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