Discrete Bispectral Darboux Transformations from Jacobi Operators

We construct families of bispectral difference operators of the form a(n)T + b(n) + c(n)T 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 Grünbaum, Haine, Horozov, and Iliev they exhaust all bispectral regular (i.e. a(n) 6= 0, c(n) 6= 0,∀n ∈ Z) operators of the form above.