×

Removing apparent singularities of linear difference systems. (English) Zbl 1470.39039

Authors’ abstract: It is well known that for a first order system of linear difference equations with rational function coefficients, a solution that is holomorphic in some left half plane can be analytically continued to a meromorphic solution in the whole complex plane. The poles stem from the singularities of the rational function coefficients of the system. Just as for systems of differential equations, not all of these singularities necessarily lead to poles in a solution, as they might be what is called removable. In our work, we show how to detect and remove these singularities and further study the connection between poles of solutions, removable singularities and the extension of numerical sequences at these points.

MSC:

39A45 Difference equations in the complex domain
39A06 Linear difference equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abramov, S. A.; Barkatou, M. A.; van Hoeij, M., Apparent singularities of linear difference equations with polynomial coefficients, Appl. Algebra Eng. Commun. Comput., 17, 2, 117-133 (June 2006) · Zbl 1106.39002
[2] Abramov, S. A.; van Hoeij, M., Desingularization of linear difference operators with polynomial coefficients, (Proceedings of ISSAC 1999 (1999)), 269-275
[3] Barkatou, M. A., Contribution à l’étude des équations différentielles et aux différences dans le champ complexe (1989), INPG: INPG Grenoble, France, PhD thesis
[4] Barkatou, M. A., Factoring systems of linear functional equations using eigenrings, (Computer Algebra 2006, Latest Advances in Symbolic Algorithms. Proceedings of the Waterloo Workshop. Computer Algebra 2006, Latest Advances in Symbolic Algorithms. Proceedings of the Waterloo Workshop, Ontario, Canada (2006))
[5] Barkatou, M. A.; Chen, G., Computing the exponential part of a formal fundamental matrix solution of a linear difference system, J. Differ. Equ. Appl., 5, 3, 1-26 (1999)
[6] Barkatou, M. A.; Chen, G., Some formal invariants of linear difference systems, J. Reine Angew. Math., 533, 2, 1-23 (2001) · Zbl 0981.39001
[7] Barkatou, M. A.; Jaroschek, M., Desingularization of first order linear difference systems with rational function coefficients, (Proceedings of ISSAC 2018 (2018)), 39-46 · Zbl 1467.39002
[8] Barkatou, M. A.; Maddah, S. S., Removing apparent singularities of systems of linear differential equations with rational function coefficients, (Proceedings of ISSAC 2015 (2015), ACM: ACM New York, NY, USA), 53-60 · Zbl 1346.68268
[9] Birkhoff, G. D., Formal theory of irregular linear difference equations, Acta Math., 54, 205-246 (1930) · JFM 56.0402.01
[10] Bronstein, M.; Petkovšek, M., An introduction to pseudo-linear algebra, Theor. Comput. Sci., 157, 3-33 (1996) · Zbl 0868.34004
[11] Chen, S.; Jaroschek, M.; Kauers, M.; Singer, M., Desingularization explains order-degree curves for Ore operators, (Proceedings of ISSAC 2013 (2013)), 157-164 · Zbl 1360.68927
[12] Chen, S.; Kauers, M.; Li, Z.; Zhang, Y., Apparent singularities of d-finite systems, J. Symb. Comput., 95, 217-237 (2019) · Zbl 1427.13036
[13] Chen, S.; Kauers, M.; Singer, M. F., Desingularization of Ore polynomials, J. Symb. Comput., 74, 617-626 (2016) · Zbl 1348.68300
[14] The Sage Developers, SageMath, the Sage mathematics software system (version 8.4) (2018)
[15] Flajolet, P.; Sedgewick, R., Analytic Combinatorics (2009), Cambridge University Press: Cambridge University Press New York, NY, USA · Zbl 1165.05001
[16] Immink, G. K., On the relation between linear difference and differential equations with polynomial coefficients, Math. Nachr., 200, 59-76 (1999) · Zbl 0923.39001
[17] Jacobson, N., Pseudo-linear transformations, Ann. Math., 33, 2, 484-507 (1937) · JFM 63.0087.01
[18] Jaroschek, M., Improved polynomial remainder sequences for Ore polynomials, J. Symb. Comput., 58, 64-76 (2013) · Zbl 1333.16054
[19] Jaroschek, M., Removable Singularities of Ore Operators (November 2013), RISC, Johannes Kepler University Linz, PhD thesis
[20] Jaroschek, M., FOS - a Sage package for first-order differential and difference systems. Development build (2018)
[21] Kauers, M.; Jaroschek, M.; Johansson, F., Ore polynomials in Sage, (Gutierrez, Jaime; Schicho, Josef; Weimann, Martin, Computer Algebra and Polynomials. Computer Algebra and Polynomials, Lecture Notes in Computer Science, vol. 8942 (2014)), 105-125 · Zbl 1439.16049
[22] Kauers, M.; Paule, P., The Concrete Tetrahedron, Text and Monographs in Symbolic Computation (2011), Springer: Springer Wien · Zbl 1225.00001
[23] Koutschan, C.; Zhang, Y., Desingularization in the q-Weyl algebra, Adv. Appl. Math., 97, 80-101 (2018) · Zbl 1418.16032
[24] Praagman, C., Fundamental solutions for meromorphic linear difference equations in the complex plane, and related problems, J. Reine Angew. Math., 369, 100-109 (1986) · Zbl 0586.30024
[25] Ramis, J.-P., 1988. Etude des solutions méromorphes des équations aux différences linéaires algébriques. Manuscript.
[26] Tsai, H., Weyl closure of a linear differential operator, J. Symb. Comput., 29, 4-5, 747-775 (2000) · Zbl 1008.16026
[27] Zhang, Y., Contraction of Ore ideals with applications, (Proceedings ISSAC 2016 (2016), ACM: ACM New York, NY, USA), 413-420 · Zbl 1361.13011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.