Constant recursive sequence

Definition

An order-d homogeneous linear recurrence with constant coefficients is an equation of the form

where the d coefficients c 1 , c 2 , … , c d are constants.

A sequence s ( 0 ) , s ( 1 ) , s ( 2 ) , … is a constant-recursive sequence if there is an order-d homogeneous linear recurrence with constant coefficients that it satisfies for all n ≥ d .

Equivalently, s ( n ) n ≥ 0 is constant-recursive if the set of sequences

is contained in a vector space whose dimension is finite.

Fibonacci sequence

The sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... of Fibonacci numbers satisfies the recurrence

with initial conditions

Explicitly, the recurrence yields the values

etc.

Lucas sequences

The sequence 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, ... of Lucas numbers satisfies the same recurrence as the Fibonacci sequence but with initial conditions

More generally, every Lucas sequence is a constant-recursive sequence.

Geometric sequences

The geometric sequence a , a r , a r 2 , … is constant-recursive, since it satisfies the recurrence s ( n ) = r s ( n − 1 ) for all n ≥ 1 .

Eventually periodic sequences

A sequence that is eventually periodic with period length ℓ is constant-recursive, since it satisfies s ( n ) = s ( n − ℓ ) for all n ≥ d for some d.

Characterization in terms of exponential polynomials

The characteristic polynomial (or "auxiliary polynomial") of the recurrence is the polynomial

whose coefficients are the same as those of the recurrence. The nth term s ( n ) of a constant-recursive sequence can be written in terms of the roots of its characteristic polynomial. If the d roots r 1 , r 2 , … , r d are all distinct, then the nth term of the sequence is

where the coefficients k_i are constants that can be determined by the initial conditions.

For the Fibonacci sequence, the characteristic polynomial is x 2 − x − 1 , whose roots ϕ = 1 + 5 2 and ϕ ¯ = 1 − 5 2 appear in Binet's formula

More generally, if a root r of the characteristic polynomial has multiplicity m, then the term r n is multiplied by a degree- ( m − 1 ) polynomial in n. That is, let r 1 , … , r e be the distinct roots of the characteristic polynomial. Then

where k i ( n ) is a polynomial of degree m i − 1 . For instance, if the characteristic polynomial factors as ( x − r ) 3 , with the same root r occurring three times, then the nth term is of the form

Conversely, if there are polynomials k i ( n ) such that

then s ( n ) n ≥ 0 is constant-recursive.

Characterization in terms of rational generating functions

A sequence is constant-recursive precisely when its generating function

is a rational function. The denominator is the polynomial obtained from the auxiliary polynomial by reversing the order of the coefficients, and the numerator is determined by the initial values of the sequence.

The generating function of the Fibonacci sequence is

In general, multiplying a generating function by the polynomial

yields a series

where

If s ( n ) satisfies the recurrence relation

then b n = 0 for all n ≥ d . In other words,

so we obtain the rational function

In the special case of a periodic sequence satisfying s ( n ) = s ( n − d ) for n ≥ d , the generating function is

by expanding the geometric series.

The generating function of the Catalan numbers is not a rational function, which implies that the Catalan numbers do not satisfy a linear recurrence with constant coefficients.

Closure properties

The termwise addition or multiplication of two constant-recursive sequences is again constant-recursive. This follows from the characterization in terms of exponential polynomials.

The Cauchy product of two constant-recursive sequences is constant-recursive. This follows from the characterization in terms of rational generating functions.

Sequences satisfying non-homogeneous recurrences

A sequence satisfying a non-homogeneous linear recurrence with constant coefficients is constant-recursive.

This is because the recurrence

can be solved for c to obtain

Substituting this into the equation

shows that s ( n ) satisfies the homogeneous recurrence

of order d + 1 .

Generalizations

A natural generalization is obtained by relaxing the condition that the coefficients of the recurrence are constants. If the coefficients are allowed to be polynomials, then one obtains holonomic sequences.

A k -regular sequence satisfies linear recurrences with constant coefficients, but the recurrences take a different form. Rather than s ( n ) being a linear combination of s ( m ) for some integers m that are close to n , each term s ( n ) in a k -regular sequence is a linear combination of s ( m ) for some integers m whose base- k representations are close to that of n . Constant-recursive sequences can be thought of as 1 -regular sequences, where the base-1 representation of n consists of n copies of the digit 1 .