# On q-asymptotics for q-difference-differential equations with Fuchsian and irregular singularities

Alberto Lastra; Stéphane Malek; Javier Sanz

Banach Center Publications (2012)

- Volume: 97, Issue: 1, page 73-90
- ISSN: 0137-6934

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topAlberto Lastra, Stéphane Malek, and Javier Sanz. "On q-asymptotics for q-difference-differential equations with Fuchsian and irregular singularities." Banach Center Publications 97.1 (2012): 73-90. <http://eudml.org/doc/282341>.

@article{AlbertoLastra2012,

abstract = {This work is devoted to the study of a Cauchy problem for a certain family of q-difference-differential equations having Fuchsian and irregular singularities. For given formal initial conditions, we first prove the existence of a unique formal power series X̂(t,z) solving the problem. Under appropriate conditions, q-Borel and q-Laplace techniques (firstly developed by J.-P. Ramis and C. Zhang) help us in order to construct actual holomorphic solutions of the Cauchy problem whose q-asymptotic expansion in t, uniformly for z in the compact sets of ℂ, is X̂(t,z). The small divisors phenomenon owing to the Fuchsian singularity causes an increase in the order of q-exponential growth and the appearance of a subexponential Gevrey growth in the asymptotics.},

author = {Alberto Lastra, Stéphane Malek, Javier Sanz},

journal = {Banach Center Publications},

keywords = {q-difference-differential equations; q-Laplace transform; formal power series solutions; q-Gevrey asymptotic expansions; small divisors; Fuchsian singularities; irregular singularities},

language = {eng},

number = {1},

pages = {73-90},

title = {On q-asymptotics for q-difference-differential equations with Fuchsian and irregular singularities},

url = {http://eudml.org/doc/282341},

volume = {97},

year = {2012},

}

TY - JOUR

AU - Alberto Lastra

AU - Stéphane Malek

AU - Javier Sanz

TI - On q-asymptotics for q-difference-differential equations with Fuchsian and irregular singularities

JO - Banach Center Publications

PY - 2012

VL - 97

IS - 1

SP - 73

EP - 90

AB - This work is devoted to the study of a Cauchy problem for a certain family of q-difference-differential equations having Fuchsian and irregular singularities. For given formal initial conditions, we first prove the existence of a unique formal power series X̂(t,z) solving the problem. Under appropriate conditions, q-Borel and q-Laplace techniques (firstly developed by J.-P. Ramis and C. Zhang) help us in order to construct actual holomorphic solutions of the Cauchy problem whose q-asymptotic expansion in t, uniformly for z in the compact sets of ℂ, is X̂(t,z). The small divisors phenomenon owing to the Fuchsian singularity causes an increase in the order of q-exponential growth and the appearance of a subexponential Gevrey growth in the asymptotics.

LA - eng

KW - q-difference-differential equations; q-Laplace transform; formal power series solutions; q-Gevrey asymptotic expansions; small divisors; Fuchsian singularities; irregular singularities

UR - http://eudml.org/doc/282341

ER -

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