Differential transcendence criteria for second-order linear difference equations and elliptic hypergeometric functions
Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 147-168.

We develop general criteria that ensure that any non-zero solution of a given second-order difference equation is differentially transcendental, which apply uniformly in particular cases of interest, such as shift difference equations, q-dilation difference equations, Mahler difference equations, and elliptic difference equations. These criteria are obtained as an application of differential Galois theory for difference equations. We apply our criteria to prove a new result to the effect that most elliptic hypergeometric functions are differentially transcendental.

Dans cet article, nous développons des critères généraux garantissant la transcendance différentielle d’une solution non nulle donnée d’une équation aux différences du deuxième ordre. Ces critères s’appliquent à de nombreuses équations, telles que les équations aux différences finies, les équations aux q-différences, les équations de Mahler, ou encore les équations aux différences elliptiques. Notre approche repose sur la théorie de Galois des équations aux différences. En guise d’application, nous démontrons que la plupart des fonctions hypergéométriques elliptiques sont différentiellement transcendantes.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.143
Classification: 39A06,  12H05
Keywords: Linear difference equations, difference Galois theory, elliptic curves, differential algebra
Carlos E. Arreche 1; Thomas Dreyfus 2; Julien Roques 3

1 The University of Texas at Dallas, Mathematical Sciences FO 35 800 West Campbell Road, Richardson, TX 75024, USA
2 Institut de Recherche Mathématique Avancée, U.M.R. 7501 Université de Strasbourg et C.N.R.S. 7, rue René Descartes 67084 Strasbourg, France
3 Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne cedex, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{JEP_2021__8__147_0,
     author = {Carlos E. Arreche and Thomas Dreyfus and Julien Roques},
     title = {Differential transcendence criteria for second-order linear difference equations and elliptic hypergeometric functions},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {147--168},
     publisher = {\'Ecole polytechnique},
     volume = {8},
     year = {2021},
     doi = {10.5802/jep.143},
     mrnumber = {4201803},
     zbl = {07315954},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.143/}
}
TY  - JOUR
TI  - Differential transcendence criteria for second-order linear difference equations and elliptic hypergeometric functions
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2021
DA  - 2021///
SP  - 147
EP  - 168
VL  - 8
PB  - École polytechnique
UR  - https://jep.centre-mersenne.org/articles/10.5802/jep.143/
UR  - https://www.ams.org/mathscinet-getitem?mr=4201803
UR  - https://zbmath.org/?q=an%3A07315954
UR  - https://doi.org/10.5802/jep.143
DO  - 10.5802/jep.143
LA  - en
ID  - JEP_2021__8__147_0
ER  - 
%0 Journal Article
%T Differential transcendence criteria for second-order linear difference equations and elliptic hypergeometric functions
%J Journal de l’École polytechnique — Mathématiques
%D 2021
%P 147-168
%V 8
%I École polytechnique
%U https://doi.org/10.5802/jep.143
%R 10.5802/jep.143
%G en
%F JEP_2021__8__147_0
Carlos E. Arreche; Thomas Dreyfus; Julien Roques. Differential transcendence criteria for second-order linear difference equations and elliptic hypergeometric functions. Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 147-168. doi : 10.5802/jep.143. https://jep.centre-mersenne.org/articles/10.5802/jep.143/

[Arr17] C. E. Arreche - “Computation of the difference-differential Galois group and differential relations among solutions for a second-order linear difference equation”, Commun. Contemp. Math. 19 (2017) no. 6, article ID 1650056, 42 pages | DOI | MR | Zbl

[AS17] C. E. Arreche & M. F. Singer - “Galois groups for integrable and projectively integrable linear difference equations”, J. Algebra 480 (2017), p. 423-449 | DOI | MR | Zbl

[DHR18] T. Dreyfus, C. Hardouin & J. Roques - “Hypertranscendence of solutions of Mahler equations”, J. Eur. Math. Soc. (JEMS) 20 (2018) no. 9, p. 2209-2238 | DOI | MR | Zbl

[DHR21] T. Dreyfus, C. Hardouin & J. Roques - “Functional relations of solutions of q-difference equations”, Math. Z. (2021), doi:10.1007/s00209-020-02669-4 | DOI

[DHRS18] T. Dreyfus, C. Hardouin, J. Roques & M. F. Singer - “On the nature of the generating series of walks in the quarter plane”, Invent. Math. 213 (2018) no. 1, p. 139-203 | DOI | MR | Zbl

[DHRS20] T. Dreyfus, C. Hardouin, J. Roques & M. F. Singer - “Walks in the quarter plane: genus zero case”, J. Combin. Theory Ser. A 174 (2020), p. 105251, 25 | DOI | MR | Zbl

[DR15] T. Dreyfus & J. Roques - “Galois groups of difference equations of order two on elliptic curves”, SIGMA Symmetry Integrability Geom. Methods Appl. 11 (2015), article ID 003, 23 pages | DOI | MR | Zbl

[DR19] T. Dreyfus & K. Raschel - “Differential transcendence & algebraicity criteria for the series counting weighted quadrant walks”, Publ. Math. Besançon (2019) no. 1, p. 41-80 | DOI | Zbl

[Hen97] P. A. Hendriks - “An algorithm for computing a standard form for second-order linear q-difference equations”, J. Pure Appl. Algebra 117/118 (1997), p. 331-352, Algorithms for algebra (Eindhoven, 1996) | DOI | MR | Zbl

[Hen98] P. A. Hendriks - “An algorithm determining the difference Galois group of second order linear difference equations”, J. Symbolic Comput. 26 (1998) no. 4, p. 445-461 | DOI | MR | Zbl

[HS08] C. Hardouin & M. F. Singer - “Differential Galois theory of linear difference equations”, Math. Ann. 342 (2008) no. 2, p. 333-377, Erratum: Ibid. 350 (2011), no. 1, p. 243–244 | DOI | MR | Zbl

[Höl86] O. Hölder - “Ueber die Eigenschaft der Gammafunction keiner algebraischen Differentialgleichung zu genügen”, Math. Ann. 28 (1886), p. 1-13 | DOI | Zbl

[Kol73] E. R. Kolchin - Differential algebra and algebraic groups, Pure and Applied Math., vol. 54, Academic Press, New York-London, 1973 | MR | Zbl

[Kol74] E. R. Kolchin - “Constrained extensions of differential fields”, Adv. Math. 12 (1974), p. 141-170 | DOI | MR | Zbl

[Mag09] A. P. Magnus - “Elliptic hypergeometric solutions to elliptic difference equations”, SIGMA Symmetry Integrability Geom. Methods Appl. 5 (2009), article ID 038, 12 pages | DOI | MR | Zbl

[Mum07] D. Mumford - Tata lectures on theta. I, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2007, Reprint of the 1983 edition | DOI | Zbl

[Rai10] E. M. Rains - “Transformations of elliptic hypergeometric integrals”, Ann. of Math. (2) 171 (2010) no. 1, p. 169-243 | DOI | MR | Zbl

[Roq11] J. Roques - “Generalized basic hypergeometric equations”, Invent. Math. 184 (2011) no. 3, p. 499-528 | DOI | MR | Zbl

[Roq18] J. Roques - “On the algebraic relations between Mahler functions”, Trans. Amer. Math. Soc. 370 (2018) no. 1, p. 321-355 | DOI | MR | Zbl

[RS20] H. Rosengren & M. J. Schlosser - “Multidimensional matrix inversions and elliptic hypergeometric series on root systems”, SIGMA Symmetry Integrability Geom. Methods Appl. 16 (2020), article ID 088, 21 pages | DOI | MR

[Spi16] V. P. Spiridonov - “Elliptic hypergeometric functions”, 2016 | arXiv

[vdB + 07] F. J. van de Bult et al. - Hyperbolic hypergeometric functions, University of Amsterdam, Amsterdam Netherlands, 2007

[vdBR09] F. J. van de Bult & E. M. Rains - “Basic hypergeometric functions as limits of elliptic hypergeometric functions”, SIGMA Symmetry Integrability Geom. Methods Appl. 5 (2009), article ID 059, 31 pages | DOI | MR | Zbl

[vdPS97] M. van der Put & M. F. Singer - Galois theory of difference equations, Lect. Notes in Math., vol. 1666, Springer-Verlag, Berlin, 1997 | DOI | MR | Zbl

Cited by Sources: