Discrete bispectral Darboux transformations from Jacobi operators

Details Discrete bispectral Darboux transformations from Jacobi operators Discrete bispectral Darboux transformations from Jacobi operators

2000-12-19

arxiv.org/abs/math/0012191v1

Mathematics

Classical Analysis and ODEs

30 pages, AMS latex

Scientific paper

We construct families of bispectral difference operators of the form a(n)T + b(n) + c(n) T^{-1} where T is the shift operator. They are obtained as discrete Darboux transformations from appropriate extensions of Jacobi operators. We conjecture that along with operators previously constructed by Grunbaum, Haine, Horozov, and Iliev they exhaust all bispectral regular (i.e. a(n)c(n) \neq 0, for all integer n) operators of the form above.

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