Transcendence of Binomial and Lucas' Formal Power Series |
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Authors: | J-P Allouche D Gouyou-Beauchamps G Skordev |
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Institution: | aCNRS and Université Paris-Sud, LRI, Bâtiment 490, F-91405, Orsay Cedex, France;bCEVIS, Universität Bremen, Universitätsallee 29, D-28359, Bremen, Germany |
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Abstract: | The formal power seriesformula]is transcendental over (X) whentis an integer ≥ 2. This is due to Stanley forteven, and independently to Flajolet and to Woodcock and Sharif for the general case. While Stanley and Flajolet used analytic methods and studied the asymptotics of the coefficients of this series, Woodcock and Sharif gave a purely algebraic proof. Their basic idea is to reduce this series modulo prime numbersp, and to use thep-Lucas property: ifn = ∑nipiis the basepexpansion of the integern, thenequation]The series reduced modulopis then proved algebraic over p(X), the field of rational functions over the Galois field p, but its degree is not a bounded function ofp. We generalize this method to characterize all formal power series that have thep-Lucas property for “many” prime numbersp, and that are furthermore algebraic over (X). |
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Keywords: | transcendence of formal power series binomial coefficients Lucas' property Legendre polynomials |
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