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The recurrence of order two satisfied by the Fibonacci numbers is the canonical example of a homogeneous linear recurrence relation with constant coefficients (see below). The Fibonacci sequence is defined using the recurrence = + with initial conditions
In mathematics (including combinatorics, linear algebra, and dynamical systems), a linear recurrence with constant coefficients [1]: ch. 17 [2]: ch. 10 (also known as a linear recurrence relation or linear difference equation) sets equal to 0 a polynomial that is linear in the various iterates of a variable—that is, in the values of the elements of a sequence.
In mathematics, and especially in numerical analysis, a homogeneous linear three-term recurrence relation (TTRR, the qualifiers "homogeneous linear" are usually taken for granted) [1] is a recurrence relation of the form
The Skolem problem is named after Thoralf Skolem, because of his 1933 paper proving the Skolem–Mahler–Lech theorem on the zeros of a sequence satisfying a linear recurrence with constant coefficients. [2] This theorem states that, if such a sequence has zeros, then with finitely many exceptions the positions of the zeros repeat regularly.
In mathematics, a homogeneous relation (also called endorelation) on a set X is a binary relation between X and itself, i.e. it is a subset of the Cartesian product X × X. [ 1 ] [ 2 ] [ 3 ] This is commonly phrased as "a relation on X " [ 4 ] or "a (binary) relation over X ".
The equation is called a linear recurrence with constant coefficients of order d. The order of the sequence is the smallest positive integer such that the sequence satisfies a recurrence of order d, or = for the everywhere-zero sequence. [citation needed]
Linear Recurrence Relations Let the signal be modeled by a series, which satisfies a linear recurrence relation s n = ∑ k = 1 r a k s n − k {\displaystyle s_{n}=\sum _{k=1}^{r}a_{k}s_{n-k}} ; that is, a series that can be represented as sums of products of exponential, polynomial and sine wave functions.
This equivalence can be used to quickly solve for the recurrence relationship for the coefficients in the power series solution of a linear differential equation. The rule of thumb (for equations in which the polynomial multiplying the first term is non-zero at zero) is that: