  Barycentric interpolation is arguably the method of choice for numerical
polynomial interpolation. The polynomial interpolant is expressed in terms of
function values using the so-called barycentric weights, which depend on the
interpolation points. Few explicit formulae for these barycentric weights are
known. In [H. Wang and S. Xiang, Math. Comp., 81 (2012), 861--877], the authors
have shown that the barycentric weights of the roots of Legendre polynomials
can be expressed explicitly in terms of the weights of the corresponding
Gaussian quadrature rule. This idea was subsequently implemented in the Chebfun
package [L. N. Trefethen and others, The Chebfun Development Team, 2011] and in
the process generalized by the Chebfun authors to the roots of Jacobi, Laguerre
and Hermite polynomials. In this paper, we explore the generality of the link
between barycentric weights and Gaussian quadrature and show that such
relationships are related to the existence of lowering operators for orthogonal
polynomials. We supply an exhaustive list of cases, in which all known formulae
are recovered and also some new formulae are derived, including the barycentric
weights for Gauss-Radau and Gauss-Lobatto points. Based on a fast ${\mathcal
O}(n)$ algorithm for the computation of Gaussian quadrature, due to Hale and
Townsend, this leads to an ${\mathcal O}(n)$ computational scheme for
barycentric weights.
