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Multisummability of formal power series solutions of linear analytic \(q\)-difference equations. (Multisommabilité des séries entières solutions formelles d’une équation aux \(q\)-différences linéaire analytique.) (French) Zbl 1063.39001
Summary: We introduce a \(q\)-analogous version of the elementary acceleration method of Écalle-Martinet-Ramis and define the \(Gq\)-multisummable power series. We show that every formal power series satisfying a linear analytic \(q\)-difference equation is \(Gq\)-multisummable.

39A10 Additive difference equations
40G99 Special methods of summability
Full Text: DOI Numdam EuDML
[1] C. R. ADAMS, On the linear ordinary q-difference equations, Ann. Math., Ser. II, 30, No. 2 (1929) 195-205. · JFM 55.0263.01
[2] W. BALSER, A different characterization of multisummable power series, Analysis, 12 (1992), 57-65. · Zbl 0759.40005
[3] W. BALSER, B.J.L. BRAAKSMA, J.-P. RAMIS et Y. SIBUYA, Multisummability of formal power series solutions of linear ordinary differential equations, Asymptotic Analysis, 5 (1991), 27-45. · Zbl 0754.34057
[4] J. ECALLE, Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, Hermann, Paris, 1992. · Zbl 1241.34003
[5] G. GASPER et M. RAHMAN, Basic hypergeometric series, Encycl. Math. Appl., Cambridge Univ. Press, Cambridge, 1990. · Zbl 0695.33001
[6] J. E. LITTLEWOOD, On the asymptotic approximation to integral functions of zero order, Proc. London Math. Soc., Serie 2, no 5 (1907), 361-410. · JFM 38.0450.01
[7] B. MALGRANGE et J.-P. RAMIS, Fonctions multisommables, Ann. Inst. Fourier, 42-1/2 (1992), 353-368. · Zbl 0759.34007
[8] F. MAROTTE et C. ZHANG, Sur la sommabilité des séries entières solutions formelles d’une équation aux q-différences, II, C. R. Acad. Sci. Paris, t. 327, Série I (1998) 715-718. · Zbl 0915.39004
[9] J. MARTINET et J.-P. RAMIS, Elementary acceleration and multisummability I, Ann. Inst. Henri Poincaré, Vol. 54, no 4 (1991) 331-401. · Zbl 0748.12005
[10] J.-P. RAMIS, LES séries k-sommables et leurs applications, Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory, Lecture Notes in Physics, 126 (1980) 178-199.
[11] J.-P. RAMIS, About the growth of entire functions solutions of linear algebraic q-difference equations, Annales de la Fac. de Toulouse, Série 6, Vol. I, no 1 (1992) 53-94. · Zbl 0796.39005
[12] W.J. TRJITZINSKY, Analytic theory of linear q-difference equations, Acta Mathematica, 61 (1933) 1-38. · JFM 59.0455.02
[13] C. ZHANG, Développements asymptotiques q-Gevrey et séries gq-sommables, Ann. Inst. Fourier, 49-1 (1999) 227-261. · Zbl 0974.39009
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