Algebraic extensions for symbolic summation by Erocal B.

By Erocal B.

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Hence, the matrix An,f = σb (σI) + Mb (f ) has only two nonzero entries per row. It can easily be computed explicitly. For instance,  −f0 σ σ −f1  −f0 0 σ 0  for n = 3. for n = 2 and  σ −f1 0 σ −f2 The general form for f ∈ E and n ∈ N is given by An,f = (ai,j )0≤i,j

Since these matrices are generated by summation problems, they have an inherent structure and the degree of the entries in the nullspace N is low. For example, while computing a recurrence in n for the expression Γ(l + 1)Γ(n)Γ(k + l + s + 2)s! (−1)k+l kj −j+n−2 l (k + l + 1)Γ(n + 1)Γ(l + s + 2)Γ(k + l + s + 3) using Wegschaider’s Mathematica package [80], we obtain a matrix M with n = 112, m = 19, degree 3, and coefficients with maximum 32 bits. Entries in the nullspace N for this matrix have degree 15 again with maximum 32-bit coefficients.

Other approaches to providing such a decomposition, especially in the rational difference field Q(x), σ with σ(x) = x + 1 can be found in [31, 50, 53], though these have not been extended to elements of ΠΣ-fields. 2. Tests for σ-radicals Given an element α of a ΠΣ-field F, σ, the σ-factorization algorithm can be used to decide if α is a σ-radical over F . This procedure will actually answer a more general problem, which comes up when going through the levels of a ΠΣ-tower recursively. Given α1 , .

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