@article{oai:soar-ir.repo.nii.ac.jp:00011741,
 author = {Odake,  Satoru and Sasaki,  Ryu},
 issue = {1},
 journal = {PROGRESS OF THEORETICAL PHYSICS},
 month = {Jul},
 note = {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., Article, PROGRESS OF THEORETICAL PHYSICS. 126(1):1-34 (2011)},
 pages = {1--34},
 title = {Dual Christoffel Transformations},
 volume = {126},
 year = {2011}
}