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Transformations of difference equations. I. (English) Zbl 1198.39001

Summary: We consider a general weighted second-order difference equation. Two transformations are studied which transform the given equation into another weighted second order difference equation of the same type, these are based on the Crum transformation. We also show how Dirichlet and non-Dirichlet boundary conditions transform as well as how the spectra and norming constants are affected.

MSC:

39A05 General theory of difference equations

Software:

OPQ
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References:

[1] Binding, PA; Browne, PJ; Watson, BA, Spectral isomorphisms between generalized Sturm-Liouville problems, 135-152, (2001), Basel, Switzerland · Zbl 1038.34027
[2] Browne, PJ; Nillsen, RV, On difference operators and their factorization, Canadian Journal of Mathematics, 35, 873-897, (1983) · Zbl 0505.39003
[3] Binding, PA; Browne, PJ; Watson, BA, Sturm-Liouville problems with reducible boundary conditions, Proceedings of the Edinburgh Mathematical Society, 49, 593-608, (2006) · Zbl 1126.34021
[4] Binding, PA; Browne, PJ; Watson, BA, Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter. II, Journal of Computational and Applied Mathematics, 148, 147-168, (2002) · Zbl 1019.34028
[5] Binding, PA; Browne, PJ; Watson, BA, Transformations between Sturm-Liouville problems with eigenvalue dependent and independent boundary conditions, The Bulletin of the London Mathematical Society, 33, 749-757, (2001) · Zbl 1030.34027
[6] Teschl G: Jacobi Operators and Completely Integrable Nonlinear Lattices, Mathematical Surveys and Monographs. Volume 72. American Mathematical Society, Providence, RI, USA; 2000:xvii+351. · Zbl 1056.39029
[7] Miller KS: Linear Difference Equations. W. A. Benjamin; 1968:x+105.
[8] Miller KS: An Introduction to the Calculus of Finite Differences and Difference Equations. Dover, New York, NY, USA; 1966:viii+167.
[9] Atkinson FV: Discrete and Continuous Boundary Value Problems. Academic Press, New York, NY, USA; 1964:xiv+570.
[10] Gautschi W: Orthogonal Polynomials: Computation and Approximation, Numerical Mathematics and Scientific Computation. Oxford University Press, New York, NY, USA; 2004:x+301. · Zbl 1130.42300
[11] Shi, Y; Chen, S, Spectral theory of second-order vector difference equations, Journal of Mathematical Analysis and Applications, 239, 195-212, (1999) · Zbl 0934.39002
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