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In this article, we prove a generalized Rodrigues formula for a wide class of holonomic Laurent series, which yields a new linear independence criterion concerning their values at algebraic points. This generalization yields a new construction of Padé approximations including those for Gauss hypergeometric functions. In particular, we obtain a linear independence criterion over a number field concerning values of Gauss hypergeometric functions, allowing the parameters of Gauss hypergeometric functions to vary.

In this paper, we derive a number of explicit lower bounds for rational approximation to certain cubic irrationalities, proving, for example, that for any non-zero integers p and q. A number of these irrationality measures improve known results, including those for . Some Diophantine consequences are briefly discussed.

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RODRIGUES FORMULA AND LINEAR INDEPENDENCE FOR VALUES OF HYPERGEOMETRIC FUNCTIONS WITH VARYING PARAMETERS

Effective measures of irrationality for certain algebraic numbers

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RODRIGUES FORMULA AND LINEAR INDEPENDENCE FOR VALUES OF HYPERGEOMETRIC FUNCTIONS WITH VARYING PARAMETERS

Effective measures of irrationality for certain algebraic numbers