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Let $p_n(x)$, $n=0,1,\dots$, be the orthogonal polynomials with respect to agiven density $d\mu(x)$. Furthermore, let $d\nu(x)$ be a density which arisesfrom $d\mu(x)$ by multiplication by a rational function in $x$. We prove aformula that expresses the Hankel determinants of moments of $d\nu(x)$ in termsof a determinant involving the orthogonal polynomials $p_n(x)$ and associatedfunctions $q_n(x)=\int p_n(u) \,d\mu(u)/(x-u)$. Uvarov's formula for theorthogonal polynomials with respect to $d\nu(x)$ is a corollary of our theorem.Our result generalises a Hankel determinant formula for the case where therational function is a polynomial that existed somehow hidden in the folkloreof the theory of orthogonal polynomials but has been stated explicitly onlyrelatively recently (see [arXiv:2101.04225]). Our theorem can be interpreted ina two-fold way: analytically or in the sense of formal series. We apply ourtheorem to derive several curious Hankel determinant evaluations.

A determinant identity for moments of orthogonal polynomials that  implies Uvarov's formula for the orthogonal polynomials of rationally related  densities