Uvarov's formula for the orthogonal polynomials with respect to
dν(x) is a corollary of our theorem. Our result generalises
a Hankel determinant formula for the case where the rational
function is a polynomial that existed somehow hidden in the folklore
of the theory of orthogonal polynomials but has been stated explicitly only
relatively recently (see [arXiv:2101.04225]).
Our theorem can be interpreted in a two-fold way: analytically
or in the sense of formal series.
We apply our theorem to derive several curious Hankel determinant
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