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In this article, we prove a generalized Rodrigues formula for a wide class ofholonomic Laurent series, which yields a new linear independence criterionconcerning their values at algebraic points. This generalization yields a newconstruction of Pad\'e approximations including those for Gauss hypergeometricfunctions. In particular, we obtain a linear independence criterion over anumber field concerning values of Gauss hypergeometric functions, allowing theparameters of Gauss hypergeometric functions to vary.

Rodrigues formula and linear independence for values of hypergeometric functions with parameters vary