{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T06:06:25Z","timestamp":1776837985523,"version":"3.51.2"},"reference-count":99,"publisher":"American Mathematical Society (AMS)","issue":"347","license":[{"start":{"date-parts":[[2024,10,23]],"date-time":"2024-10-23T00:00:00Z","timestamp":1729641600000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100001665","name":"Agence Nationale de la Recherche","doi-asserted-by":"publisher","award":["ANR-19- CE40-0018"],"award-info":[{"award-number":["ANR-19- CE40-0018"]}],"id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    A power series being given as the solution of a linear differential equation with appropriate initial conditions, minimization consists in finding a non-trivial linear differential equation of minimal order having this power series as a solution. This problem exists in both homogeneous and inhomogeneous variants; it is distinct from, but related to, the classical problem of factorization of differential operators. Recently, minimization has found applications in Transcendental Number Theory, more specifically in the computation of non-zero algebraic points where Siegel\u2019s\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper E\">\n                        <mml:semantics>\n                          <mml:mi>E<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">E<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -functions take algebraic values. We present algorithms and implementations for these questions, and discuss examples and experiments.\n                  <\/p>","DOI":"10.1090\/mcom\/3912","type":"journal-article","created":{"date-parts":[[2023,10,4]],"date-time":"2023-10-04T09:58:16Z","timestamp":1696413496000},"page":"1427-1472","source":"Crossref","is-referenced-by-count":7,"title":["Minimization of differential equations and algebraic values of \ud835\udc38-functions"],"prefix":"10.1090","volume":"93","author":[{"given":"Alin","family":"Bostan","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tanguy","family":"Rivoal","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bruno","family":"Salvy","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2023,10,23]]},"reference":[{"issue":"11","key":"1","doi-asserted-by":"publisher","first-page":"1611","DOI":"10.1016\/S0041-5553(89)80002-3","article-title":"Rational solutions of linear differential and difference equations with polynomial coefficients","volume":"29","author":"Abramov, S. A.","year":"1989","journal-title":"Zh. Vychisl. Mat. i Mat. Fiz.","ISSN":"https:\/\/id.crossref.org\/issn\/0044-4669","issn-type":"print"},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"117","DOI":"10.1007\/s00200-005-0193-9","article-title":"Apparent singularities of linear difference equations with polynomial coefficients","volume":"17","author":"Abramov, S. A.","year":"2006","journal-title":"Appl. Algebra Engrg. Comm. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0938-1279","issn-type":"print"},{"issue":"1","key":"3","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1112\/plms.12038","article-title":"M\u00e9thode de Mahler: relations lin\u00e9aires, transcendance et applications aux nombres automatiques","volume":"115","author":"Adamczewski, Boris","year":"2017","journal-title":"Proc. Lond. Math. Soc. (3)","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6115","issn-type":"print"},{"issue":"2","key":"4","doi-asserted-by":"publisher","first-page":"557","DOI":"10.5802\/jtnb.1039","article-title":"M\u00e9thode de Mahler, transcendance et relations lin\u00e9aires: aspects effectifs","volume":"30","author":"Adamczewski, Boris","year":"2018","journal-title":"J. Th\\'{e}or. Nombres Bordeaux","ISSN":"https:\/\/id.crossref.org\/issn\/1246-7405","issn-type":"print"},{"key":"5","doi-asserted-by":"crossref","unstructured":"[AK91] S. A. Abramov and K. Yu. Kvansenko, Fast algorithms for the search of the rational solutions of linear differential equations with polynomial coefficients, ISSAC\u201991\u2014Proceedings of the 16th International Symposium on Symbolic and Algebraic Computation, ACM Press, 1991, pp. 267\u2013270.","DOI":"10.1145\/120694.120735"},{"issue":"2","key":"6","doi-asserted-by":"publisher","first-page":"705","DOI":"10.2307\/121045","article-title":"S\u00e9ries Gevrey de type arithm\u00e9tique. I. Th\u00e9or\u00e8mes de puret\u00e9 et de dualit\u00e9","volume":"151","author":"Andr\u00e9, Yves","year":"2000","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"issue":"2","key":"7","doi-asserted-by":"publisher","first-page":"741","DOI":"10.2307\/121046","article-title":"S\u00e9ries Gevrey de type arithm\u00e9tique. II. Transcendance sans transcendance","volume":"151","author":"Andr\u00e9, Yves","year":"2000","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"issue":"2","key":"8","doi-asserted-by":"publisher","first-page":"449","DOI":"10.24033\/asens.2218","article-title":"Solution algebras of differential equations and quasi-homogeneous varieties: a new differential Galois correspondence","volume":"47","author":"Andr\u00e9, Yves","year":"2014","journal-title":"Ann. Sci. \\'{E}c. Norm. Sup\\'{e}r. (4)","ISSN":"https:\/\/id.crossref.org\/issn\/0012-9593","issn-type":"print"},{"issue":"4","key":"9","doi-asserted-by":"publisher","first-page":"697","DOI":"10.1112\/blms.12168","article-title":"Exceptional values of \ud835\udc38-functions at algebraic points","volume":"50","author":"Adamczewski, Boris","year":"2018","journal-title":"Bull. Lond. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6093","issn-type":"print"},{"key":"10","series-title":"National Bureau of Standards Applied Mathematics Series, No. 55","volume-title":"Handbook of mathematical functions with formulas, graphs, and mathematical tables","author":"Abramowitz, Milton","year":"1964"},{"key":"11","doi-asserted-by":"crossref","unstructured":"[Bar95] A. Barkatou, A rational version of Moser\u2019s algorithm, ISSAC\u201995\u2014Proceedings of the 20th International Symposium on Symbolic and Algebraic Computation, ACM Press, 1995, pp. 297\u2013302.","DOI":"10.1145\/220346.220385"},{"issue":"1","key":"12","doi-asserted-by":"crossref","first-page":"181","DOI":"10.24033\/asens.1488","article-title":"\u00c9quations diff\u00e9rentielles lin\u00e9aires et majorations de multiplicit\u00e9s","volume":"18","author":"Bertrand, Daniel","year":"1985","journal-title":"Ann. Sci. \\'{E}cole Norm. Sup. (4)","ISSN":"https:\/\/id.crossref.org\/issn\/0012-9593","issn-type":"print"},{"issue":"2","key":"13","doi-asserted-by":"publisher","first-page":"279","DOI":"10.2307\/2007054","article-title":"Siegel normality","volume":"127","author":"Beukers, Frits","year":"1988","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"issue":"4","key":"14","doi-asserted-by":"publisher","first-page":"661","DOI":"10.1007\/s00026-016-0328-7","article-title":"On 3-dimensional lattice walks confined to the positive octant","volume":"20","author":"Bostan, Alin","year":"2016","journal-title":"Ann. Comb.","ISSN":"https:\/\/id.crossref.org\/issn\/0218-0006","issn-type":"print"},{"key":"15","isbn-type":"print","doi-asserted-by":"publisher","first-page":"203","DOI":"10.1145\/1837934.1837975","article-title":"Complexity of creative telescoping for bivariate rational functions","author":"Bostan, Alin","year":"2010","ISBN":"https:\/\/id.crossref.org\/isbn\/9781450301503"},{"key":"16","isbn-type":"print","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1145\/1277548.1277553","article-title":"Differential equations for algebraic functions","author":"Bostan, Alin","year":"2007","ISBN":"https:\/\/id.crossref.org\/isbn\/9781595937438"},{"issue":"2","key":"17","doi-asserted-by":"crossref","first-page":"241","DOI":"10.5802\/afst.1069","article-title":"Effective estimates for global relations on Euler-type series","volume":"13","author":"Bertrand, Daniel","year":"2004","journal-title":"Ann. Fac. Sci. Toulouse Math. (6)","ISSN":"https:\/\/id.crossref.org\/issn\/0240-2963","issn-type":"print"},{"issue":"2","key":"18","doi-asserted-by":"publisher","first-page":"278","DOI":"10.1007\/BF01446541","article-title":"Die Irreducibilit\u00e4t der homogenen linearen Differentialgleichungen","volume":"45","author":"Beke, Emanuel","year":"1894","journal-title":"Math. Ann.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5831","issn-type":"print"},{"key":"19","unstructured":"[Ben92] I. Bendixson, Sur les \u00e9quations diff\u00e9rentielles lin\u00e9aires homog\u00e8nes, \u00d6fversigt af Kongl. Vetenskaps-Akademiens F\u00f6rhandlingar 49 (1892), 91\u2013105."},{"key":"20","first-page":"61","article-title":"Exposants des syst\u00e8mes diff\u00e9rentiels, vecteurs cycliques et majorations de multiplicit\u00e9s","author":"Bertrand, Daniel","year":"1988"},{"key":"21","first-page":"51","article-title":"On Andr\u00e9\u2019s proof of the Siegel-Shidlovsky theorem","author":"Bertrand, Daniel","year":"1999"},{"issue":"1","key":"22","doi-asserted-by":"publisher","first-page":"369","DOI":"10.4007\/annals.2006.163.369","article-title":"A refined version of the Siegel-Shidlovskii theorem","volume":"163","author":"Beukers, F.","year":"2006","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"key":"23","doi-asserted-by":"publisher","first-page":"351","DOI":"10.1016\/j.jsc.2016.04.002","article-title":"Multiple binomial sums","volume":"80","author":"Bostan, Alin","year":"2017","journal-title":"J. Symbolic Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0747-7171","issn-type":"print"},{"key":"24","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1145\/309831.309929","article-title":"Shifted normal forms of polynomial matrices","author":"Beckermann, Bernhard","year":"1999"},{"key":"25","isbn-type":"print","first-page":"53","article-title":"Removing apparent singularities of systems of linear differential equations with rational function coefficients","author":"Barkatou, Moulay A.","year":"2015","ISBN":"https:\/\/id.crossref.org\/isbn\/9781450334358"},{"key":"26","doi-asserted-by":"crossref","unstructured":"[Bro92] M. Bronstein, Linear ordinary differential equations: Breaking through the order 2 barrier, ISSAC\u201992\u2014Proceedings of the 17th International Symposium on Symbolic and Algebraic Computation, ACM Press, 1992, pp. 42\u201348.","DOI":"10.1145\/143242.143264"},{"key":"27","doi-asserted-by":"crossref","unstructured":"[Bro94] M. Bronstein, An improved algorithm for factoring linear ordinary differential operators, ISSAC\u201994\u2014Proceedings of the 19th International Symposium on Symbolic and Algebraic Computation, ACM Press, 1994, pp. 336\u2013340.","DOI":"10.1145\/190347.190436"},{"issue":"1","key":"28","doi-asserted-by":"publisher","first-page":"53","DOI":"10.1112\/blms.12396","article-title":"Explicit degree bounds for right factors of linear differential operators","volume":"53","author":"Bostan, Alin","year":"2021","journal-title":"Bull. Lond. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6093","issn-type":"print"},{"key":"29","isbn-type":"print","first-page":"73","article-title":"Symbolic-numeric factorization of differential operators","author":"Chyzak, Fr\u00e9d\u00e9ric","year":"[2022] \\copyright2022","ISBN":"https:\/\/id.crossref.org\/isbn\/9781450386883"},{"key":"30","isbn-type":"print","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1145\/2465506.2465510","article-title":"Desingularization explains order-degree curves for Ore operators","author":"Chen, Shaoshi","year":"2013","ISBN":"https:\/\/id.crossref.org\/isbn\/9781450320597"},{"key":"31","isbn-type":"print","doi-asserted-by":"publisher","first-page":"191","DOI":"10.1142\/9789812778437_0007","article-title":"Cyclic vectors","author":"Churchill, R. C.","year":"2002","ISBN":"https:\/\/id.crossref.org\/isbn\/9810247036"},{"key":"32","isbn-type":"print","doi-asserted-by":"publisher","first-page":"122","DOI":"10.1145\/2442829.2442850","article-title":"Order-degree curves for hypergeometric creative telescoping","author":"Chen, Shaoshi","year":"2012","ISBN":"https:\/\/id.crossref.org\/isbn\/9781450312691"},{"issue":"8","key":"33","doi-asserted-by":"publisher","first-page":"968","DOI":"10.1016\/j.jsc.2012.02.002","article-title":"Trading order for degree in creative telescoping","volume":"47","author":"Chen, Shaoshi","year":"2012","journal-title":"J. Symbolic Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0747-7171","issn-type":"print"},{"key":"34","doi-asserted-by":"publisher","first-page":"617","DOI":"10.1016\/j.jsc.2015.11.001","article-title":"Desingularization of Ore operators","volume":"74","author":"Chen, Shaoshi","year":"2016","journal-title":"J. Symbolic Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0747-7171","issn-type":"print"},{"issue":"1","key":"35","doi-asserted-by":"publisher","first-page":"130","DOI":"10.2307\/2371061","article-title":"Formal Solutions of Irregular Linear Differential Equations. Part II","volume":"58","author":"Cope, Frances Thorndike","year":"1936","journal-title":"Amer. J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9327","issn-type":"print"},{"issue":"2","key":"36","doi-asserted-by":"publisher","first-page":"187","DOI":"10.1006\/jsco.1998.0207","article-title":"Non-commutative elimination in Ore algebras proves multivariate identities","volume":"26","author":"Chyzak, Fr\u00e9d\u00e9ric","year":"1998","journal-title":"J. Symbolic Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0747-7171","issn-type":"print"},{"issue":"3","key":"37","doi-asserted-by":"publisher","first-page":"1043","DOI":"10.1016\/j.jsc.2003.11.004","article-title":"A modular algorithm for computing the exponential solutions of a linear differential operator","volume":"38","author":"Cluzeau, Thomas","year":"2004","journal-title":"J. Symbolic Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0747-7171","issn-type":"print"},{"key":"38","doi-asserted-by":"crossref","unstructured":"[DAS15] A. Dehghan, S. M. Assari, and M. Shah, Gmmcp tracker: Globally optimal generalized maximum multi clique problem for multiple object tracking, 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2015, pp. 4091\u20134099.","DOI":"10.1109\/CVPR.2015.7299036"},{"key":"39","isbn-type":"print","first-page":"273","article-title":"An algorithm to obtain formal solutions of a linear homogeneous differential equation at an irregular singular point","author":"Della Dora, J.","year":"1982","ISBN":"https:\/\/id.crossref.org\/isbn\/3540116079"},{"key":"40","unstructured":"[Fab85] E. Fabry, Sur les int\u00e9grales des \u00e9quations diff\u00e9rentielles lin\u00e9aires \u00e0 coefficients rationnels, Th\u00e8se de doctorat \u00e8s sciences math\u00e9matiques, Facult\u00e9 des Sciences de Paris, July 1885."},{"key":"41","doi-asserted-by":"crossref","first-page":"135","DOI":"10.24033\/bsmf.369","article-title":"R\u00e9ductibilit\u00e9 des \u00e9quations diff\u00e9rentielles lin\u00e9aires","volume":"16","author":"Fabry","year":"1888","journal-title":"Bull. Soc. Math. France","ISSN":"https:\/\/id.crossref.org\/issn\/0037-9484","issn-type":"print"},{"key":"42","unstructured":"[Fab88b] E. Fabry, R\u00e9ducibilit\u00e9 des \u00e9quations diff\u00e9rentielles lin\u00e9aires, C. R. Acad. Sci. Paris 106 (1888), 732\u2013734."},{"issue":"3","key":"43","doi-asserted-by":"publisher","first-page":"349","DOI":"10.1007\/BF01388432","article-title":"Endlichkeitss\u00e4tze f\u00fcr abelsche Variet\u00e4ten \u00fcber Zahlk\u00f6rpern","volume":"73","author":"Faltings, G.","year":"1983","journal-title":"Invent. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-9910","issn-type":"print"},{"issue":"3","key":"44","doi-asserted-by":"publisher","first-page":"903","DOI":"10.4007\/annals.2021.194.3.7","article-title":"A non-hypergeometric \ud835\udc38-function","volume":"194","author":"Fres\u00e1n, Javier","year":"2021","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"issue":"2","key":"45","doi-asserted-by":"publisher","first-page":"177","DOI":"10.4064\/aa105-2-4","article-title":"On the irreducibility of the generalized Laguerre polynomials","volume":"105","author":"Filaseta, M.","year":"2002","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"issue":"2","key":"46","doi-asserted-by":"publisher","first-page":"313","DOI":"10.4171\/CMH\/321","article-title":"On the values of \ud835\udc3a-functions","volume":"89","author":"Fischler, St\u00e9phane","year":"2014","journal-title":"Comment. Math. Helv.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-2571","issn-type":"print"},{"key":"47","doi-asserted-by":"publisher","first-page":"31","DOI":"10.5802\/jep.28","article-title":"Arithmetic theory of \ud835\udc38-operators","volume":"3","author":"Fischler, St\u00e9phane","year":"2016","journal-title":"J. \\'{E}c. polytech. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/2429-7100","issn-type":"print"},{"issue":"2","key":"48","doi-asserted-by":"publisher","first-page":"174","DOI":"10.1007\/BF02686240","article-title":"On the algebraic independence of the values of \ud835\udc38-functions at singular points and Siegel\u2019s conjecture","volume":"67","author":"Gorelov, V. A.","year":"2000","journal-title":"Mat. Zametki","ISSN":"https:\/\/id.crossref.org\/issn\/0025-567X","issn-type":"print"},{"issue":"4","key":"49","doi-asserted-by":"publisher","first-page":"549","DOI":"10.1023\/B:MATN.0000023333.36063.12","article-title":"On the Siegel conjecture for the case of second-order linear homogeneous differential equations","volume":"75","author":"Gorelov, V. A.","year":"2004","journal-title":"Mat. Zametki","ISSN":"https:\/\/id.crossref.org\/issn\/0025-567X","issn-type":"print"},{"issue":"1","key":"50","doi-asserted-by":"publisher","first-page":"7","DOI":"10.1016\/S0747-7171(08)80034-X","article-title":"Complexity of factoring and calculating the GCD of linear ordinary differential operators","volume":"10","author":"Grigor\u2032ev, D. Yu.","year":"1990","journal-title":"J. Symbolic Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0747-7171","issn-type":"print"},{"key":"51","doi-asserted-by":"crossref","unstructured":"[Hil15] E. Hilb, Lineare Differentialgleichungen im komplexen Gebiet, pp. 471\u2013562, 1915, Chapter II.B.5 of \u201cB. Analysis der komplexen Gr\u00f6\u00dfen\u201d in Encyklop\u00e4die der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen: Zweiter Band in Drei Teilen Analysis, Vieweg+Teubner Verlag, 1921.","DOI":"10.1007\/978-3-663-16022-9_5"},{"key":"52","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1210-2","volume-title":"Diophantine geometry","volume":"201","author":"Hindry, Marc","year":"2000","ISBN":"https:\/\/id.crossref.org\/isbn\/0387989757"},{"issue":"4","key":"53","doi-asserted-by":"publisher","first-page":"1127","DOI":"10.5802\/aif.2208","article-title":"Specializations of one-parameter families of polynomials","volume":"56","author":"Hajir, Farshid","year":"2006","journal-title":"Ann. Inst. Fourier (Grenoble)","ISSN":"https:\/\/id.crossref.org\/issn\/0373-0956","issn-type":"print"},{"key":"54","volume-title":"Ordinary Differential Equations","author":"Ince, E. L.","year":"1944"},{"key":"55","isbn-type":"print","doi-asserted-by":"publisher","first-page":"211","DOI":"10.1145\/2465506.2465513","article-title":"Finding hyperexponential solutions of linear ODEs by numerical evaluation","author":"Johansson, Fredrik","year":"2013","ISBN":"https:\/\/id.crossref.org\/isbn\/9781450320597"},{"key":"56","doi-asserted-by":"publisher","first-page":"192","DOI":"10.1016\/j.jsc.2019.07.011","article-title":"Fast computation of approximant bases in canonical form","volume":"98","author":"Jeannerod, Claude-Pierre","year":"2020","journal-title":"J. Symbolic Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0747-7171","issn-type":"print"},{"key":"57","isbn-type":"print","doi-asserted-by":"publisher","first-page":"288","DOI":"10.1145\/2608628.2608634","article-title":"Bounds for D-finite closure properties","author":"Kauers, Manuel","year":"2014","ISBN":"https:\/\/id.crossref.org\/isbn\/9781450325011"},{"key":"58","first-page":"83","article-title":"Le th\u00e9or\u00e8me d\u2019Andr\u00e9-Chudnovsky-Katz \u201cau sense large\u201d","volume":"7","author":"Lepetit, Gabriel","year":"2021","journal-title":"North-West. Eur. J. Math."},{"key":"59","doi-asserted-by":"crossref","unstructured":"[Lio33] J. Liouville, Second m\u00e9moire sur la d\u00e9termination des int\u00e9grales dont la valeur est alg\u00e9brique, J. \u00c9c. Polytech. 14 (1833), 149\u2013193.","DOI":"10.1515\/crll.1833.10.342"},{"issue":"3","key":"60","doi-asserted-by":"publisher","first-page":"551","DOI":"10.1155\/S0161171295000706","article-title":"Transcendentality of zeros of higher derivatives of functions involving Bessel functions","volume":"18","author":"Lorch, Lee","year":"1995","journal-title":"Internat. J. Math. Math. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/0161-1712","issn-type":"print"},{"issue":"1","key":"61","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1006\/jabr.2000.8448","article-title":"Elimination of variables in linear solvable polynomial algebras and \u2202-holonomicity","volume":"234","author":"Li, Huishi","year":"2000","journal-title":"J. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0021-8693","issn-type":"print"},{"key":"62","unstructured":"[Mal79] B. Malgrange, Sur la r\u00e9duction formelle des \u00e9quations diff\u00e9rentielles \u00e0 singularit\u00e9s irr\u00e9guli\u00e8res, Preprint, 1979."},{"key":"63","unstructured":"[Mar91a] A. Markoff, Sur la th\u00e9orie des \u00e9quations diff\u00e9rentielles lin\u00e9aires, C. R. Acad. Sci. Paris 113 (1891), 790\u2013791."},{"key":"64","unstructured":"[Mar91b] A. Markoff, Sur la th\u00e9orie des \u00e9quations diff\u00e9rentielles lin\u00e9aires, C. R. Acad. Sci. Paris 113 (1891), 1024\u20131025."},{"key":"65","unstructured":"[Mar91c] A. Markoff, Sur les \u00e9quations diff\u00e9rentielles lin\u00e9aires, C. R. Acad. Sci. Paris 113 (1891), 685\u2013688."},{"key":"66","unstructured":"[Mor22] L. J. Mordell, On the rational solutions of the indeterminate equations of the third and fourth degrees, Math. Proc. Cambridge Philos. Soc. 21 (1922), 179\u2013192."},{"key":"67","doi-asserted-by":"publisher","first-page":"379","DOI":"10.1007\/BF01162962","article-title":"The order of a singularity in Fuchs\u2019 theory","volume":"72","author":"Moser, J\u00fcrgen","year":"1959","journal-title":"Math. Z.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5874","issn-type":"print"},{"issue":"8","key":"68","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1070\/SM1996v187n08ABEH000152","article-title":"On the linear independence of values of \ud835\udc38-functions","volume":"187","author":"Nesterenko, Yu. V.","year":"1996","journal-title":"Mat. Sb.","ISSN":"https:\/\/id.crossref.org\/issn\/0368-8666","issn-type":"print"},{"key":"69","doi-asserted-by":"publisher","first-page":"2","DOI":"10.3406\/rhs.1967.2512","article-title":"Les lettres de Ch. Hermite \u00e0 A. Markoff, 1885\u20131899","volume":"20","author":"Ogigova, H.","year":"1967","journal-title":"Rev. Histoire Sci. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0048-7996","issn-type":"print"},{"key":"70","unstructured":"[OOL{\\etalchar{+}}22] F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and eds. M. A. McClain, NIST digital library of mathematical functions, \\url{http:\/\/dlmf.nist.gov\/}, 2022."},{"issue":"3","key":"71","doi-asserted-by":"publisher","first-page":"480","DOI":"10.2307\/1968173","article-title":"Theory of non-commutative polynomials","volume":"34","author":"Ore, Oystein","year":"1933","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"key":"72","unstructured":"[Pai91] P. Painlev\u00e9, Remarque sur une Communication de M. Markoff, relative \u00e0 des \u00e9quations diff\u00e9rentielles lin\u00e9aires, C. R. Acad. Sci. Paris 113 (1891), 739\u2013740 (French)."},{"key":"73","volume-title":"Introduction to the theory of linear differential equations","author":"Poole, E. G. C.","year":"1960"},{"key":"74","doi-asserted-by":"crossref","first-page":"252","DOI":"10.1137\/0310020","article-title":"Invariant description of linear, time-invariant controllable systems","volume":"10","author":"Popov, V. M.","year":"1972","journal-title":"SIAM J. Control"},{"key":"75","unstructured":"[Riv16] T. Rivoal, Valeurs alg\u00e9briques de \ud835\udc38-fonctions aux points alg\u00e9briques, Unpublished note (2016), 4 pages."},{"key":"76","unstructured":"[Sch97] Ludwig Schlesinger, Handbuch der Theorie der linearen Differentialgleichungen. In zwei B\u00e4nden. Band II, Theil 1, Teubner, 1897."},{"key":"77","unstructured":"[Sch09] L. Schlesinger, Bericht \u00fcber die Entwickelung der Theorie der linearen Differentialgleichungen seit 1865, Jahresber. Deutsch. Math.-Verein. 18 (1909), 133\u2013266."},{"key":"78","doi-asserted-by":"publisher","first-page":"52","DOI":"10.1515\/crll.1931.165.52","article-title":"Affektlose Gleichungen in der Theorie der Laguerreschen und Hermiteschen Polynome","volume":"165","author":"Schur, J.","year":"1931","journal-title":"J. Reine Angew. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0075-4102","issn-type":"print"},{"key":"79","series-title":"Wiley-Interscience Series in Discrete Mathematics","isbn-type":"print","volume-title":"Theory of linear and integer programming","author":"Schrijver, Alexander","year":"1986","ISBN":"https:\/\/id.crossref.org\/isbn\/0471908541"},{"key":"80","doi-asserted-by":"crossref","unstructured":"[Sch89] F. Schwarz, A factorization algorithm for linear ordinary differential equations, ISSAC\u201989\u2014Proceedings of the 14th International Symposium on Symbolic and Algebraic Computation, ACM Press, 1989, pp. 17\u201325.","DOI":"10.1145\/74540.74544"},{"key":"81","series-title":"De Gruyter Studies in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1515\/9783110889055","volume-title":"Transcendental numbers","volume":"12","author":"Shidlovskii, Andrei Borisovich","year":"1989","ISBN":"https:\/\/id.crossref.org\/isbn\/3110115689"},{"key":"82","isbn-type":"print","first-page":"81","article-title":"\u00dcber einige Anwendungen diophantischer Approximationen [reprint of Abhandlungen der Preu\u00dfischen Akademie der Wissenschaften. Physikalisch-mathematische Klasse 1929, Nr. 1]","author":"Siegel, Carl L.","year":"2014","ISBN":"https:\/\/id.crossref.org\/isbn\/9788876425196"},{"issue":"2","key":"83","doi-asserted-by":"crossref","first-page":"343","DOI":"10.2140\/pjm.1993.160.343","article-title":"Moduli of linear differential equations on the Riemann sphere with fixed Galois groups","volume":"160","author":"Singer, Michael F.","year":"1993","journal-title":"Pacific J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0030-8730","issn-type":"print"},{"issue":"2","key":"84","doi-asserted-by":"publisher","first-page":"77","DOI":"10.1007\/BF01191378","article-title":"Testing reducibility of linear differential operators: a group-theoretic perspective","volume":"7","author":"Singer, Michael F.","year":"1996","journal-title":"Appl. Algebra Engrg. Comm. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0938-1279","issn-type":"print"},{"key":"85","isbn-type":"print","first-page":"1","article-title":"Introduction to the Galois theory of linear differential equations","author":"Singer, Michael F.","year":"2009","ISBN":"https:\/\/id.crossref.org\/isbn\/9780521720083"},{"key":"86","series-title":"American Mathematical Society Colloquium Publications, Vol. XXIII","volume-title":"Orthogonal polynomials","author":"Szeg\u0151, G\u00e1bor","year":"1975","edition":"4"},{"issue":"1","key":"87","first-page":"45","article-title":"Some problems that arise in the factorization of linear ordinary differential operators","author":"Tsar\u00ebv, S. P.","year":"1994","journal-title":"Programmirovanie","ISSN":"https:\/\/id.crossref.org\/issn\/0132-3474","issn-type":"print"},{"issue":"4-5","key":"88","doi-asserted-by":"publisher","first-page":"747","DOI":"10.1006\/jsco.1999.0400","article-title":"Weyl closure of a linear differential operator","volume":"29","author":"Tsai, Harrison","year":"2000","journal-title":"J. Symbolic Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0747-7171","issn-type":"print"},{"key":"89","isbn-type":"print","volume-title":"Approximation algorithms","author":"Vazirani, Vijay V.","year":"2001","ISBN":"https:\/\/id.crossref.org\/isbn\/3540653678"},{"issue":"1-4","key":"90","doi-asserted-by":"publisher","first-page":"451","DOI":"10.1007\/BF02141952","article-title":"A general module-theoretic framework for vector M-Pad\u00e9 and matrix rational interpolation","volume":"3","author":"Van Barel, M.","year":"1992","journal-title":"Numer. Algorithms","ISSN":"https:\/\/id.crossref.org\/issn\/1017-1398","issn-type":"print"},{"issue":"1-2","key":"91","doi-asserted-by":"publisher","first-page":"236","DOI":"10.1016\/j.jsc.2006.03.007","article-title":"Around the numeric-symbolic computation of differential Galois groups","volume":"42","author":"van der Hoeven, Joris","year":"2007","journal-title":"J. Symbolic Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0747-7171","issn-type":"print"},{"issue":"2","key":"92","doi-asserted-by":"publisher","first-page":"105","DOI":"10.1007\/s00200-015-0269-0","article-title":"On the complexity of skew arithmetic","volume":"27","author":"van der Hoeven, Joris","year":"2016","journal-title":"Appl. Algebra Engrg. Comm. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0938-1279","issn-type":"print"},{"key":"93","series-title":"Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-55750-7","volume-title":"Galois theory of linear differential equations","volume":"328","author":"van der Put, Marius","year":"2003","ISBN":"https:\/\/id.crossref.org\/isbn\/3540442286"},{"key":"94","doi-asserted-by":"crossref","unstructured":"[vH96] M. van Hoeij, Rational solutions of the mixed differential equation and its application to factorization of differential operators, ISSAC\u201996\u2014Proceedings of the 21st International Symposium on Symbolic and Algebraic Computation, ACM Press, 1996, pp. 219\u2013225.","DOI":"10.1145\/236869.237078"},{"issue":"5","key":"95","doi-asserted-by":"publisher","first-page":"537","DOI":"10.1006\/jsco.1997.0151","article-title":"Factorization of differential operators with rational functions coefficients","volume":"24","author":"van Hoeij, Mark","year":"1997","journal-title":"J. Symbolic Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0747-7171","issn-type":"print"},{"issue":"1","key":"96","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1006\/jsco.1997.0110","article-title":"Formal solutions and factorization of differential operators with power series coefficients","volume":"24","author":"van Hoeij, Mark","year":"1997","journal-title":"J. Symbolic Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0747-7171","issn-type":"print"},{"issue":"1","key":"97","doi-asserted-by":"publisher","first-page":"109","DOI":"10.2206\/kyushujm.61.109","article-title":"Degenerate Gauss hypergeometric functions","volume":"61","author":"Vid\u016bnas, Raimundas","year":"2007","journal-title":"Kyushu J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1340-6116","issn-type":"print"},{"issue":"2","key":"98","doi-asserted-by":"publisher","first-page":"392","DOI":"10.1016\/j.jalgebra.2005.02.027","article-title":"On the genus of generalized Laguerre polynomials","volume":"288","author":"Wong, Siman","year":"2005","journal-title":"J. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0021-8693","issn-type":"print"},{"issue":"3","key":"99","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1016\/S0747-7171(08)80044-2","article-title":"The method of creative telescoping","volume":"11","author":"Zeilberger, Doron","year":"1991","journal-title":"J. Symbolic Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0747-7171","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2024-93-347\/S0025-5718-2023-03912-6\/mcom3912_AM.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/www.ams.org\/mcom\/2024-93-347\/S0025-5718-2023-03912-6\/S0025-5718-2023-03912-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:17:30Z","timestamp":1776835050000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2024-93-347\/S0025-5718-2023-03912-6\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,23]]},"references-count":99,"journal-issue":{"issue":"347","published-print":{"date-parts":[[2024,5]]}},"alternative-id":["S0025-5718-2023-03912-6"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3912","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2023,10,23]]}}}