[Critères de transcendance différentielle pour les équations aux différences du deuxième ordre et les fonctions hypergéométriques elliptiques]
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 -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.
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, -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.
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DOI : 10.5802/jep.143
Keywords: Linear difference equations, difference Galois theory, elliptic curves, differential algebra
Mot clés : Équations aux différences linéaires, théorie de Galois aux différences, courbe elliptiques, algèbre différentielle
Carlos E. Arreche 1 ; Thomas Dreyfus 2 ; Julien Roques 3
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@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 AU - Carlos E. Arreche AU - Thomas Dreyfus AU - Julien Roques TI - Differential transcendence criteria for second-order linear difference equations and elliptic hypergeometric functions JO - Journal de l’École polytechnique — Mathématiques PY - 2021 SP - 147 EP - 168 VL - 8 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.143/ DO - 10.5802/jep.143 LA - en ID - JEP_2021__8__147_0 ER -
%0 Journal Article %A Carlos E. Arreche %A Thomas Dreyfus %A Julien Roques %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://jep.centre-mersenne.org/articles/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, Tome 8 (2021), pp. 147-168. doi : 10.5802/jep.143. https://jep.centre-mersenne.org/articles/10.5802/jep.143/
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