Fast Algorithms for Refined Parameterized Telescoping in Difference Fields

Parameterized telescoping (including telescoping and creative telescoping) and refined versions of it play a central role in the research area of symbolic summation. Karr introduced 1981 $ΠΣ$-fields, a general class of difference fields, that enables one to consider this problem for indefinite nested sums and products covering as special cases, e.g., the ($q$--)hypergeometric case and their mixed versions. This survey article presents the available algorithms in the framework of $ΠΣ$-extensions and elaborates new results concerning efficiency.

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