Symbolic Summation | Some Recent Developments
نویسنده
چکیده
In recent years, the problem of symbolic summation has received much attention due to the exciting applications of Zeilberger’s method for definite hypergeometric summation. This lead to renewed interest in the central part of the algorithmic machinery, Gosper’s “classical” method for indefinite hypergeometric summation. We review some of the recent (partly unpublished as yet) work done in this field, with a particular emphasis on the unifying and guiding rôle of normal forms for polynomials and rational functions, especially adapted to summation algorithms.
منابع مشابه
Symbolic computation and finite element methods
During the past few decades there have been many examples where computer algebra methods have been applied successfully in the analysis and construction of numerical schemes, including the computation of approximate solutions to partial differential equations. The methods range from Gröbner basis computations and Cylindrical Algebraic Decomposition to algorithms for symbolic summation and integ...
متن کاملWhat’s New in Symbolic Summation
Combinatorics is the primary application area of symbolic summation algorithms. The classical hypergeometric summation machinery consisting of Gosper’s algorithm [7], Zeilberger’s algorithm [23, 25], and Petkovšek’s algorithm [16] (see [17, 10, 9] for introductory textbooks covering these algorithms) is nowadays widely known and widely used. Also q-analogs of these algorithms [15, 18, 1, 19], m...
متن کاملSymbolic Summation and Higher Orders in Perturbation Theory
Higher orders in perturbation theory require the calculation of Feynman integrals at multiple loops. We report on an approach to systematically solve Feynman integrals by means of symbolic summation and discuss the underlying algorithms. Examples such as the non-planar vertex at two loops, or integrals from the recent calculation of the three-loop QCD corrections to structure functions in deep-...
متن کاملA symbolic operator approach to several summation formulas for power series II
This paper deals with the summation problem of power series of the form Sb a(f ;x) = ∑ a≤k≤b f(k)x k, where 0 ≤ a < b ≤ ∞, and {f(k)} is a given sequence of numbers with k ∈ [a, b) or f(t) is a differentiable function defined on [a, b). We present a symbolic summation operator with its various expansions, and construct several summation formulas with estimable remainders for Sb a(f ;x), by the ...
متن کاملمبانی افتراق تفاسیر عرفانی در مقایسه با تفاسیر عرفانی متقدّم
Theosophical and symbolic commentaries are among the major sustaining commentaries on the Quran. These commentaries approaches date at least back to the third century and they have still preserved many proponents till today. Thereby, some of the recent works have followed theosophical and symbolic trends in the course of deducing glorified Quranic verses. The commentary specifications of the wo...
متن کامل