Moment representations of exceptional X1 orthogonal polynomials

We generalize the representations of X1 exceptional orthogonal polynomials through determinants of matrices that have certain adjusted moments as entries. We start out directly from the Darboux transformation, allowing for a universal perspective, rather than one dependent upon the particular system (Jacobi or Type of Laguerre polynomials). We include a recursion formula for the adjusted moments and provide the initial adjusted moments for each system. Throughout we relate to the various examples of X1 exceptional orthogonal polynomials. We especially focus on and provide complete proofs for the Jacobi and the Type III Laguerre case, as they are less prevalent in literature. Lastly, we include a preliminary discussion explaining that the higher codimension setting becomes more involved. The number of possibilities and choices is exemplified, but only starts, with the lack of a canonical flag.