Differential transcendence & algebraicity criteria for the series counting weighted quadrant walks

Dreyfus, Thomas; Raschel, Kilian

Publications mathématiques de Besançon. Algèbre et théorie des nombres, no. 1 (2019), pp. 41-80.

Nous considérons des marches à petits pas avec poids dans le quadrant positif, et nous fournissons des résultats d’algébricité et de transcendance différentielle pour les fonctions génératrices sous-jacentes  : nous prouvons que, selon les probabilités des pas permis, certaines fonctions génératrices sont algébriques sur le corps des fonctions rationnelles, alors que d’autres ne satisfont aucune équation différentielle algébrique. Nos techniques utilisent la théorie de Galois différentielle des équations aux différences ainsi que l’analyse complexe (notamment la paramétrisation de Weierstrass des courbes elliptiques). Nous étendons aussi au cas pondéré plusieurs résultats intermédiaires clé, comme un théorème de prolongement analytique des fonctions génératrices.

We consider weighted small step walks in the positive quadrant, and provide algebraicity and differential transcendence results for the underlying generating functions: we prove that depending on the probabilities of allowed steps, certain of the generating functions are algebraic over the field of rational functions, while some others do not satisfy any algebraic differential equation with rational function coefficients. Our techniques involve differential Galois theory for difference equations as well as complex analysis (Weierstrass parameterization of elliptic curves). We also extend to the weighted case many key intermediate results, as a theorem of analytic continuation of the generating functions.

Classification : 05A15, 30D05, 39A06
Mots clés : Random walks in the quarter plane, Difference Galois theory, Elliptic functions, Transcendence, Algebraicity

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