Hankel operators plus orthogonal polynomials yield combinatorial identities

A Hankel operator \( H = [h_{i+j}] \) can be factored as \( H = MM^* \), where \( M \) maps a space of \( L^2 \) functions to the corresponding moment sequences. Furthermore, a necessary and sufficient condition for a sequence to be in the range of \( M \) can be expressed in terms of an expansion in orthogonal polynomials. Combining these two results yields a wealth of combinatorial identities that incorporate both the matrix entries \( h_{i+j} \) and the coefficients of the orthogonal polynomials.