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A famous example is the recurrence for the Fibonacci numbers, = + where the order is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients , because the coefficients of the linear function (1 and 1) are constants that do not depend on n . {\displaystyle n.}
The equation = (+) = is called a linear recurrence equation with polynomial coefficients (all recurrence equations in this article are of this form). If p 0 {\textstyle p_{0}} and p r {\textstyle p_{r}} are both nonzero, then r {\textstyle r} is called the order of the equation.
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.
If the {} and {} are constant and independent of the step index n, then the TTRR is a Linear recurrence with constant coefficients of order 2. Arguably the simplest, and most prominent, example for this case is the Fibonacci sequence , which has constant coefficients a n = b n = 1 {\displaystyle a_{n}=b_{n}=1} .
The equation is called a linear recurrence with constant coefficients of order d. The order of the sequence is the smallest positive integer d {\displaystyle d} such that the sequence satisfies a recurrence of order d , or d = 0 {\displaystyle d=0} for the everywhere-zero sequence.
The next approximation x k is now one of the roots of the p k,m, i.e. one of the solutions of p k,m (x)=0. Taking m =1 we obtain the secant method whereas m =2 gives Muller's method. Muller calculated that the sequence { x k } generated this way converges to the root ξ with an order μ m where μ m is the positive solution of x m + 1 − x m ...
The previous example involved an indicial polynomial with a repeated root, which gives only one solution to the given differential equation. In general, the Frobenius method gives two independent solutions provided that the indicial equation's roots are not separated by an integer (including zero).
The series converges for | | < (note, x may be complex), as may be seen by applying the ratio test to the recurrence. The recurrence may be started with arbitrary values of a 0 and a 1, leading to the two-dimensional space of solutions that arises from second order differential equations. The standard choices are: