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Source: Constant-recursive sequence

In mathematics, an infinite sequence of numbers

s

0

,

s

1

,

s

2

,

s

3

,
…

{\displaystyle s_{0},s_{1},s_{2},s_{3},\ldots }

is called constant-recursive if it satisfies an equation of the form

s

n

=

c

1

s

n
−
1

+

c

2

s

n
−
2

+
⋯
+

c

d

s

n
−
d

,

{\displaystyle s_{n}=c_{1}s_{n-1}+c_{2}s_{n-2}+\dots +c_{d}s_{n-d},}

for all

n
≥
d

{\displaystyle n\geq d}

, where

c

i

{\displaystyle c_{i}}

are constants. The equation is called a linear recurrence relation.
The concept is also known as a linear recurrence sequence, linear-recursive sequence, linear-recurrent sequence, or a C-finite sequence.
For example, the Fibonacci sequence

0
,
1
,
1
,
2
,
3
,
5
,
8
,
13
,
…

{\displaystyle 0,1,1,2,3,5,8,13,\ldots }

,
is constant-recursive because it satisfies the linear recurrence

F

n

=

F

n
−
1

+

F

n
−
2

{\displaystyle F_{n}=F_{n-1}+F_{n-2}}

: each number in the sequence is the sum of the previous two.
Other examples include the power of two sequence

1
,
2
,
4
,
8
,
16
,
…

{\displaystyle 1,2,4,8,16,\ldots }

, where each number is the sum of twice the previous number, and the square number sequence

0
,
1
,
4
,
9
,
16
,
25
,
…

{\displaystyle 0,1,4,9,16,25,\ldots }

. All arithmetic progressions, all geometric progressions, and all polynomials are constant-recursive. However, not all sequences are constant-recursive; for example, the factorial sequence

1
,
1
,
2
,
6
,
24
,
120
,
…

{\displaystyle 1,1,2,6,24,120,\ldots }

is not constant-recursive.
Constant-recursive sequences are studied in combinatorics and the theory of finite differences. They also arise in algebraic number theory, due to the relation of the sequence to polynomial roots; in the analysis of algorithms, as the running time of simple recursive functions; and in the theory of formal languages, where they count strings up to a given length in a regular language. Constant-recursive sequences are closed under important mathematical operations such as term-wise addition, term-wise multiplication, and Cauchy product.
The Skolem–Mahler–Lech theorem states that the zeros of a constant-recursive sequence have a regularly repeating (eventually periodic) form. The Skolem problem, which asks for an algorithm to determine whether a linear recurrence has at least one zero, is an unsolved problem in mathematics.

Definition

constant

Examples

= Fibonacci and Lucas sequences

= Arithmetic progressions

constant

recursive

= Geometric progressions

constant

recursive

sequence

= Eventually periodic sequences

sequence

constant

recursive

= Polynomial sequences

= Enumeration of words in a regular language

= Other examples

constant

recursive

= Non-examples

Equivalent definitions

= In terms of matrices

= In terms of non-homogeneous linear recurrences

= In terms of generating functions

.
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:

∑

n
=
0

∞

s

n

x

n

=

b

0

+

b

1

x

1

+

b

2

x

2

+
⋯
+

b

d
−
1

x

d
−
1

1
−

c

1

x

1

−

c

2

x

2

−
⋯
−

c

d

x

d

,

{\displaystyle \sum _{n=0}^{\infty }s_{n}x^{n}={\frac {b_{0}+b_{1}x^{1}+b_{2}x^{2}+\dots +b_{d-1}x^{d-1}}{1-c_{1}x^{1}-c_{2}x^{2}-\dots -c_{d}x^{d}}},}

where

b

n

=

s

n

−

c

1

s

n
−
1

−

c

2

s

n
−
2

−
⋯
−

c

d

s

n
−
d

.

{\displaystyle b_{n}=s_{n}-c_{1}s_{n-1}-c_{2}s_{n-2}-\dots -c_{d}s_{n-d}.}

It follows from the above that the denominator

q
(
x
)

{\displaystyle q(x)}

must be a polynomial not divisible by

x

{\displaystyle x}

(and in particular nonzero).

= In terms of sequence spaces

=

A sequence

(

s

n

)

n
=
0

∞

{\displaystyle (s_{n})_{n=0}^{\infty }}

is constant-recursive if and only if the set of sequences

{

(

s

n
+
r

)

n
=
0

∞

:
r
≥
0

}

{\displaystyle \left\{(s_{n+r})_{n=0}^{\infty }:r\geq 0\right\}}

is contained in a sequence space (vector space of sequences) whose dimension is finite. That is,

(

s

n

)

n
=
0

∞

{\displaystyle (s_{n})_{n=0}^{\infty }}

is contained in a finite-dimensional subspace of

C

N

{\displaystyle \mathbb {C} ^{\mathbb {N} }}

closed under the left-shift operator.
This characterization is because the order-

d

{\displaystyle d}

linear recurrence relation can be understood as a proof of linear dependence between the sequences

(

s

n
+
r

)

n
=
0

∞

{\displaystyle (s_{n+r})_{n=0}^{\infty }}

for

r
=
0
,
…
,
d

{\displaystyle r=0,\ldots ,d}

. An extension of this argument shows that the order of the sequence is equal to the dimension of the sequence space generated by

(

s

n
+
r

)

n
=
0

∞

{\displaystyle (s_{n+r})_{n=0}^{\infty }}

for all

r

{\displaystyle r}

.

Closed-form characterization

Constant-recursive sequences admit the following unique closed form characterization using exponential polynomials: every constant-recursive sequence can be written in the form

s

n

=

z

n

+

k

1

(
n
)

r

1

n

+

k

2

(
n
)

r

2

n

+
⋯
+

k

e

(
n
)

r

e

n

,

{\displaystyle s_{n}=z_{n}+k_{1}(n)r_{1}^{n}+k_{2}(n)r_{2}^{n}+\cdots +k_{e}(n)r_{e}^{n},}

for all

n
≥
0

{\displaystyle n\geq 0}

, where

The term

z

n

{\displaystyle z_{n}}

is a sequence which is zero for all

n
≥
d

{\displaystyle n\geq d}

(where

d

{\displaystyle d}

is the order of the sequence);
The terms

k

1

(
n
)
,

k

2

(
n
)
,
…
,

k

e

(
n
)

{\displaystyle k_{1}(n),k_{2}(n),\ldots ,k_{e}(n)}

are complex polynomials; and
The terms

r

1

,

r

2

,
…
,

r

k

{\displaystyle r_{1},r_{2},\ldots ,r_{k}}

are distinct complex constants.
This characterization is exact: every sequence of complex numbers that can be written in the above form is constant-recursive.
For example, the Fibonacci number

F

n

{\displaystyle F_{n}}

is written in this form using Binet's formula:

F

n

=

1

5

φ

n

−

1

5

ψ

n

,

{\displaystyle F_{n}={\frac {1}{\sqrt {5}}}\varphi ^{n}-{\frac {1}{\sqrt {5}}}\psi ^{n},}

where

φ
=
(
1
+

5

)

/

2
≈
1.61803
…

{\displaystyle \varphi =(1+{\sqrt {5}})\,/\,2\approx 1.61803\ldots }

is the golden ratio and

ψ
=
−
1

/

φ

{\displaystyle \psi =-1\,/\,\varphi }

. These are the roots of the equation

x

2

−
x
−
1
=
0

{\displaystyle x^{2}-x-1=0}

. In this case,

e
=
2

{\displaystyle e=2}

,

z

n

=
0

{\displaystyle z_{n}=0}

for all

n

{\displaystyle n}

,

k

1

(
n
)
=

k

2

(
n
)
=
1

/

5

{\displaystyle k_{1}(n)=k_{2}(n)=1\,/\,{\sqrt {5}}}

are both constant polynomials,

r

1

=
φ

{\displaystyle r_{1}=\varphi }

, and

r

2

=
ψ

{\displaystyle r_{2}=\psi }

.
The term

z

n

{\displaystyle z_{n}}

is only needed when

c

d

≠
0

{\displaystyle c_{d}\neq 0}

; if

c

d

=
0

{\displaystyle c_{d}=0}

then it corrects for the fact that some initial values may be exceptions to the general recurrence. In particular,

z

n

=
0

{\displaystyle z_{n}=0}

for all

n
≥
d

{\displaystyle n\geq d}

.
The complex numbers

r

1

,
…
,

r

n

{\displaystyle r_{1},\ldots ,r_{n}}

are the roots of the characteristic polynomial of the recurrence:

x

d

−

c

1

x

d
−
1

−
⋯
−

c

d
−
1

x
−

c

d

{\displaystyle x^{d}-c_{1}x^{d-1}-\dots -c_{d-1}x-c_{d}}

whose coefficients are the same as those of the recurrence.
We call

r

1

,
…
,

r

n

{\displaystyle r_{1},\ldots ,r_{n}}

the characteristic roots of the recurrence. If the sequence consists of integers or rational numbers, the roots will be algebraic numbers.
If the

d

{\displaystyle d}

roots

r

1

,

r

2

,
…
,

r

d

{\displaystyle r_{1},r_{2},\dots ,r_{d}}

are all distinct, then the polynomials

k

i

(
n
)

{\displaystyle k_{i}(n)}

are all constants, which can be determined from the initial values of the sequence.
If the roots of the characteristic polynomial are not distinct, and

r

i

{\displaystyle r_{i}}

is a root of multiplicity

m

{\displaystyle m}

, then

k

i

(
n
)

{\displaystyle k_{i}(n)}

in the formula has degree

m
−
1

{\displaystyle m-1}

. For instance, if the characteristic polynomial factors as

(
x
−
r

)

3

{\displaystyle (x-r)^{3}}

, with the same root r occurring three times, then the

n

{\displaystyle n}

th term is of the form

s

n

=
(
a
+
b
n
+
c

n

2

)

r

n

.

{\displaystyle s_{n}=(a+bn+cn^{2})r^{n}.}

Closure properties

= Examples

=
The sum of two constant-recursive sequences is also constant-recursive. For example, the sum of

s

n

=

2

n

{\displaystyle s_{n}=2^{n}}

and

t

n

=
n

{\displaystyle t_{n}=n}

is

u

n

=

2

n

+
n

{\displaystyle u_{n}=2^{n}+n}

(

1
,
3
,
6
,
11
,
20
,
…

{\displaystyle 1,3,6,11,20,\ldots }

), which satisfies the recurrence

u

n

=
4

u

n
−
1

−
5

u

n
−
2

+
2

u

n
−
3

{\displaystyle u_{n}=4u_{n-1}-5u_{n-2}+2u_{n-3}}

. The new recurrence can be found by adding the generating functions for each sequence.
Similarly, the product of two constant-recursive sequences is constant-recursive. For example, the product of

s

n

=

2

n

{\displaystyle s_{n}=2^{n}}

and

t

n

=
n

{\displaystyle t_{n}=n}

is

u

n

=
n
⋅

2

n

{\displaystyle u_{n}=n\cdot 2^{n}}

(

0
,
2
,
8
,
24
,
64
,
…

{\displaystyle 0,2,8,24,64,\ldots }

), which satisfies the recurrence

u

n

=
4

u

n
−
1

−
4

u

n
−
2

{\displaystyle u_{n}=4u_{n-1}-4u_{n-2}}

.
The left-shift sequence

u

n

=

s

n
+
1

{\displaystyle u_{n}=s_{n+1}}

and the right-shift sequence

u

n

=

s

n
−
1

{\displaystyle u_{n}=s_{n-1}}

(with

u

0

=
0

{\displaystyle u_{0}=0}

) are constant-recursive because they satisfy the same recurrence relation. For example, because

s

n

=

2

n

{\displaystyle s_{n}=2^{n}}

is constant-recursive, so is

u

n

=

2

n
+
1

{\displaystyle u_{n}=2^{n+1}}

.

= List of operations

=
In general, constant-recursive sequences are closed under the following operations, where

s
=
(

s

n

)

n
∈

N

,
t
=
(

t

n

)

n
∈

N

{\displaystyle s=(s_{n})_{n\in \mathbb {N} },t=(t_{n})_{n\in \mathbb {N} }}

denote constant-recursive sequences,

f
(
x
)
,
g
(
x
)

{\displaystyle f(x),g(x)}

are their generating functions, and

d
,
e

{\displaystyle d,e}

are their orders, respectively.

The closure under term-wise addition and multiplication follows from the closed-form characterization in terms of exponential polynomials. The closure under Cauchy product follows from the generating function characterization. The requirement

s

0

=
1

{\displaystyle s_{0}=1}

for Cauchy inverse is necessary for the case of integer sequences, but can be replaced by

s

0

≠
0

{\displaystyle s_{0}\neq 0}

if the sequence is over any field (rational, algebraic, real, or complex numbers).

Behavior

= Zeros

=
Despite satisfying a simple local formula, a constant-recursive sequence can exhibit complicated global behavior. Define a zero of a constant-recursive sequence to be a nonnegative integer

n

{\displaystyle n}

such that

s

n

=
0

{\displaystyle s_{n}=0}

. The Skolem–Mahler–Lech theorem states that the zeros of the sequence are eventually repeating: there exists constants

M

{\displaystyle M}

and

N

{\displaystyle N}

such that for all

n
>
M

{\displaystyle n>M}

,

s

n

=
0

{\displaystyle s_{n}=0}

if and only if

s

n
+
N

=
0

{\displaystyle s_{n+N}=0}

. This result holds for a constant-recursive sequence over the complex numbers, or more generally, over any field of characteristic zero.

= Decision problems

=
The pattern of zeros in a constant-recursive sequence can also be investigated from the perspective of computability theory. To do so, the description of the sequence

s

n

{\displaystyle s_{n}}

must be given a finite description; this can be done if the sequence is over the integers or rational numbers, or even over the algebraic numbers.
Given such an encoding for sequences

s

n

{\displaystyle s_{n}}

, the following problems can be studied:

Because the square of a constant-recursive sequence

s

n

2

{\displaystyle s_{n}^{2}}

is still constant-recursive (see closure properties), the existence-of-a-zero problem in the table above reduces to positivity, and infinitely-many-zeros reduces to eventual positivity. Other problems also reduce to those in the above table: for example, whether

s

n

=
c

{\displaystyle s_{n}=c}

for some

n

{\displaystyle n}

reduces to existence-of-a-zero for the sequence

s

n

−
c

{\displaystyle s_{n}-c}

. As a second example, for sequences in the real numbers, weak positivity (is

s

n

≥
0

{\displaystyle s_{n}\geq 0}

for all

n

{\displaystyle n}

?) reduces to positivity of the sequence

−

s

n

{\displaystyle -s_{n}}

(because the answer must be negated, this is a Turing reduction).
The Skolem-Mahler-Lech theorem would provide answers to some of these questions, except that its proof is non-constructive. It states that for all

n
>
M

{\displaystyle n>M}

, the zeros are repeating; however, the value of

M

{\displaystyle M}

is not known to be computable, so this does not lead to a solution to the existence-of-a-zero problem. On the other hand, the exact pattern which repeats after

n
>
M

{\displaystyle n>M}

is computable. This is why the infinitely-many-zeros problem is decidable: just determine if the infinitely-repeating pattern is empty.
Decidability results are known when the order of a sequence is restricted to be small. For example, the Skolem problem is decidable for algebraic sequences of order up to 4. It is also known to be decidable for reversible integer sequences up to order 7, that is, sequences that may be continued backwards in the integers.
Decidability results are also known under the assumption of certain unproven conjectures in number theory. For example, decidability is known for rational sequences of order up to 5 subject to the Skolem conjecture (also known as the exponential local-global principle). Decidability is also known for all simple rational sequences (those with simple characteristic polynomial) subject to the Skolem conjecture and the weak p-adic Schanuel conjecture.

= Degeneracy

=
Let

r

1

,
…
,

r

n

{\displaystyle r_{1},\ldots ,r_{n}}

be the characteristic roots of a constant recursive sequence

s

{\displaystyle s}

. We say that the sequence is degenerate if any ratio

r

i

/

r

j

{\displaystyle r_{i}/r_{j}}

is a root of unity, for

i
≠
j

{\displaystyle i\neq j}

. It is often easier to study non-degenerate sequences, in a certain sense one can reduce to this using the following theorem: if

s

{\displaystyle s}

has order

d

{\displaystyle d}

and is contained in a number field

K

{\displaystyle K}

of degree

k

{\displaystyle k}

over

Q

{\displaystyle \mathbb {Q} }

, then there is a constant

M
(
k
,
d
)
≤

{

exp
⁡
(
2
d
(
3
log
⁡
d

)

1

/

2

)

if

k
=
1
,

2

k
d
+
1

if

k
≥
2

{\displaystyle M(k,d)\leq {\begin{cases}\exp(2d(3\log d)^{1/2})&{\text{if }}k=1,\\2^{kd+1}&{\text{if }}k\geq 2\end{cases}}}

such that for some

M
≤
M
(
k
,
d
)

{\displaystyle M\leq M(k,d)}

each subsequence

s

M
n
+
ℓ

{\displaystyle s_{Mn+\ell }}

is either identically zero or non-degenerate.

Generalizations

A D-finite or holonomic sequence is a natural generalization where the coefficients of the recurrence are allowed to be polynomial functions of

n

{\displaystyle n}

rather than constants.
A

k

{\displaystyle k}

-regular sequence satisfies a linear recurrences with constant coefficients, but the recurrences take a different form. Rather than

s

n

{\displaystyle s_{n}}

being a linear combination of

s

m

{\displaystyle s_{m}}

for some integers

m

{\displaystyle m}

that are close to

n

{\displaystyle n}

, each term

s

n

{\displaystyle s_{n}}

in a

k

{\displaystyle k}

-regular sequence is a linear combination of

s

m

{\displaystyle s_{m}}

for some integers

m

{\displaystyle m}

whose base-

k

{\displaystyle k}

representations are close to that of

n

{\displaystyle n}

. Constant-recursive sequences can be thought of as

1

{\displaystyle 1}

-regular sequences, where the base-1 representation of

n

{\displaystyle n}

consists of

n

{\displaystyle n}

copies of the digit

1

{\displaystyle 1}

.

Notes

References

External links

"OEIS Index Rec". OEIS index to a few thousand examples of linear recurrences, sorted by order (number of terms) and signature (vector of values of the constant coefficients)

Definition

Examples

= Fibonacci and Lucas sequences

= Arithmetic progressions

= Geometric progressions

= Eventually periodic sequences

= Polynomial sequences

= Enumeration of words in a regular language

= Other examples

= Non-examples

Equivalent definitions

= In terms of matrices

= In terms of non-homogeneous linear recurrences

= In terms of generating functions

= In terms of sequence spaces

Closed-form characterization

Closure properties

= Examples

= List of operations

Behavior

= Zeros

= Decision problems

= Degeneracy

Generalizations

Notes

References

External links

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