Dual Christoffel transformations
  Crum's theorem and its modification a la Krein-Adler are formulated for the discrete quantum mechanics with real shifts, whose eigenfunctions consist of orthogonal polynomials of a discrete variable. The modification produces the associated polynomials with a finite number of degrees deleted. This in turn provides the well known Christoffel transformation for the dual orthogonal polynomials with the corresponding positions deleted. 
Satoru Odake and Ryu Sasaki
math-ph hep-th math.CA math.MP nlin.SI quant-ph