We consider weighted small step walks in the positive quadrant, and providealgebraicity and differential transcendence results for the underlyinggenerating functions: we prove that depending on the probabilities of allowedsteps, certain of the generating series are algebraic over the field ofrational functions, while some others do not satisfy any algebraic differentialequation with rational functions coefficients. Our techniques involvedifferential Galois theory for difference equations as well as complex analysis(Weierstrass parameterization of elliptic curves). We also extend to theweighted case many key intermediate results, as a theorem of analyticcontinuation of the generating functions.
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