On q -summation and confluence

Lucia Di Vizio[1]; Changgui Zhang[2]

  • [1] Institut de Mathématiques de Jussieu Topologie et géométrie algébriques, Case 7012 2, place Jussieu 75251 Paris Cedex 05 (France)
  • [2] Laboratoire P. Painleve U.F.R. de Mathématiques Pures et Appliquées USTL, Cité scientifique 59655 Villeneuve d’Ascq Cedex (France)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 1, page 347-392
  • ISSN: 0373-0956

Abstract

top
This paper is divided in two parts. In the first part we study a convergent q -analog of the divergent Euler series, with q ∈ ( 0 , 1 ) , and we show how the Borel sum of a generic Gevrey formal solution to a differential equation can be uniformly approximated on a convenient sector by a meromorphic solution of a corresponding q -difference equation. In the second part, we work under the assumption q ∈ ( 1 , + ∞ ) . In this case, at least four different q -Borel sums of a divergent power series solution of an irregular singular analytic q -difference equations are spread in the literature: under convenient assumptions we clarify the relations among them.

How to cite

top

Di Vizio, Lucia, and Zhang, Changgui. "On $q$-summation and confluence." Annales de l’institut Fourier 59.1 (2009): 347-392. <http://eudml.org/doc/10395>.

@article{DiVizio2009,
abstract = {This paper is divided in two parts. In the first part we study a convergent $q$-analog of the divergent Euler series, with $q\in (0,1)$, and we show how the Borel sum of a generic Gevrey formal solution to a differential equation can be uniformly approximated on a convenient sector by a meromorphic solution of a corresponding $q$-difference equation. In the second part, we work under the assumption $q\in (1,+\infty )$. In this case, at least four different $q$-Borel sums of a divergent power series solution of an irregular singular analytic $q$-difference equations are spread in the literature: under convenient assumptions we clarify the relations among them.},
affiliation = {Institut de Mathématiques de Jussieu Topologie et géométrie algébriques, Case 7012 2, place Jussieu 75251 Paris Cedex 05 (France); Laboratoire P. Painleve U.F.R. de Mathématiques Pures et Appliquées USTL, Cité scientifique 59655 Villeneuve d’Ascq Cedex (France)},
author = {Di Vizio, Lucia, Zhang, Changgui},
journal = {Annales de l’institut Fourier},
keywords = {Summation; confluence; $q$-difference equations; Euler series; summation; -difference equations},
language = {eng},
number = {1},
pages = {347-392},
publisher = {Association des Annales de l’institut Fourier},
title = {On $q$-summation and confluence},
url = {http://eudml.org/doc/10395},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Di Vizio, Lucia
AU - Zhang, Changgui
TI - On $q$-summation and confluence
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 1
SP - 347
EP - 392
AB - This paper is divided in two parts. In the first part we study a convergent $q$-analog of the divergent Euler series, with $q\in (0,1)$, and we show how the Borel sum of a generic Gevrey formal solution to a differential equation can be uniformly approximated on a convenient sector by a meromorphic solution of a corresponding $q$-difference equation. In the second part, we work under the assumption $q\in (1,+\infty )$. In this case, at least four different $q$-Borel sums of a divergent power series solution of an irregular singular analytic $q$-difference equations are spread in the literature: under convenient assumptions we clarify the relations among them.
LA - eng
KW - Summation; confluence; $q$-difference equations; Euler series; summation; -difference equations
UR - http://eudml.org/doc/10395
ER -

References

top
  1. G. E. Andrew, R. Askey, R. Roy, Special functions, 71 (1999), Cambridge University Press, Cambridge Zbl0920.33001MR1688958
  2. Y. André, Séries Gevrey de type arithmétique. I. Théorèmes de pureté et de dualité, Ann. of Math. 151 (2000), 705-740 Zbl1037.11049MR1765707
  3. E. W. Barnes, A new development of the theory of the hypergeometric functions, Proceedings of the London Mathematical Society 6 (1908), 141-177 Zbl39.0506.01
  4. J.-P. Bézivin, Sur les équations fonctionnelles aux q -différences, Aequationes Mathematicae 43 (1992), 159-176 Zbl0757.39002MR1158724
  5. A. De Sole, V. G. Kac, On integral representations of q -gamma and q -beta functions, Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Serie IX. Matematica e Applicazioni 16 (2005), 11-29 Zbl1225.33017MR2225920
  6. L. Di Vizio, Arithmetic theory of q -difference equations. The q -analogue of Grothendieck-Katz’s conjecture on p -curvatures, Inventiones Mathematicae 150 (2002), 517-578 Zbl1023.12004MR1946552
  7. Higher transcendental functions, I (1981), ErdélyiA.A., Melbourne, Fla. 
  8. G. Gasper, M. Rahman, Basic hypergeometric series, 35 (1990), Cambridge University Press, Cambridge Zbl0695.33001MR1052153
  9. M. Loday-Richaud, Introduction à la multisommabilité, Gazette des Mathématiciens (1990), 41-63 Zbl0722.34005MR1060425
  10. M. Loday-Richaud, Solutions formelles des systèmes différentiels linéaires méromorphes et sommation, Expositiones Mathematicae. International Journal 13 (1995), 116-162 Zbl0831.34002MR1346200
  11. B. Malgrange, Sommation des séries divergentes, Expositiones Mathematicae. International Journal 13 (1995), 163-222 Zbl0836.40004MR1346201
  12. F. Marotte, C. Zhang, Multisommabilité des séries entières solutions formelles d’une équation aux q -différences linéaire analytique, Annales de l’Institut Fourier 50 (2000), 1859-1890 Zbl1063.39001MR1817385
  13. J.-P. Ramis, About the growth of entire functions solutions of linear algebraic q -difference equations, Toulouse. Faculté des Sciences. Annales. Mathématiques. Série 6 1 (1992), 53-94 Zbl0796.39005MR1191729
  14. J.-P. Ramis, J. Martinet, Théorie de Galois différentielle et resommation, (1990), 117-214, Academic Press Zbl0722.12007MR1038060
  15. J.-P. Ramis, C. Zhang, Développement asymptotique q -Gevrey et fonction thêta de Jacobi, C. R., Math. 335 (2002), 899-902 Zbl1025.39014MR1952546
  16. J. Sauloy, Systèmes aux q -différences singuliers réguliers: classification, matrice de connexion et monodromie, Annales de l’Institut Fourier 50 (2000), 1021-1071 Zbl0957.05012MR1799737
  17. J. Sauloy, La filtration canonique par les pentes d’un module aux q -différences, (2002) Zbl0998.39011
  18. J. Sauloy, La filtration canonique par les pentes d’un module aux q -différences et le gradué associé, Annales de l’Institut Fourier 54 (2004), 181-210 Zbl1061.39013MR2069126
  19. E. T. Whittaker, G. N. Watson, A Course of Modern Analysis, (1988), Cambridge University Press, Cambridge Zbl0951.30002MR1424469
  20. C. Zhang, Développements asymptotiques q -Gevrey et séries G q -sommables, Annales de l’Institut Fourier 49 (1999), 227-261 Zbl0974.39009MR1688144
  21. C. Zhang, Transformations de q -Borel-Laplace au moyen de la fonction thêta de Jacobi, Comptes Rendus de l’Académie des Sciences Série I. Mathématique 331 (2000), 31-34 Zbl1101.33307MR1780181
  22. C. Zhang, Sur la fonction q -gamma de Jackson, Aequationes Mathematicae 4 (2001), 60-78 Zbl0990.39018MR1849140
  23. C. Zhang, Une sommation discrète pour des équations aux q -différences linéaires et à coefficients analytiques: théorie générale et exemples, (2002), 309-329, World Sci. Publishing, River Edge, NJ Zbl1041.39013MR2067338
  24. C. Zhang, Sur les fonctions q -Bessel de Jackson, Journal of Approximation Theory 122 (2003), 208-223 Zbl1023.33012MR1988300

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.