q-Analogues of Laplace and Borel transforms by means of q-exponentials
Annales de l'Institut Fourier, Volume 67 (2017) no. 5, p. 1865-1903

The article discusses certain q-analogues of Laplace and Borel transforms, and shows a new inversion formula between q-Laplace and q-Borel transforms. q-Analogues of Watson type lemma and convolution operators are also discussed. These results give a new framework of the summability of formal power series solutions of q-difference equations.

Nous considérons certaines q-analogues des transformées de Laplace et Borel et montrons une nouvelle formule d’inversion entre les transformées de q-Laplace et de q-Borel. Des q-analogues des lemmes de type Watson et des opérateurs de convolution sont aussi discutés. Ces résultats donnent un nouveau cadre pour la sommabilité des séries formelles qui sont solutions d’équations aux q-différences.

Received : 2015-01-23
Revised : 2016-12-05
Accepted : 2016-12-08
Published online : 2017-11-17
DOI : https://doi.org/10.5802/aif.3124
Classification:  44A10,  39A13,  40G10
Keywords: q-analogue, q-Laplace transform, q-Borel transform, q-difference equation 
@article{AIF_2017__67_5_1865_0,
     author = {Tahara, Hidetoshi},
     title = {$q$-Analogues of Laplace and Borel transforms by means of $q$-exponentials},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {5},
     year = {2017},
     pages = {1865-1903},
     doi = {10.5802/aif.3124},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2017__67_5_1865_0}
}

Tahara, Hidetoshi. $q$-Analogues of Laplace and Borel transforms by means of $q$-exponentials. Annales de l'Institut Fourier, Volume 67 (2017) no. 5, pp. 1865-1903. doi : 10.5802/aif.3124. http://www.numdam.org/item/AIF_2017__67_5_1865_0/

[1] Abdi, Wazir Hasan On q-Laplace transforms, Proc. Natl. Acad. Sci. India, Sect. A, Tome 29 (1960), pp. 389-408 | Zbl 0168.38303

[2] Abdi, Wazir Hasan On certain q-difference equations and q-Laplace transform, Proc. Natl. Acad. Sci. India, Sect. A, Tome 28 (1962), pp. 1-15 | Zbl 0105.30601

[3] Di Vizio, Lucia; Zhang, Changgui On q-summation and confluence, Ann. Inst. Fourier, Tome 59 (2009) no. 1, pp. 347-392 | Article | Zbl 1175.34111

[4] Dreyfus, Thomas Building meromorphic solutions of q-difference equations using a Borel-Laplace summation, Int. Math. Res. Not., Tome 2015 (2015) no. 15, pp. 6562-6587 | Article | Zbl 1357.39007

[5] Gasper, George; Rahman, Mizan Basic hypergeometric series, Cambridge University Press, Encyclopedia of Mathematics and Its Applications, Tome 96 (2004), xxvi+428 pages | Zbl 1129.33005

[6] Hahn, Wolfgang Beitrage zur Theorie der Heineschen Reihen, Math. Nachr., Tome 2 (1949), pp. 340-379 | Article | Zbl 0033.05703

[7] Jackson, Frederick H. On q-definite integrals, Quart. J., Tome 41 (1910), pp. 193-203 | Zbl 41.0317.04

[8] Kac, Victor; Cheung, Pokman Quantum calculus, Springer, Universitext (2002), ix+112 pages | Zbl 0986.05001

[9] Lastra, Alberto; Malek, Stéphane On q-Gevrey asymptotics for singularly perturbed q-difference-differential problems with an irregular singularity, Abstr. Appl. Anal., Tome 2012 (2012) (Art. ID 860716, 35 p.) | Article | Zbl 1239.34104

[10] Lastra, Alberto; Malek, Stéphane; Sanz, Javier On q-asymptotics for linear q-difference-differential equations with Fuchsian and irregular singularities, J. Differ. Equations, Tome 252 (2012) no. 10, pp. 5185-5216 | Article | Zbl 1251.35016

[11] Malek, Stéphane On singularly perturbed q-difference-differential equations with irregular singularity, J. Dyn. Control Syst., Tome 17 (2011) no. 2, pp. 243-271 | Article | Zbl 1219.35020

[12] Marotte, Fabienne; Zhang, Changgui Multisommabilité des séries entières solutions formelles d’une équation aux q-différences linéaire analytique, Ann. Inst. Fourier, Tome 50 (2000) no. 6, pp. 1859-1890 | Article | Zbl 1063.39001

[13] Olde Daalhuis, Adri B. Asymptotic expansions for q-gamma, q-exponential, and q-Bessel functions, J. Math. Anal. Appl., Tome 186 (1994) no. 3, pp. 896-913 | Article | Zbl 0809.33008

[14] Olver, Frank William John Asymptotics and special functions, Academic Press (1974), xii+297 pages | Zbl 0308.41023

[15] Ramis, Jean-Pierre About the growth of the entire functions solutions of linear algebraic q-difference equations, Ann. Fac. Sci. Toulouse, Tome 1 (1992) no. 1, pp. 53-94 | Article | Zbl 0792.39005

[16] Ramis, Jean-Pierre Séries divergentes et théories asymptotiques, Société Mathématique de France, Panoramas et Synthèses, Tome 121 (1993), 74 pages | Zbl 0830.34045

[17] Ramis, Jean-Pierre; Zhang, Changgui Développement asymptotique q-Gevrey et fonction thêta de Jacobi, C. R. Math. Acad. Sci. Paris, Tome 335 (2002) no. 11, pp. 899-902 | Article | Zbl 1025.39014

[18] Tahara, Hidetoshi; Yamazawa, Hiroshi q-Analogue of summability of formal solutions of some linear q-difference-differential equations, Opusc. Math., Tome 35 (2015) no. 5, pp. 713-738 | Article | Zbl 1329.35104

[19] Zhang, Changgui Développements asymptotiques q-Gevrey et séries Gq-sommables, Ann. Inst. Fourier, Tome 49 (1999) no. 1, pp. 227-261 | Article | Zbl 0974.39009

[20] Zhang, Changgui Transformations de q-Borel-Laplace au moyen de la fonction thêta de Jacobi, C. R. Acad. Sci. Paris, Tome 331 (2000), pp. 31-34 | Article | Zbl 1101.33307

[21] Zhang, Changgui Une sommation discrète pour des équations aux q-différences linéaires et à coefficients analytiques: théorie générale et exemples, Differential equations and the Stokes phenomenon, Groningen (2001), World Scientific (2002), pp. 309-329 | Zbl 1041.39013