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A new algorithm for the recursion of hypergeometric multisums with improved universal denominator. (English) Zbl 1207.33025

Amdeberhan, Tewodros (ed.) et al., Gems in experimental mathematics. AMS special session on experimental mathematics, Washington, DC, January 5, 2009. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4869-2/pbk). Contemporary Mathematics 517, 143-156 (2010).
Summary: The purpose of the paper is to introduce two new algorithms. The first algorithm computes a linear recursion for proper hypergeometric multisums, by treating one summation variable at a time, and provides rational certificates along the way. A key part in the search for a linear recursion is an improved second universal denominator algorithm that constructs all rational solutions \(x(n)\) of the equation
\[ \frac{a_m(n)}{b_m(n)} x(n+m)+\cdots+ \frac{a_0(n)}{b_0(n)} x(n)=c(n), \]
where \(a_i(n)\), \(b_i(n)\), \(c(n)\) are polynomials. Our second algorithm improves Abramov’s universal denominator.
For the entire collection see [Zbl 1193.00060].

MSC:

33F10 Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.)
05E99 Algebraic combinatorics

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