Vyacheslav Spiridonov et al 1997 J. Phys. A: Math. Gen. 30 7621 doi:10.1088/0305-4470/30/21/030
Spectral transformations, self-similar reductions and orthogonal polynomials
Vyacheslav Spiridonov
, Luc Vinet
and Alexei Zhedanov§
We study spectral transformations in the theory of orthogonal polynomials which are similar to Darboux transformations for the Schrödinger equation. Linear transformations of the Stieltjes function with coefficients that are rational in the argument are constructed as iterations of the Christoffel and Geronimus transformations. We describe a characteristic property of semi-classical orthogonal polynomials (SCOP) on the uniform and the exponential lattice; namely, that all these polynomials can be obtained through simple quasi-periodic and q-periodic (self-similar) closures of the chain of linear spectral transformations. In the self-similar setting, a characterization of the Laguerre - Hahn polynomials on linear and q-linear lattices is obtained by considering rational transformations of the Stieltjes function generated by transitions to the associated polynomials.
- PACS
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03.65.Ge Solutions of wave equations: bound states
- MSC
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15A04 Linear transformations, semilinear transformations
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
- Subjects
- Dates
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Issue 21 (7 November 1997)
Received 7 April 1997
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Spectral transformations, self-similar reductions and orthogonal polynomials
Vyacheslav Spiridonov et al 1997 J. Phys. A: Math. Gen. 30 7621
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