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\(q\)-Borel-Laplace summation for \(q\)-difference equations with two slopes. (English) Zbl 1364.39008

The authors study \(q\)-difference equations with two slopes of the Newton polygon, and coefficients which are germs of meromorphic functions. The authors prove the existence and uniqueness of meromorphic gauge transformations having \(q^{\frac{d{\mathbb{Z}}}{n}}\)-spiral of poles of multiplicity at most \(1\), and explain how to compute the latter meromorphic gauge transformations by using a \(q\)-analogue of the Borel-Laplace summation. Finally, they provide an illustrative example.

MSC:

39A13 Difference equations, scaling (\(q\)-differences)
39A45 Difference equations in the complex domain
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[1] Abdi W.H., Proc. Nat. Acad. Sci. India Sect. A 29 pp 389– (1960)
[2] DOI: 10.1007/BF01111201 · Zbl 0123.30102 · doi:10.1007/BF01111201
[3] DOI: 10.1007/BFb0073564 · doi:10.1007/BFb0073564
[4] V. Bugeaud, Groupe de Galois local des équations aux q-différences irrégulières, PhD thesis, Institut de Mathématiques de Toulouse, 2012.
[5] DOI: 10.1093/imrn/rnu137 · Zbl 1357.39007 · doi:10.1093/imrn/rnu137
[6] DOI: 10.5802/aif.2937 · Zbl 1331.39006 · doi:10.5802/aif.2937
[7] Di Vizio L., Gaz. Math. (96) pp 20– (2003)
[8] DOI: 10.5802/aif.2433 · Zbl 1175.34111 · doi:10.5802/aif.2433
[9] DOI: 10.1002/mana.19490020604 · Zbl 0033.05703 · doi:10.1002/mana.19490020604
[10] Malgrange B., Sommation des séries divergentes. Exposition. Math. 13 (2) pp 163– (1995) · Zbl 0836.40004
[11] DOI: 10.5802/aif.1809 · Zbl 1063.39001 · doi:10.5802/aif.1809
[12] Praagman C., J. Reine Angew. Math. 369 pp 101– (1986)
[13] Ramis J.-P., C. R. Acad. Sci. Paris Sér. I Math. 301 (5) pp 165– (1985)
[14] DOI: 10.5802/afst.739 · Zbl 0796.39005 · doi:10.5802/afst.739
[15] Ramis J.-P., Astérisque 323 pp 301– (2009)
[16] Ramis J.-P., Astérisque 355 pp vi+151–
[17] DOI: 10.1016/S1631-073X(02)02586-4 · Zbl 1025.39014 · doi:10.1016/S1631-073X(02)02586-4
[18] DOI: 10.5802/aif.1784 · Zbl 0957.05012 · doi:10.5802/aif.1784
[19] DOI: 10.5802/afst.1164 · Zbl 1213.39011 · doi:10.5802/afst.1164
[20] DOI: 10.1007/978-3-642-55750-7 · doi:10.1007/978-3-642-55750-7
[21] DOI: 10.5802/aif.1672 · Zbl 0974.39009 · doi:10.5802/aif.1672
[22] DOI: 10.1016/S0764-4442(00)00327-X · Zbl 1101.33307 · doi:10.1016/S0764-4442(00)00327-X
[23] DOI: 10.1007/PL00000144 · Zbl 0990.39018 · doi:10.1007/PL00000144
[24] DOI: 10.1016/S0021-9045(03)00073-X · Zbl 1023.33012 · doi:10.1016/S0021-9045(03)00073-X
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