Simplifying Sums in Πς∗-extensions
نویسنده
چکیده
We present algorithms which split a rational expression in terms of indefinite nested sums and products into a summable part which can be summed by telescoping and into a non-summable part which is degree-optimal with respect to one of the most nested sums or products. If possible, our algorithms find a non-summable part where all these most nested sums and products are eliminated.
منابع مشابه
Simplifying Sums in Πς ∗ - Extensions Carsten
We present telescoping algorithms which compute optimal sum representations of indefinite nested sums. More precisely, given a rational summand expression in terms of nested sums and products, the algorithm splits the summand into a summable part, which can be summed by telescoping, and into a non-summable part, which is degree-optimal with respect to one of the most nested sums or products. If...
متن کاملResolution of Sign Ambiguities in Jacobi and Jacobsthal Sums
Let p be a prime = 1 (mod 16). We obtain extensions of known congruences involving parameters of bioctic Jacobi sums (modp). These extensions are used to give an elementary proof of an important congruence of Ήasse relating parameters of quartic and octic Jacobi sums (mod p). This proof leads directly to an elementary resolution of sign ambiguities of parameters of certain quartic, octic, and b...
متن کاملSummation Theory II: Characterizations of $\boldsymbol{R\Pi\Sigma^*}$-extensions and algorithmic aspects
Recently, RΠΣ∗-extensions have been introduced which extend Karr’s ΠΣ∗-fields substantially: one can represent expressions not only in terms of transcendental sums and products, but one can work also with products over primitive roots of unity. Since one can solve the parameterized telescoping problem in such rings, covering as special cases the summation paradigms of telescoping and creative t...
متن کاملDiscrete Analogues in Harmonic Analysis , I : ` 2 Estimates for Singular Radon Transforms
This paper studies the discrete analogues of singular Radon transforms. We prove the `2 boundedness for those operators that are “quasi-translation-invariant.” The approach used is related to the “circle-method” of Hardy and Littlewood, and requires multi-dimensional extensions of Weyl sums and Gauss sums, as well as variants that replace scalar sums by operator sums.
متن کاملGeneralized Harmonic, Cyclotomic, and Binomial Sums, their Polylogarithms and Special Numbers
A survey is given on mathematical structures which emerge in multi-loop Feynman diagrams. These are multiply nested sums, and, associated to them by an inverse Mellin transform, specific iterated integrals. Both classes lead to sets of special numbers. Starting with harmonic sums and polylogarithms we discuss recent extensions of these quantities as cyclotomic, generalized (cyclotomic), and bin...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2006