Factorization Of Linear Differential Operators Research Articles

A sage package for the symbolic-numeric factorization of linear differential operators

We present a SageMath implementation of the symbolic-numeric algorithm introduced by van der Hoeven in 2007 for factoring linear differential operators whose coefficients are rational functions.

Explicit degree bounds for right factors of linear differential operators

If a linear differential operator with rational function coefficients is reducible, its factors may have coefficients with numerators and denominatorsof very high degree. When the base field is $\mathbb C$, we give a completely explicit bound for the degrees of the monic right factors in terms of the degree and the order of the original operator, as well as the largest modulus of the local exponents at all its singularities. As a consequence, if a differential operator $L$ has rational function coefficients over a number field, we get degree bounds for its monic right factors in terms of the degree, the order and the height of $L$, and of the degree of the number field.

Around the numeric–symbolic computation of differential Galois groups

Let L ∈ K ( z ) [ ∂ ] be a linear differential operator, where K is an effective algebraically closed subfield of C . It can be shown that the differential Galois group of L is generated (as a closed algebraic group) by a finite number of monodromy matrices, Stokes matrices and matrices in local exponential groups. Moreover, there exist fast algorithms for the approximation of the entries of these matrices. In this paper, we present a numeric–symbolic algorithm for the computation of the closed algebraic subgroup generated by a finite number of invertible matrices. Using the above results, this yields an algorithm for the computation of differential Galois groups, when computing with a sufficient precision. Even though there is no straightforward way to find a “sufficient precision” for guaranteeing the correctness of the end result, it is often possible to check a posteriori whether the end result is correct. In particular, we present a non-heuristic algorithm for the factorization of linear differential operators.

DARBOUX TRANSFORMS AND ORTHOGONAL POLYNOMIALS

We give a new interpretation of Darboux transforms in the context of orthogonal polynomials and find conditions in or-der for any Darboux transform to yield a new set of orthogonal polynomials. We also discuss connections between Darboux trans-forms and factorization of linear differential operators which have orthogonal polynomial eigenfunctions.

Some algebraic, geometric, and system-theoretic properties of the Special Functions of mathematical physics

It is known that many of the Special Functions of mathematical physics appear as matrix elements of Lie group representations. This paper is concerned with a beginning attack on the converse problem, i.e., finding conditions that a given function be a matrix element. The methods used are based on a combination of ideas from system theory, functional analysis, Lie theory, differential algebra, and linear ordinary differential equation theory. A key idea is to attach a symbol as an element of a commutative algebra. In favorable cases, this symbol defines a Riemann surface, and a meromorphic differential form on that surface. The topological and analytical invariants attached to this form play a key role in system theory. The Lie algebras of the groups appear as linear differential operators on this Riemann surface. Finally, it is shown how the Picard-Vessiot-Infeld-Hull theory of factorization of linear differential operators leads to realization of many Special Functions as matrix representations of group representations.