Regular systems of linear functional equations and applications

The algorithmic classification of singularities of linear differential systems via the computation of Moser- and super-irreducible forms as introduced in [21] and [16] respectively has been widely studied in Computer Algebra ([8, 12, 22, 6, 10]). Algorithms have subsequently been given for other forms of systems such as linear difference systems [4, 3] and the perturbed algebraic eigenvalue problem [18]. In this paper, we extend these concepts to the general class of systems of linear functional equations. We derive a definition of regularity for these type of equations, and an algorithm for recognizing regular systems. When specialised to q-difference systems, our results lead to new algorithms for computing polynomial solutions and regular formal solutions.

[1]  Moulay A. Barkatou.  FACTORING SYSTEMS OF LINEAR FUNCTIONAL EQUATIONS USING EIGENRINGS , 2007 .

[2]  Moulay A. Barkatou,et al.  An Algorithm Computing the Regular Formal Solutions of a System of Linear Differential Equations , 1999, J. Symb. Comput..

[3]  J. Moser,et al.  The order of a singularity in Fuchs' theory , 1959 .

[4]  Moulay A. Barkatou,et al.  An algorithm to compute the exponential part of a formal fundamental matrix solution of a linear differential system , 2009, Applicable Algebra in Engineering, Communication and Computing.

[5]  Moulay A. Barkatou,et al.  On the Moser- and super-reduction algorithms of systems of linear differential equations and their complexity , 2009, J. Symb. Comput..

[6]  Moulay A. Barkatou,et al.  A rational version of Moser's algorithm , 1995, ISSAC '95.

[7]  Manuel Bronstein,et al.  An Introduction to Pseudo-Linear Algebra , 1996, Theor. Comput. Sci..

[8]  Eckhard Pflügel,et al.  Effective Formal Reduction of Linear Differential Systems , 2000, Applicable Algebra in Engineering, Communication and Computing.

[9]  Volker Dietrich.  Zur Reduktion von linearen Differentialgleichungssystemen , 1978 .

[10]  Claude-Pierre Jeannerod,et al.  A reduction algorithm for matrices depending on a parameter , 1999, ISSAC '99.

[11]  N. Jacobson,et al.  Pseudo-Linear Transformations , 1937 .

[12]  M. Barkatou.  On super-irreducible forms of linear differential systems with rational function coefficients , 2004 .

[13]  A. Hilali,et al.  Formes super-irréductibles des systèmes différentiels linéaires , 1986 .

[14]  Pamela B. Lawhead,et al.  Super-irreducible form of linear differential systems , 1986 .

[15]  Moulay A. Barkatou,et al.  On Rational Solutions of Systems of Linear Differential Equations , 1999, J. Symb. Comput..

[16]  Moulay A. Barkatou,et al.  Computing super-irreducible forms of systems of linear differential equations via moser-reduction: a new approach , 2007, ISSAC '07.

[17]  Sergei A. Abramov.  Rational solutions of linear difference and q-difference equations with polynomial coefficients , 1995, ISSAC '95.

[18]  R. Tennant.  Algebra , 1941, Nature.

[19]  Moulay A. Barkatou,et al.  Rational solutions of first order linear difference systems , 1998, ISSAC '98.

[20]  Moulay A. Barkatou,et al.  On the reduction of linear systems of difference equations , 1989, ISSAC '89.

[21]  My Abdelfattah Barkatou.  Contribution à l'étude des équations différentielles et aux différences dans le champ complexe. (Contributions to the study of linear differential equations and difference equations in the complex domain) , 1989 .