Confluence of meromorphic solutions of q-difference equations
Annales de l'Institut Fourier, Volume 65 (2015) no. 2, pp. 431-507.

In this paper, we consider a q-analogue of the Borel-Laplace summation where q>1 is a real parameter. In particular, we show that the Borel-Laplace summation of a divergent power series solution of a linear differential equation can be uniformly approximated on a convenient sector, by a meromorphic solution of a corresponding family of linear q-difference equations. We perform the computations for the basic hypergeometric series. Following Sauloy, we prove how a basis of solutions of a linear differential equation can be uniformly approximated on a convenient domain by a basis of solutions of a corresponding family of linear q-difference equations. This leads us to the approximations of Stokes matrices and monodromy matrices of the linear differential equation by matrices with entries that are invariants by the multiplication by q.

Dans cet article, nous considérons un q-analogue du processus de sommation de Borel-Laplace, avec q>1 paramètre réel. En particulier, nous prouvons que la sommation de Borel-Laplace d’une série formelle solution d’une équation différentielle linéaire peut être approchée, dans un secteur convenable, par une solution méromorphe d’une certaine famille d’équations aux q-différences linéaire. Nous faisons les calculs pour les séries hypergéométriques. En s’inspirant de Sauloy, nous prouvons comment une base de solutions d’une equation différentielle linéaire peut être approchée, sur un secteur convenable, par une base de solutions d’une famille correspondante d’équations aux q-différences. Cela nous mène à l’approximation des matrices de Stokes et de monodromies de l’équation différentielle, par des matrices dont les entrées sont invariantes par multiplication par q.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.2937
Classification: 39A13,  34M40
Keywords: Stokes phenomenon, Borel-Laplace transformations, q-difference equations, Confluence, Basic hypergeometric series, Confluent hypergeometric series.
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Dreyfus, Thomas. Confluence of meromorphic solutions of $q$-difference equations. Annales de l'Institut Fourier, Volume 65 (2015) no. 2, pp. 431-507. doi : 10.5802/aif.2937. https://aif.centre-mersenne.org/articles/10.5802/aif.2937/

[1] Adams, C. R. Linear q-difference equations, Bull. Amer. Math. Soc., Tome 37 (1931) no. 6, pp. 361-400 | Article | MR: 1562160 | Zbl: 0002.19103

[2] André, Yves Séries Gevrey de type arithmétique. II. Transcendance sans transcendance, Ann. of Math. (2), Tome 151 (2000) no. 2, pp. 741-756 | Article | MR: 1765708 | Zbl: 1037.11050

[3] Andrews, George E.; Askey, Richard; Roy, Ranjan Special functions, Encyclopedia of Mathematics and its Applications, Tome 71, Cambridge University Press, Cambridge, 1999, xvi+664 pages | MR: 1688958 | Zbl: 0920.33001

[4] Balser, Werner From divergent power series to analytic functions, Lecture Notes in Mathematics, Tome 1582, Springer-Verlag, Berlin, 1994, x+108 pages (Theory and application of multisummable power series) | MR: 1317343 | Zbl: 0810.34046

[5] Bertrand, Daniel Groupes algébriques et équations différentielles linéaires, Astérisque (1992) no. 206, pp. Exp. No. 750, 4, 183-204 (Séminaire Bourbaki, Vol. 1991/92) | Numdam | MR: 1206068 | Zbl: 0813.12004

[6] Bézivin, Jean-Paul Sur les équations fonctionnelles aux q-différences, Aequationes Math., Tome 43 (1992) no. 2-3, pp. 159-176 | Article | MR: 1158724 | Zbl: 0757.39002

[7] Carmichael, R. D. The General Theory of Linear q-Difference Equations, Amer. J. Math., Tome 34 (1912) no. 2, pp. 147-168 | Article | MR: 1506145

[8] De Sole, Alberto; Kac, Victor G. On integral representations of q-gamma and q-beta functions, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., Tome 16 (2005) no. 1, pp. 11-29 | MR: 2225920 | Zbl: 1225.33017

[9] Di Vizio, L.; Ramis, J.-P.; Sauloy, J.; Zhang, C. Équations aux q-différences, Gaz. Math. (2003) no. 96, pp. 20-49 | MR: 1988639 | Zbl: 1063.39015

[10] Di Vizio, Lucia Arithmetic theory of q-difference equations: the q-analogue of Grothendieck-Katz’s conjecture on p-curvatures, Invent. Math., Tome 150 (2002) no. 3, pp. 517-578 | Article | MR: 1946552 | Zbl: 1023.12004

[11] Di Vizio, Lucia Local analytic classification of q-difference equations with |q|=1, J. Noncommut. Geom., Tome 3 (2009) no. 1, pp. 125-149 | Article | MR: 2457039 | Zbl: 1168.39002

[12] Di Vizio, Lucia; Hardouin, Charlotte Descent for differential Galois theory of difference equations: confluence and q-dependence, Pacific J. Math., Tome 256 (2012) no. 1, pp. 79-104 | Article | MR: 2928542 | Zbl: 1258.12004

[13] Di Vizio, Lucia; Zhang, Changgui On q-summation and confluence, Ann. Inst. Fourier (Grenoble), Tome 59 (2009) no. 1, pp. 347-392 | Article | Numdam | MR: 2514868 | Zbl: 1175.34111

[14] Duval, Anne; Roques, Julien Familles fuchsiennes d’équations aux (q-)différences et confluence, Bull. Soc. Math. France, Tome 136 (2008) no. 1, pp. 67-96 | Numdam | MR: 2415336 | Zbl: 1168.39003

[15] Fauvet, F.; Ramis, J.-P.; Richard-Jung, F.; Thomann, J. Stokes phenomenon for the prolate spheroidal wave equation, Appl. Numer. Math., Tome 60 (2010) no. 12, pp. 1309-1319 | Article | MR: 2735163 | Zbl: 1213.34113

[16] Fauvet, F.; Richard-Jung, F.; Thomann, J. Automatic computation of Stokes matrices, Numer. Algorithms, Tome 50 (2009) no. 2, pp. 179-213 | Article | MR: 2487235 | Zbl: 1167.65391

[17] Gasper, George; Rahman, Mizan Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, Tome 96, Cambridge University Press, Cambridge, 2004, xxvi+428 pages (With a foreword by Richard Askey) | MR: 2128719 | Zbl: 1129.33005

[18] van der Hoeven, Joris Around the numeric-symbolic computation of differential Galois groups, J. Symbolic Comput., Tome 42 (2007) no. 1-2, pp. 236-264 | Article | MR: 2284295

[19] Kriegl, Andreas; Michor, Peter W. The convenient setting of global analysis, Mathematical Surveys and Monographs, Tome 53, American Mathematical Society, Providence, RI, 1997, x+618 pages | MR: 1471480 | Zbl: 0889.58001

[20] Loday-Richaud, Michèle Introduction à la multisommabilité, Gaz. Math. (1990) no. 44, pp. 41-63 | Zbl: 0722.34005

[21] Loday-Richaud, Michèle Solutions formelles des systèmes différentiels linéaires méromorphes et sommation, Exposition. Math., Tome 13 (1995) no. 2-3, pp. 116-162 | MR: 1346200 | Zbl: 0831.34002

[22] Loday-Richaud, Michèle; Remy, Pascal Resurgence, Stokes phenomenon and alien derivatives for level-one linear differential systems, J. Differential Equations, Tome 250 (2011) no. 3, pp. 1591-1630 | Article | MR: 2737219 | Zbl: 1214.34087

[23] Malgrange, B.; Ramis, J.-P. Fonctions multisommables, Ann. Inst. Fourier (Grenoble), Tome 42 (1992) no. 1-2, pp. 353-368 | Article | Numdam | MR: 1162566 | Zbl: 0759.34007

[24] Malgrange, Bernard Sommation des séries divergentes, Exposition. Math., Tome 13 (1995) no. 2-3, pp. 163-222 | MR: 1346201 | Zbl: 0836.40004

[25] Marotte, F.; Zhang, C. Multisommabilité des séries entières solutions formelles d’une équation aux q-différences linéaire analytique, Ann. Inst. Fourier (Grenoble), Tome 50 (2000) no. 6, p. 1859-1890 (2001) | Article | Numdam | MR: 1817385 | Zbl: 1063.39001

[26] Martinet, Jean; Ramis, Jean-Pierre Théorie de Galois différentielle et resommation, Computer algebra and differential equations (Comput. Math. Appl.), Academic Press, London, 1990, pp. 117-214 | MR: 1038060 | Zbl: 0722.12007

[27] Mezzarobba, Marc Autour de l’évaluation numérique des fonctions D-finies. Thèse de doctorat. (2001) (Ph. D. Thesis)

[28] Mezzarobba, Marc; Salvy, Bruno Effective bounds for P-recursive sequences, J. Symbolic Comput., Tome 45 (2010) no. 10, pp. 1075-1096 | Article | MR: 2679389 | Zbl: 1201.65219

[29] van der Put, Marius; Reversat, Marc Galois theory of q-difference equations, Ann. Fac. Sci. Toulouse Math. (6), Tome 16 (2007) no. 3, pp. 665-718 | Article | Numdam | MR: 2379057 | Zbl: 1213.39011

[30] van der Put, Marius; Singer, Michael F. Galois theory of linear differential equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Tome 328, Springer-Verlag, Berlin, 2003, xviii+438 pages | MR: 1960772 | Zbl: 1036.12008

[31] Ramis, J.-P.; Sauloy, J.; Zhang, C. Local analytic classification of q-difference equations. To appear in Astérisque. (http://arxiv.org/abs/0903.0853) | MR: 3185985

[32] Ramis, Jean-Pierre About the growth of entire functions solutions of linear algebraic q-difference equations, Ann. Fac. Sci. Toulouse Math. (6), Tome 1 (1992) no. 1, pp. 53-94 | Article | Numdam | MR: 1191729 | Zbl: 0796.39005

[33] Ramis, Jean-Pierre Séries divergentes et théories asymptotiques, Bull. Soc. Math. France, Tome 121 (1993) no. Panoramas et Syntheses, suppl., 74 pages | MR: 1272100 | Zbl: 0830.34045

[34] Ramis, Jean-Pierre; Zhang, Changgui Développement asymptotique q-Gevrey et fonction thêta de Jacobi, C. R. Math. Acad. Sci. Paris, Tome 335 (2002) no. 11, pp. 899-902 | Article | MR: 1952546 | Zbl: 1025.39014

[35] Remy, Pascal Matrices de Stokes-Ramis et constantes de connexion pour les systèmes différentiels linéaires de niveau unique, Ann. Fac. Sci. Toulouse Math. (6), Tome 21 (2012) no. 1, pp. 93-150 | Article | MR: 2954106 | Zbl: 1244.34109

[36] Roques, Julien Classification rationnelle et confluence des systèmes aux différences singuliers réguliers, Ann. Inst. Fourier (Grenoble), Tome 56 (2006) no. 6, pp. 1663-1699 | Article | Numdam | MR: 2282672 | Zbl: 1125.39019

[37] Roques, Julien Galois groups of the basic hypergeometric equations, Pacific J. Math., Tome 235 (2008) no. 2, pp. 303-322 | Article | MR: 2386226 | Zbl: 1153.39026

[38] Roques, Julien Generalized basic hypergeometric equations, Invent. Math., Tome 184 (2011) no. 3, pp. 499-528 | Article | MR: 2800693 | Zbl: 1241.33017

[39] Sauloy, Jacques Systèmes aux q-différences singuliers réguliers: classification, matrice de connexion et monodromie, Ann. Inst. Fourier (Grenoble), Tome 50 (2000) no. 4, pp. 1021-1071 | Article | Numdam | MR: 1799737 | Zbl: 0957.05012

[40] Singer, Michael F. Introduction to the Galois theory of linear differential equations, Algebraic theory of differential equations (London Math. Soc. Lecture Note Ser.) Tome 357, Cambridge Univ. Press, Cambridge, 2009, pp. 1-82 | MR: 2484905 | Zbl: 1176.12005

[41] Trjitzinsky, W. J. Analytic theory of linear q-difference equations, Acta Math., Tome 61 (1933) no. 1, pp. 1-38 | Article | MR: 1555369 | Zbl: 0007.21103

[42] Zhang, Changgui Développements asymptotiques q-Gevrey et séries Gq-sommables, Ann. Inst. Fourier (Grenoble), Tome 49 (1999) no. 1, p. vi-vii, x, 227–261 | Article | Numdam | MR: 1688144 | Zbl: 0974.39009

[43] Zhang, Changgui Transformations de q-Borel-Laplace au moyen de la fonction thêta de Jacobi, C. R. Acad. Sci. Paris Sér. I Math., Tome 331 (2000) no. 1, pp. 31-34 | Article | MR: 1780181 | Zbl: 1101.33307

[44] Zhang, Changgui Sur la fonction q-gamma de Jackson, Aequationes Math., Tome 62 (2001) no. 1-2, pp. 60-78 | Article | MR: 1849140 | Zbl: 0990.39018

[45] Zhang, Changgui Une sommation discrète pour des équations aux q-différences linéaires et à coefficients analytiques: théorie générale et exemples, Differential equations and the Stokes phenomenon, World Sci. Publ., River Edge, NJ, 2002, pp. 309-329 | MR: 2067338 | Zbl: 1041.39013

[46] Zhang, Changgui Sur les fonctions q-Bessel de Jackson, J. Approx. Theory, Tome 122 (2003) no. 2, pp. 208-223 | Article | MR: 1988300 | Zbl: 1023.33012

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