By Erocal B.

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**Extra info for Algebraic extensions for symbolic summation**

**Example text**

1. However, it is possible to find a 1 new extension which contains an element h with the shift behavior σ(h) = −h + i+1 and the same action of σ on the indeterminates added to the component fields. 39. Take σ(h) = −h + 1 i+1 , s0 = h and s1 = −h. Then 1 , i+1 1 σ(s1 ) = σ(−h) = h − . i+1 σ(s0 ) = σ(h) = −h + 1 contains the element s = he0 − he1 = eh, The extension R[h], σ with σ(h) = −h + i+1 (−1)j i j=1 j . which models the sum The following lemma describes how to find the action of σ on the new variable h.

The ring ⊕0≤i

Let (R, σ, δ) be a D-ring and X an indeterminate over R. The left skew polynomial ring over R, denoted R[X; σ, δ] is the ring of polynomials in X over R with the usual polynomial addition and multiplication given by Xa = σ(a)X + δ(a) for any a ∈ R. The multiplication in a skew polynomial ring can be uniquely extended to multiplication of monomials by (aX n )(bX m ) = (aX n−1 )(Xb)X m = (aX n−1 )(σ(b)X m+1 + δ(b)X m ) for m, n > 0 and to arbitrary polynomials by distributivity. 18. (i) For a difference ring R, σ, the skew polynomial ring R[S; σ, 0] is the ring of linear ordinary recurrence operators, where S denotes the shift operator.

### Algebraic extensions for symbolic summation by Erocal B.

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