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

(1)

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

(2)

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

(3)

with
a constant, as well as the logistic equation

(4)

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

(5)

for
and with .

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.

Let

(6)

where the generalized power sum for , 1, ... is given by

(7)

with distinct nonzero roots , coefficients which are polynomials
of degree
for positive integers , and . Then the sequence with satisfies the recurrence
relation

(8)

(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 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
be a bound so that a nondegenerate integer recurrence
sequence of order
takes the value zero at least times. Then , , and (Myerson and van der Poorten 1995). The maximal
case for
is

(9)

with

(10)

(11)

The zeros are

(12)

(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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Cite this as:
Weisstein, Eric W. "Recurrence Equation."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/RecurrenceEquation.html

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