Hankel operators plus orthogonal polynomials yield combinatorial identities

Abstract

A Hankel operator $H=[h_{i+j}]$ can be factored as $H=M{M}^{\ast }$, 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.