Rekürans

Rekürans, in mathematics and related fields, refers to a relation that defines each term of a sequence in terms of previous terms. A rekürans, or recurrence relation, specifies the n-th term using earlier terms and often initial values that seed the sequence.

A classic example is the Fibonacci recurrence: a_n = a_{n-1} + a_{n-2}, with given a_0 and a_1. More generally, a linear rekürans of order k has the form a_n = c_1 a_{n-1} + c_2 a_{n-2} + ... + c_k a_{n-k} + f(n), where c_i are constants and f(n) is an external input. The sequence is determined by the initial k terms.

Rekürans can be categorized by several properties. Linear versus nonlinear, homogeneous (f(n) = 0) versus non-homogeneous, and first-order versus higher-order. In many cases, especially with constant coefficients, the goal is to find a closed-form expression for a_n. This often involves solving a characteristic equation to obtain the roots that shape the explicit form of the sequence. Other approaches include generating functions and iterative expansion.

Applications of rekürans are widespread. They appear in algorithm analysis (for example, time or space complexity expressed as T(n) = a T(n/b) + f(n)), combinatorics, and dynamic modeling in economics, biology, and other sciences. They provide a framework for understanding how complex, sequential processes evolve from simpler, repeated rules.

In everyday use, rekürans also describes the recurrence of a condition, such as disease recurrence or cancer recurrence, where symptoms or tumors reappear after a period of improvement, guiding monitoring and treatment decisions. See also recurrence relation, recursion, generating function.