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{{short description|Infinite sequence of numbers satisfying a linear equation}} |
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{{Further information|Linear difference equation}} |
{{Further information|Linear difference equation}} |
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[[File:Fibonacci sequence.jpg|thumb|300px|right|The Fibonacci sequence is constant-recursive: each element of the sequence is the sum of the previous two.]] |
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In [[mathematics]], a '''constant-recursive sequence''', '''C-finite sequence''', or '''linear recurrence sequence''' is a sequence satisfying a linear [[recurrence relation | recurrence]] with constant coefficients. |
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In [[mathematics]] and [[theoretical computer science]], a '''constant-recursive sequence''' or '''linear recurrence sequence''' is an [[infinite sequence]] of [[number]]s in which each number in the sequence is equal to a [[linear combination]] of one or more of its immediate predecessors. The most famous example of a constant-recursive sequence is the [[Fibonacci number|Fibonacci sequence]], in which each number is the sum of the previous two; formally, we say that it satisfies the [[Linear difference equation|linear recurrence relation]] <math>F(n) = F(n-1) + F(n-2)</math>, where <math>F(n)</math> is the <math>n</math> Fibonacci number. Generalizing this idea, a constant-recursive sequence <math>s(n)</math> satisfies a formula of the form |
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<math display=block>s(n) = c_1 s(n-1) + c_2 s(n-2) + \dots + c_d s(n-d),</math> |
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where <math>c_i</math> are [[constant]]s. In addition to the Fibonacci sequence, constant-recursive sequences include many other well-known sequences, such as [[arithmetic progression]]s, [[geometric progression]]s, and [[polynomial]]s. |
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Constant-recursive sequences arise in many areas, including [[combinatorics]], due to the [[Combinatorial principles#Recurrence relation|combinatorial interpretation of recurrence relations]]; in [[algebraic number theory]], due to the relation of the sequence to the roots of a polynomial; in the theory of [[finite difference]]s; in the [[analysis of algorithms]] in the running time or space usage of a [[Recursion (computer science)|recursive function]]; and in other settings, for example in [[formal language theory]], where they count the number of strings in a [[regular language]]. Constant-recursive sequences are closed under important mathematical operations such as [[pointwise]] [[addition]], pointwise [[multiplication]], and [[Cauchy product]]. A seminal result is the [[Skolem–Mahler–Lech theorem]], which states that the [[zeros of a function|zeros]] of a constant-recursive sequence have a regularly-repeating (eventually-periodic) form. On the other hand, the [[Skolem problem]], which asks for [[algorithm]] to determine whether a linear recurrence has at least one zero, remains one of the [[List of unsolved problems in mathematics|unsolved problems in mathematics]]. |
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==Definition== |
==Definition== |
Revision as of 02:57, 8 November 2021
In mathematics and theoretical computer science, a constant-recursive sequence or linear recurrence sequence is an infinite sequence of numbers in which each number in the sequence is equal to a linear combination of one or more of its immediate predecessors. The most famous example of a constant-recursive sequence is the Fibonacci sequence, in which each number is the sum of the previous two; formally, we say that it satisfies the linear recurrence relation , where is the Fibonacci number. Generalizing this idea, a constant-recursive sequence satisfies a formula of the form
Constant-recursive sequences arise in many areas, including combinatorics, due to the combinatorial interpretation of recurrence relations; in algebraic number theory, due to the relation of the sequence to the roots of a polynomial; in the theory of finite differences; in the analysis of algorithms in the running time or space usage of a recursive function; and in other settings, for example in formal language theory, where they count the number of strings in a regular language. Constant-recursive sequences are closed under important mathematical operations such as pointwise addition, pointwise multiplication, and Cauchy product. A seminal result is the Skolem–Mahler–Lech theorem, which states that the zeros of a constant-recursive sequence have a regularly-repeating (eventually-periodic) form. On the other hand, the Skolem problem, which asks for algorithm to determine whether a linear recurrence has at least one zero, remains one of the unsolved problems in mathematics.
Definition
An order-d homogeneous linear recurrence relation is an equation of the form
where the d coefficients are constants. The coefficients are typically taken to range over the integers, rational numbers, algebraic numbers, real numbers, or complex numbers.
A sequence (ranging over the same domain as the coefficients) is a constant-recursive sequence if there is an order-d homogeneous linear recurrence with constant coefficients that it satisfies for all . Note that this definition allows eventually-periodic sequences such as and , which are disallowed by some authors.[1] These sequences can be disallowed by requiring that .
Equivalently, is constant-recursive if the set of sequences
is contained in a vector space whose dimension is finite.
The order of a constant-recursive sequence is the smallest such that the sequence satisfies an order-d homogeneous linear recurrence. Nonhomogenous linear recurrence relations can be allowed without changing the set of constant-recursive sequences, but the definition of order would change.
Examples
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, ... (sequence A000032 in the OEIS) 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 is constant-recursive, since it satisfies the recurrence for all .
Eventually periodic sequences
A sequence that is eventually periodic with period length is constant-recursive, since it satisfies for all for some .
Polynomial sequences
For any polynomial s(n), the sequence of its values is a constant-recursive sequence. If the degree of the polynomial is d, the sequence satisfies a recurrence of order , with coefficients given by the corresponding element of the binomial transform.[2] The first few such equations are
- for a degree 0 (that is, constant) polynomial,
- for a degree 1 or less polynomial,
- for a degree 2 or less polynomial, and
- for a degree 3 or less polynomial.
A sequence obeying the order-d equation also obeys all higher order equations. These identities may be proved in a number of ways, including via the theory of finite differences.[citation needed] Each individual equation may also be verified by substituting the degree-d polynomial
where the coefficients are symbolic. Any sequence of integer, real, or complex values can be used as initial conditions for a constant-recursive sequence of order . If the initial conditions lie on a polynomial of degree or less, then the constant-recursive sequence also obeys a lower order equation.
Enumeration of words in a regular language
Let be a regular language, and let be the number of words of length in . Then is constant-recursive.
Other examples
The sequences of Jacobsthal numbers, Padovan numbers, Pell numbers, and Perrin numbers are constant-recursive.
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. If , the nth term of a constant-recursive sequence can be written in terms of the roots of its characteristic polynomial. If the d roots are all distinct, then the nth term of the sequence is
where the coefficients ki are constants that can be determined from the initial conditions.
For the Fibonacci sequence, the characteristic polynomial is , whose roots and appear in Binet's formula
More generally, if a root r of the characteristic polynomial has multiplicity m, then the term is multiplied by a degree- polynomial in n. That is, let be the distinct roots of the characteristic polynomial. Then
where is a polynomial of degree . For instance, if the characteristic polynomial factors as , with the same root r occurring three times, then the nth term is of the form
Conversely, if there are polynomials such that
then 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.[4]
The generating function of the Fibonacci sequence is
In general, multiplying a generating function by the polynomial
yields a series
where
If satisfies the recurrence relation
then for all . In other words,
so we obtain the rational function
In the special case of a periodic sequence satisfying for , 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 to obtain
Substituting this into the equation
shows that satisfies the homogeneous recurrence
of order .
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 -regular sequence satisfies linear recurrences with constant coefficients, but the recurrences take a different form. Rather than being a linear combination of for some integers that are close to , each term in a -regular sequence is a linear combination of for some integers whose base- representations are close to that of . Constant-recursive sequences can be thought of as -regular sequences, where the base-1 representation of consists of copies of the digit .
Notes
- ^ Kauers, Manuel; Paule, Peter (2010-12-01). The Concrete Tetrahedron: Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates. Springer Vienna. ISBN 978-3-7091-0444-6.
- ^ Boyadzhiev, Boyad (2012). "Close Encounters with the Stirling Numbers of the Second Kind" (PDF). Math. Mag. 85: 252–266.
- ^ Greene, Daniel H.; Knuth, Donald E. (1982), "2.1.1 Constant coefficients – A) Homogeneous equations", Mathematics for the Analysis of Algorithms (2nd ed.), Birkhäuser, p. 17.
- ^ Martino, Ivan; Martino, Luca (2013-11-14). "On the variety of linear recurrences and numerical semigroups". Semigroup Forum. 88 (3): 569–574. arXiv:1207.0111. doi:10.1007/s00233-013-9551-2. ISSN 0037-1912.
References
- Brousseau, Alfred (1971). Linear Recursion and Fibonacci Sequences. Fibonacci Association.
- Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1994). Concrete Mathematics: A Foundation for Computer Science (2 ed.). Addison-Wesley. ISBN 978-0-201-55802-9.
See also
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)