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Parameterized Telescoping Proves Algebraic Independence of Sums
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Usually creative telescoping is used to derive recurrences for sums. In this article we show that the non-existence of a creative telescoping solution, and more generally, of a parameterized telescoping solution, proves algebraic independence of certain types of sums. Combining this fact with summation-theory shows transcendence of whole classes of sums. Moreover, this result throws new light on the question why, e.g., Zeilberger's algorithm fails to find a recurrence with minimal order.
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Telescoping Algorithms for $\Sigma^*$-Extensions via Complete Reductions
Complete reductions can be built recursively in towers of Sigma*-extensions, yielding faster refined and parameterized telescoping algorithms for nested harmonic sums.
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