Projekt ohne Drittmittelfinanzierung

Elimination in operator algebras

Details zum Projekt

Projektlaufzeit: 06/2007–12/2014

Zusammenfassung

Let F be a function in several discrete and continuous variables. F is called holonomic,if it satisfies a mixed homogeneous linear (partial) difference-differentialequation system with polynomial coefficients in all variables. The holonomic systemtogether with suitably many initial values characterize F uniquely.If we interpret the (partial) differentiations and shifts that occur as operators,and the representing holonomic equations as operator equations, then these form apolynomial system in a noncommutative operator algebra. These polynomials generatein the considered operator algebra a left ideal, by means of which a holonomicsystem may be regarded as an algebraic object.Furthermore, This algebraic approach to holonomic systems permitted to prove,among other, that the class of holonmic functions is closed under addition, multiplication,and indefinite integration and summation with respect to a variable.Performing one of the previous operations leads to a new holonomic system, whichis obtained by a transformation of the given ones. This transformation represents anelimination problem in a considered noncommutative operator algebra. This eliminationprocess may be used to verify existing and find new special function identitiesalgorithmically.In this dissertation, algorithms will be developed and implemented in the computeralgebra system Singular to solve the elimination problem in operator algebras.This implementation will be used to find new special function identities.