Special formal series solutions of linear operator equations

Special formal series solutions of linear operator equations

The transformation which assigns to a linear operator L the recurrence satis ed by coe cient se quences of the polynomial series in its kernel is shown to be an isomorphism of the corresponding operator algebras We use this fact to help factoring q di erence and recurrence operators and to nd nice power series solutions of linear di erential equations In particular we characterize generalized h...

Hypergeometric series solutions of linear operator equations

Formal solutions of linear differential equations

A new approach to formal solutions of (first order systems of) linear differential equations stressing as much as possible the similarity with systems of linear equations. This approach leads to a fine structure of formal solutions, comparable to [1] but from a different point of view. A quick introduction to characteristic classes (cf. [3]) will be given. 1 Linear algebra; recalling Jordan dec...

Index theory for linear self-adjoint operator equations and nontrivial solutions for asymptotically linear operator equations

We will first establish an index theory for linear self-adjoint operator equations. And then with the help of this index theory we will discuss existence and multiplicity of solutions for asymptotically linear operator equations by making use of the dual variational methods and Morse theory. Finally, some interesting examples concerning second order Hamiltonian systems, first order Hamiltonian ...

Formal series solutions of singular systems of linear differential equations and singular matrix pencils

A system of n linear differential equations xy'{x) =x~8B{x)y(x) is considered, in which s is a positive integer and B(x) is a formal power series. In a preceding paper [6] the author has developed a method for evaluating a formal fundamental matrix solution to the system. This method leads to the algebraic treatment of certain singular matrix pencils. In the present paper the author provides so...