By Peter Paule, Manuel Kauers
The publication treats 4 mathematical options which play a primary position in lots of diverse parts of arithmetic: symbolic sums, recurrence (difference) equations, producing features, and asymptotic estimates.
Their key positive aspects, in isolation or together, their mastery by way of paper and pencil or through desktop courses, and their purposes to difficulties in natural arithmetic or to "real global problems" (e.g. the research of algorithms) are studied. The publication is meant as an algorithmic complement to the bestselling "Concrete Mathematics" through Graham, Knuth and Patashnik.
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Additional info for The Concrete Tetrahedron: Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates (Texts & Monographs in Symbolic Computation)
Example text
Therefore, for n = i + j we find i+ j ci+ j = ∑ ak bi+ j−k = ai b j . k=0 Since both ai and b j are nonzero, so is ci+ j , and therefore c(x) cannot be the zero series. n k If a(x) = ∑∞ n=0 an x is a formal power series, we will use the notation [x ]a(x) := ak k k to refer the coefficient of x . For fixed k ∈ , coefficient extraction [x ] : [[x]] → is a linear map. For k = 0, also the notations a(x)|x=0 := a(0) := [x0 ]a(x) are used. The coefficient of x0 is called the constant term of a(x). The definition of the Cauchy product may seem somewhat unmotivated at first sight, but there is a natural combinatorial interpretation for it.
1. For the moment, we record ∞ 1 n n k = ∑ x y 1 − x − xy n,k=0 k as its bivariate generating function. Like in the univariate case, operations on multivariate power series correspond to operations on the (multivariate) coefficient sequence. For example, multiplication by x corresponds to a shift in n and multiplication by y corresponds to shift in k. Our result about the bivariate generating function of the binomial coefficients is therefore just a reformulation of the Pascal triangle relation: (1 − x − xy) ∞ ∑ n,k=0 n n k x y =1 k =⇒ n n−1 n−1 − − = 0 (n, k > 0).
In that case we write a(x) = limk→∞ ak (x), a notation which is justified because the limit of a convergent sequence of power series is unique. ∞ n n If ak (x) = ∑∞ n=0 an,k x and a(x) = ∑n=0 an x , then convergence of the sequence ∞ (ak (x))k=0 to a(x) means that every coefficient sequence an,0 , an,1 , an,2 , an,3 , . . differs from an only for finitely many indices. 5 Let (an (x))∞ n=0 and (bn (x))n=0 be two convergent sequences in and let a(x) = limn→∞ an (x) and b(x) = limn→∞ bn (x). Then: Ã[[x]] 1.