### Abstract

Let q > 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 X(t; z ) one can construct an actual holomorphic solution which admits X(t; z ) as a q-Gevrey asymptotic expansion of order 1.

Original language | English |
---|---|

Pages (from-to) | 713-738 |

Number of pages | 26 |

Journal | Opuscula Mathematica |

Volume | 35 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2015 |

### Keywords

- Formal power series solutions
- Q-difference-differential equations
- Q-Gevrey asymptotic expansions
- Q-Laplace transform
- Summability

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Q-analogue of summability of formal solutions of some linear q-difference-differential equations.** / Tahara, Hidetoshi; Yamazawa, Hiroshi.

Research output: Contribution to journal › Article

*Opuscula Mathematica*, vol. 35, no. 5, pp. 713-738. https://doi.org/10.7494/OpMath.2015.35.5.713

}

TY - JOUR

T1 - Q-analogue of summability of formal solutions of some linear q-difference-differential equations

AU - Tahara, Hidetoshi

AU - Yamazawa, Hiroshi

PY - 2015

Y1 - 2015

N2 - Let q > 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 X(t; z ) one can construct an actual holomorphic solution which admits X(t; z ) as a q-Gevrey asymptotic expansion of order 1.

AB - Let q > 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 X(t; z ) one can construct an actual holomorphic solution which admits X(t; z ) as a q-Gevrey asymptotic expansion of order 1.

KW - Formal power series solutions

KW - Q-difference-differential equations

KW - Q-Gevrey asymptotic expansions

KW - Q-Laplace transform

KW - Summability

UR - http://www.scopus.com/inward/record.url?scp=84928406636&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84928406636&partnerID=8YFLogxK

U2 - 10.7494/OpMath.2015.35.5.713

DO - 10.7494/OpMath.2015.35.5.713

M3 - Article

AN - SCOPUS:84928406636

VL - 35

SP - 713

EP - 738

JO - Opuscula Mathematica

JF - Opuscula Mathematica

SN - 1232-9274

IS - 5

ER -