Hypergeometric Equation and Differential-Difference Bispectrality

The bispectral problem was posed by Duistermaat and Grünbaum in 1986. Since then, many interesting links of this problem with nonlinear integrable PDEs, algebraic geometry, orthogonal polynomials and special functions have been found. Bispectral operators of rank one are related to the KP equation and have been completely classified by G. Wilson. For rank greater than 1 some large families related to Bessel functions are known, although the classification problem remains open.

If one generalises the bispectral problem by allowing difference operators in the spectral variable, then this has a clear parallel with the three-term recurrence relation in the theory of orthogonal polynomials. This differential-difference version of the bispectral problem has also been studied extensively, more recently in the context of the exceptional orthogonal polynomials. However, the associated special functions have not been treated in such a way, until now.

In our work we make a step in that direction by constructing a large family of bispectral operators related to the hypergeometric equation. In this thesis, we will fully explain our construction.