## Ana Foulquié

## The Toda Lattices through the theory of orthogonal polynomials

We show how the theory of orthogonal polynomials is used to solve the Toda Lattices. We extend this study to Discrete Toda Lattices. The correspondence between dynamics of the Delta Toda and Delta Volterra equations for the coefficients of the Jacobi operator and its resolvent equation is established. The main ingredient are orthogonal polynomials which satisfy an Appell condition with respect to the Delta diference operator and a Lax Theorem for the point spectrum of the Jacobian operator associated with these equations.

Nelson Vieira

## Fischer Decomposition in Fractional Clifford Analysis

What is nowadays called (classic) Clifford analysis consists in the establishment of a function theory for functions belonging to the kernel of the Dirac operator. While such functions can very well describe problems of a particle with internalSU(2)-symmetries, higher order symmetries are beyond this theory. Although many modifications (such as Yang-Mills theory) were suggested over the years they could not address the principal problem, the need of a n-fold factorization of the d'Alembert operator. While Dirac operators with fractional derivatives could achieve this they are more difficult to work with. The main reason is that they do not allow a construction of a Howe dual pair. Hereby, the principal problem is not the invariance under a fractional spin group, but the construction of a Super-Lie-algebra osp(1|2) for general fractional Dirac operators. This requires the application of a new approach and new methods, in particular the establishment of fractional Sommen-Weyl relations. In this talk we will present the building blocks of a function theory for fractional Dirac operators based on Gelfond-Leontiev operators of generalized differentiation. These operators were heavily studied in the 1970's and 1980's by a group of mathematicians from Yerevan (Armenia) and allow particular realizations in form of the classical Caputo and Riemann/Liouville fractional derivatives.

This is a joint work with P. Cerejeiras, A. Fonseca, and U. Kähler.

This is a joint work with P. Cerejeiras, A. Fonseca, and U. Kähler.