A Fast Approach to Creative Telescoping

In this note we reinvestigate the task of computing creative telescoping relations in differential-difference operator algebras. Our approach is based on an ansatz that explicitly includes the denominators of the delta parts. We contribute several ideas of how to make an implementation of this approach reasonably fast and provide such an implementation. A selection of examples shows that it can be superior to existing methods by a large factor.Comment: 9 pages, 1 table, final version as it appeared in the journa

Denominator Bounds and Polynomial Solutions for Systems of q-Recurrences over K(t) for Constant K

We consider systems A_\ell(t) y(q^\ell t) + ... + A_0(t) y(t) = b(t) of higher order q-recurrence equations with rational coefficients. We extend a method for finding a bound on the maximal power of t in the denominator of arbitrary rational solutions y(t) as well as a method for bounding the degree of polynomial solutions from the scalar case to the systems case. The approach is direct and does not rely on uncoupling or reduction to a first order system. Unlike in the scalar case this usually requires an initial transformation of the system.Comment: 8 page

Polynomial solutions of algebraic difference equations and homogeneous symmetric polynomials

This article addresses the problem of computing an upper bound of the degree d of a polynomial solution P(x) of an algebraic differ- ence equation of the form Gx)(P(x −τ1), . . . , P(x −τs) + G0(x) = 0 when such P(x) with the coefficients in a field K of character- istic zero exists and where G is a non-linear s-variable polynomial with coefficients in K[x] and G0 is a polynomial with coefficients in K. It will be shown that if G is a quadratic polynomial with constant coefficients then one can construct a countable family of polynomi- als fl(u0) such that if there exists a (minimal) index l0 with fl0(u0) being a non-zero polynomial, then the degree d is one of its roots or d ≤ l0, or d < deg(G0). Moreover, the existence of such l0 will be proven for K being the field of real numbers. These results are based on the properties of the modules generated by special fami- lies of homogeneous symmetric polynomials. A sufficient condition for the existence of a similar bound of the degree of a polynomial solution for an algebraic difference equation with G of arbitrary total degree and with variable coefficients will be proven as well

Using morphism computations for factoring and decomposing linear functional systems

Within a constructive homological algebra approach, we study the factorization and decomposition problems for general linear functional systems (determined, under-determined, over-determined). Using the concept of Ore algebras of functional operators (e.g., ordinary/partial differential operators, shift operators, time-delay operators), we first concentrate on the computation of morphisms from a finitely presented left module $M$ over an Ore algebra to another one $M'$, where $M$ (resp., $M'$) is a module intrinsically associated with the linear functional system $R \, y=0$ (resp., $R' \, z=0$). These morphisms define applications sending solutions of the system $R' \, z=0$ to the ones of $R \, y=0$. We explicitly characterize the kernel, image, cokernel and coimage of a general morphism. We then show that the existence of a non-injective endomorphism of the module $M$ is equivalent to the existence of a non-trivial factorization $R=R_2\,R_1$ of the system matrix $R$. The corresponding system can then be integrated in cascade. Under certain conditions, we also show that the system $R \, y=0$ is equivalent to a system $R'\, z=0$, where $R'$ is a block-triangular matrix of the same size as $R$. We show that the existence of projectors of the ring of endomorphisms of the module $M$ allows us to reduce the integration of the system $R\,y=0$ to the integration of two independent systems $R_1 \, y_1=0$ and $R_2 \, y_2=0$. Furthermore, we prove that, under certain conditions, idempotents provide decompositions of the system $R\,y=0$, i.e., they allow us to compute an equivalent system $R'\, z=0$, where $R'$ is a block-diagonal matrix of the same size as $R$. Many applications of these results in mathematical physics and control theory are given. Finally, the different algorithms of the paper are implemented in a Maple package Morphisms based on the library OreModules