It is hoped that this exposition will encourage the use of Hopf algebraic ideas in the study of certain combinatorial areas of mathematics. One way to think of recursive algorithms, is as algorithms which reduce tasks to instances of the same task with a smaller number of inputs. We stress the classical connection (say over the real or complex numbers) with formal power series and the theory of linear homogeneous ordinary differential equations. (If the sequence starts with a1, then this means for n 1.) The integer k is called the order of the linear recurrence. The computation involves inverting the Hankel matrix of the sequence. In particular, we give an algorithm for diagonalizing a sequence in terms of the basis of the coalgebra it generates formed by its images under the difference-operator shift. ![]() This example shows that the recurrence relation can depend on n, as well as on the values of the preceding terms. We were asked for a 6, and we know that a 5 27, so a 6 a 5 + 2 ( 6) 1 27 + 11 38. We discuss possible bases for the solution space from the point of view of diagonalization. So the sequence can be defined by a 1 3 and an a n 1 + ( 2 n 1), for every n 2. Such a sequence satisfies a minimal recursive relation, whose solution space is the subcoalgebra generated by the sequence. In this case, our first term has the value a 1 2 and represents the first term of our recursive sequence. Step 2: The first term, represented by a 1, is and will always be given to us. Step 1: First, let’s decode what these formulas are saying. The algebra structure is that of divided-power sequences, so we concentrate on the perhaps lesser-known coalgebra (diagonalization) structure. Example 1: Arithmetic Recursive Sequence. ![]() We explain how the space of linearly recursive sequences over a field can be considered as a Hopf algebra. A recursive sequence is a sequence in which terms are defined using one or more previous terms which are given.
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