Comparing two matrices of generalized moments defined by continued fraction expansions

We study two matrices $N$ and $M$ defined by the parameters of equivalent $S$- and $J$-continued fraction expansions, and compare them by examining the product $N^{-1}M$. Using examples based on the Catalan numbers, the little Schr\"oder numbers and powers of $q$, we indicate that this matrix product is an object worthy of study. In the case of the little Schr\"oder numbers, we find that the matrix $N$ has an interleaved structure based on two Riordan arrays.