On sequences preserving q-Gevrey asymptotic expansions
Tóm tắt
The modification of the coefficients of formal power series is analyzed in order that such variation preserves q-Gevrey asymptotic properties, in particular q-Gevrey asymptotic expansions. A characterization of such sequences is determined, providing a handy tool in practice. The sequence of q-factorials is proved to preserve q-Gevrey asymptotic expansions.
Từ khóa
Tài liệu tham khảo
Balser, W.: Formal power series and linear systems of meromorphic ordinary differential equations. In: Universitext. Springer, New York (2000)
Balser, W., Yoshino, M.: Gevrey order of formal power series solutions of inhomogeneous partial differential equations with constant coefficients. Funkcial. Ekvac. 53(3), 411–434 (2010)
di Vizio, L., Zhang, C.: On \(q\)-summation and confluence. Ann. Inst. Fourier 59(1), 347–392 (2009)
di Vizio, L., Ramis, J.-P., Sauloy, J., Zhang, C.: Équations aux \(q\)-différences. Gaz. Math. Soc. Math. Fr. 96, 20–49 (2003)
Dreyfus, T.: Building meromorphic solutions of q-difference equations using a Borel–Laplace summation. Int. Math. Res. Not. 2015(15), 6562–6587 (2015)
Dreyfus, T., Eloy, A.: \(q\)-Borel–Laplace summation for \(q\)-difference equations with two slopes. J. Differ. Equ. Appl. 22(10), 1501–1511 (2016)
Dreyfus, T., Lastra, A., Malek, S.: On the multiple-scale analysis for some linear partial q-difference and differential equations with holomorphic coefficients. Adv. Differ. Equ. 2019, 1–42 (2019)
Ernst, T.: A Comprehensive Treatment of \(q\)-Calculus. Birkhäuser, Basel (2012)
Gasper, G., Rahman, M.: Basic Hypergeometric Series. Cambridge University Press, Cambridge (2004)
Ichinobe, K., Michalik, S.: On the summability and convergence of formal solutions of linear \(q\)-difference–differential equations with constant coefficients. Math. Ann. (2023). https://doi.org/10.1007/s00208-023-02672-0
Jiménez-Garrido, J., Kamimoto, S., Lastra, A., Sanz, J.: Multisummability in Carleman ultraholomorphic classes by means of nonzero proximate orders. J. Math. Anal. Appl. 472(1), 627–686 (2019)
Lastra, A., Malek, S.: On \(q\)-Gevrey asymptotics for singularly perturbed \(q\)-difference-differential problems with an irregular singularity. Abstr. Appl. Anal. 2012, 860716 (2012)
Lastra, A., Malek, S.: On parametric multilevel \(q\)-Gevrey asymptotics for some linear \(q\)-difference-differential equations. Adv. Differ. Equ. 2015, 1–52 (2015)
Lastra, A., Malek, S.: On multiscale Gevrey and \(q\)-Gevrey asymptotics for some linear \(q\)-difference differential initial value Cauchy problems. J. Differ. Equ. Appl. 23(8), 1397–1457 (2017)
Lastra, A., Malek, S., Sanz, J.: On \(q\)-asymptotics for linear \(q\)-difference–differential equations with Fuchsian and irregular singularities. J. Differ. Equ. 252(10), 5185–5216 (2012)
Lastra, A., Malek, S., Sanz, J.: Strongly regular multi-level solutions of singularly perturbed linear partial differential equations. Results Math. 70(3–4), 581–614 (2016)
Lastra, A., Michalik, S., Suwińska, M.: Summability of formal solutions for some generalized moment partial differential equations. Results Math. 76(1), 22 (2021)
Lastra, A., Michalik, S., Suwińska, M.: Estimates of formal solutions for some generalized moment partial differential equations. J. Math. Anal. Appl. 500(1), 125094 (2021)
Lastra, A., Michalik, S., Suwińska, M.: Summability of formal solutions for a family of generalized moment integro-differential equations. Fract. Calc. Appl. Anal. 24, 1445–1476 (2021)
Lastra, A., Michalik, S., Suwińska, M.: Multisummability of formal solutions of a family of generalized singularly perturbed moment differential equations. Results Math. 78(1), 49 (2023)
Loday-Richaud, M.: Divergent series, summability and resurgence. II. In: Simple and Multiple Summability, Lecture Notes in Mathematics, vol. 2154. Springer (2016)
Malek, S.: On parametric Gevrey asymptotics for a \(q-\)analog of some linear initial value problem. Funkcial. Ekvac. 60(1), 21–63 (2017)
Malek, S.: On a partial \(q\)-analog of a singularly perturbed problem with Fuchsian and irregular time singularities. Abstr. Appl. Anal. 2020, 1–32 (2020)
Malek, S.: Asymptotics and confluence for some linear \(q\)-difference–differential Cauchy problem. J. Geom. Anal. 32(3), 93 (2022)
Marotte, F., Zhang, C.: Multisommabilité des séries entières solutions formelles d’une équation aux \(q-\)différences linéaire analytique. Ann. Inst. Fourier 50(6), 1859–1890 (2000)
Ramis, J.-P., Sauloy, J., Zhang, C.: Local analytic classification of \(q\)-difference equations. In: Astérisque 355. Société Mathématique de France (SMF) VI, Paris (2013)
Ramis, J.-P., Sauloy, J., Zhang, C.: Développement asymptotique et sommabilité des solutions des équations linéaires aux \(q\)-différences. C. R. Math. Acad. Sci. Paris 342(7), 515–518 (2006)
Ruan, Y.B., Wen, Y.X.: Quantum K-theory and q-difference equations. Acta Math. Sin. Engl. Ser. 38, 1677–1704 (2022)
Sanz, J.: Asymptotic analysis and summability of formal power series. In: Analytic, Algebraic and Geometric Aspects of Differential Equations, pp. 199–262. Trends Math, Birkhäuser/Springer, Cham (2017)
Tahara, H.: \(q\)-analogues of Laplace and Borel transforms by means of \(q\)-exponentials. Ann. Inst. Fourier 67(5), 1865–1903 (2017)
Tahara, H.: On the summability of formal solutions of some linear \(q\)-difference–differential equations. Funkc. Ekvacioj 63(2), 259–291 (2020)
Tahara, H., Yamazawa, H.: \(q\)-analogue of summability of formal solutions of some linear \(q\)-difference–differential equations. Opuscula Math. 35(5), 713–738 (2015)
Thilliez, V.: Division by flat ultradifferentiable functions and sectorial extensions. Results Math. 44, 169–188 (2003)
Vinod, G.: Quantum groups, \(q\)-oscillators and \(q\)-deformed quantum mechanics. In: Sreelatha, K.S., Jacob, V. (eds.) Modern Perspectives in Theoretical Physics. Springer, Singapore (2021)
Zhang, C.: Transformations de \(q\)-Borel–Laplace au moyen de la fonction thêta de Jacobi. C R Acad Sci Paris Sér I Math 331(1), 31–34 (2000)
Zhang, C.: On Jackson’s \(q\)-gamma function. Aequationes Math. 62(1–2), 60–78 (2001)