Opuscula Math. 35, no. 5 (2015), 713-738
http://dx.doi.org/10.7494/OpMath.2015.35.5.713

Opuscula Mathematica

# q-analogue of summability of formal solutions of some linear q-difference-differential equations

Hidetoshi Tahara
Hiroshi Yamazawa

Abstract. Let $$q\gt 1$$. The paper considers a linear $$q$$-difference-differential equation: it is a $$q$$-difference equation in the time variable $$t$$, and a partial differential equation in the space variable $$z$$. Under suitable conditions and by using $$q$$-Borel and $$q$$-Laplace transforms (introduced by J.-P. Ramis and C. Zhang), the authors show that if it has a formal power series solution $$\hat{X}(t,z)$$ one can construct an actual holomorphic solution which admits $$\hat{X}(t,z)$$ as a $$q$$-Gevrey asymptotic expansion of order $$1$$.

Keywords: $$q$$-difference-differential equations, summability, formal power series solutions, $$q$$-Gevrey asymptotic expansions, $$q$$-Laplace transform.

Mathematics Subject Classification: 35C10, 35C20, 39A13.

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• Hidetoshi Tahara
• Sophia University, Department of Information and Communication Sciences, Kioicho, Chiyoda-ku, Tokyo 102-8554, Japan
• Hiroshi Yamazawa
• Shibaura Institute of Technology, College of Engineer and Design, Minuma-ku, Saitama-shi, Saitama 337-8570, Japan
• Communicated by P.A. Cojuhari.
• Revised: 2014-07-05.
• Accepted: 2014-07-27.
• Published online: 2015-04-27.