Recurrence Equation

A recurrence equation (also called a difference equation) is the discrete analog of a differential equation. A difference equation involves an integer function f(n) in a form like


where g is some integer function. The above equation is the discrete analog of the first-order ordinary differential equation


Examples of difference equations often arise in dynamical systems. Examples include the iteration involved in the Mandelbrot and Julia set definitions,


with c a constant, as well as the logistic equation


with r a constant. Perhaps the most famous example of a recurrence relation is the one defining the Fibonacci numbers,


for n>=3 and with F_1=F_2=1.

Recurrence equations can be solved using RSolve[eqn, a[n], n]. The solutions to a linear recurrence equation can be computed straightforwardly, but quadratic recurrence equations are not so well understood.

The sequence generated by a recurrence relation is called a recurrence sequence.



where the generalized power sum a(h) for h=0, 1, ... is given by


with distinct nonzero roots alpha_i, coefficients A_i(h) which are polynomials of degree n_i-1 for positive integers n_i, and i in [1,m]. Then the sequence {a_h} with a_h=a(h) satisfies the recurrence relation


(Myerson and van der Poorten 1995).

The terms in a general recurrence sequence belong to a finitely generated ring over the integers, so it is impossible for every rational number to occur in any finitely generated recurrence sequence. If a recurrence sequence vanishes infinitely often, then it vanishes on an arithmetic progression with a common difference 1 that depends only on the roots. The number of values that a recurrence sequence can take on infinitely often is bounded by some integer l that depends only on the roots. There is no recurrence sequence in which each integer occurs infinitely often, or in which every Gaussian integer occurs (Myerson and van der Poorten 1995).

Let mu(n) be a bound so that a nondegenerate integer recurrence sequence of order n takes the value zero at least mu(n) times. Then mu(2)=1, mu(3)=6, and mu(4)>=9 (Myerson and van der Poorten 1995). The maximal case for mu(3) is




The zeros are


(Beukers 1991).

See also

Argument Addition Relation, Argument Multiplication Relation, Binet Forms, Binet's Fibonacci Number Formula, Clenshaw Recurrence Formula, Difference-Differential Equation, Fast Fibonacci Transform, Fibonacci Number, Finite Difference, Indicial Equation, Linear Recurrence Equation, Lucas Sequence, Ordinary Differential Equation, Quadratic Recurrence Equation, Quotient-Difference Table, Reflection Relation, Translation Relation, Skolem-Mahler-Lech Theorem

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Recurrence Equation

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Weisstein, Eric W. "Recurrence Equation." From MathWorld--A Wolfram Web Resource.

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