Rekurrenzrelation

Rekurrenzrelation is a mathematical term referring to an equation that defines a sequence recursively, meaning each term of the sequence is defined as a function of the preceding terms. A rekurrenzrelation typically involves one or more initial values, known as base cases, to start the sequence. For example, the Fibonacci sequence is a classic example of a rekurrenzrelation, defined by F(n) = F(n-1) + F(n-2) with base cases F(0) = 0 and F(1) = 1. This means each number in the sequence is the sum of the two preceding ones.

Rekurrenzrelationen are crucial in various fields of mathematics and computer science. In combinatorics, they are used to count arrangements or combinations. In algorithm analysis, they help determine the time complexity of recursive algorithms. Solving a rekurrenzrelation means finding a closed-form expression for the nth term of the sequence, which does not depend on previous terms. Techniques for solving rekurrenzrelationen include characteristic equations for linear homogeneous relations with constant coefficients, and substitution or iteration for simpler cases. Understanding rekurrenzrelationen allows for the prediction and analysis of sequences and recursive processes.