We study the Fourier expansion of a function in orthogonal polynomial series with respect to the weight functions x α1−1/2 1 · · · xαd−1/2 d (1 − |x|1)αd+1−1/2 on the standard simplex Σd in Rd. It is proved that such an expansion is uniformly (C, δ) summable on the simplex for any continuous function if and only if δ > |α|1 + (d − 1)/2. Moreover, it is shown that (C, |α|1 + (d + 1)/2) means def...

Let x1 and xk be the least and the largest zeros of the Laguerre or Jacobi polynomial of degree k. We shall establish sharp inequalities of the form x1 < A, xk > B, which are uniform in all the parameters involved. Together with inequalities in the opposite direction, recently obtained by the author, this locates the extreme zeros of classical orthogonal polynomials with the relative precision,...

We express a weighted generalization of the Delannoy numbers in terms of shifted Jacobi polynomials. A specialization of our formulas extends a relation between the central Delannoy numbers and Legendre polynomials, observed over 50 years ago [12, 17, 19], to all Delannoy numbers and certain Jacobi polynomials. Another specialization provides a weighted lattice path enumeration model for shifte...

We prove a central limit theorem for the real part of logarithm characteristic polynomial random Jacobi matrices. Our results cover G $$\beta $$ E models >0$$ .

This paper centers on the derivation of a Rodrigues-type formula for Gegenbauer matrix polynomial. A connection between Gegenbauer and Jacobi matrix polynomials is given.

We survey some recent developments in the theory of orthogonal polynomials defined by differential equations. The key finding is that there exist orthogonal polynomials defined by 2nd order differential equations that fall outside the classical families of Jacobi, Laguerre, and Hermite polynomials. Unlike the classical families, these new examples, called exceptional orthogonal polynomials, fea...

Here we give a simple proof of a new representation for orthogonal polynomials over triangular domains which overcomes the need to make symmetry destroying choices to obtain an orthogonal basis for polynomials of fixed degree by employing redundancy. A formula valid for simplices with Jacobi weights is given, and we exhibit its symmetries by using the Bernstein–Bézier form. From it we obtain th...