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Zbl 1119.39017
Dobrogowska, Alina; Odzijewicz, Anatol
Second order $q$-difference equations solvable by factorization method.
(English)
[J] J. Comput. Appl. Math. 193, No. 1, 319-346 (2006). ISSN 0377-0427

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.
[Jacques Sauloy (Toulouse)]
MSC 2000:
*39A13 Difference equations, scaling ($q$-differences)
33D45 Basic hypergeometric functions and integrals in several variables
39A12 Discrete version of topics in analysis

Keywords: discretization; $q$-Hahn orthogonal polynomials

Citations: Zbl 0043.38602

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