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Second order $q$-difference equations solvable by factorization method. (English) Zbl 1119.39017
The factorization method to solve ordinary differential equations due to Darboux has many applications to orthogonal polynomials and quantum mechanics. The authors propose an extension to $q$-difference equations. It is a $q$-analogue, since, in the continuous limit $q \to 1$, it boils down to the classical factorization method (at least in the usual intuitive description of “continuous limit”). The paper contains the proof in a particular important case (indeed, a generalisation of a famous study by {\it L. Infeld} and {\it T. E. Hull}, Rev. Mod. Phys. 23, 21--68 (1951; Zbl 0043.38602) that the $q$-Hahn orthogonal polynomials are among the solutions. They also consider other interesting examples.

MSC:
39A13Difference equations, scaling ($q$-differences)
33D45Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
39A12Discrete version of topics in analysis
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References:
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