Editing Recurrence relation

{{Short description|Pattern defining an infinite sequence of numbers}}
<!-- Lead needs additional context and examples to provide an introduction for beginners. (January 2022) -->
In [[mathematics]], a '''recurrence relation''' is an [[equation]] according to which the <math>n</math>th term of a [[sequence]] of numbers is equal to some combination of the previous terms. Often, only <math>k</math> previous terms of the sequence appear in the equation, for a parameter <math>k</math> that is independent of <math>n</math>; this number <math>k</math> is called the ''order'' of the relation. If the values of the first <math>k</math> numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation.

In ''linear recurrences'', the {{mvar|n}}th term is equated to a [[linear function]] of the <math>k</math> previous terms. A famous example is the recurrence for the [[Fibonacci number]]s,
<math display=block>F_n=F_{n-1}+F_{n-2}</math>
where the order <math>k</math> is two and the linear function merely adds the two previous terms. This example is a [[linear recurrence with constant coefficients]], because the coefficients of the linear function (1 and 1) are constants that do not depend on <math>n.</math> For these recurrences, one can express the general term of the sequence as a [[closed-form expression]] of <math>n</math>. As well, [[P-recursive equation|linear recurrences with polynomial coefficients]] depending on <math>n</math> are also important, because many common [[elementary functions]] and [[special functions]] have a [[Taylor series]] whose coefficients satisfy such a recurrence relation (see [[holonomic function]]).

Solving a recurrence relation means obtaining a [[closed-form solution]]: a non-recursive function of <math>n</math>.

The concept of a recurrence relation can be extended to [[multidimensional array]]s, that is, [[indexed families]] that are indexed by [[tuple]]s of [[natural number]]s.

==Definition==
A ''recurrence relation'' is an equation that expresses each element of a [[sequence]] as a function of the preceding ones. More precisely, in the case where only the immediately preceding element is involved, a recurrence relation has the form
:<math>u_n=\varphi(n, u_{n-1})\quad\text{for}\quad n>0,</math>
where 
:<math>\varphi:\mathbb N\times X \to X</math>
is a function, where {{mvar|X}} is a set to which the elements of a sequence must belong. For any <math>u_0\in X</math>, this defines a unique sequence with <math>u_0</math> as its first element, called the ''initial value''.<ref>Jacobson, Nathan, Basic Algebra 2 (2nd ed.), § 0.4. pg 16.</ref>

It is easy to modify the definition for getting sequences starting from the term of index 1 or higher.

This defines recurrence relation of ''first order''. A recurrence relation of ''order'' {{mvar|k}} has the form 
:<math>u_n=\varphi(n, u_{n-1}, u_{n-2}, \ldots, u_{n-k})\quad\text{for}\quad n\ge k,</math>

where <math>\varphi: \mathbb N\times X^k \to X</math> is a function that involves {{mvar|k}} consecutive elements of the sequence.
In this case, {{mvar|k}} initial values are needed for defining a sequence.

==Examples==
===Factorial===
The [[factorial]] is defined by the recurrence relation
:<math>n!=n\cdot (n-1)!\quad\text{for}\quad n>0,</math>
and the initial condition
:<math>0!=1.</math>
This is an example of a ''linear recurrence with polynomial coefficients'' of order 1, with the simple polynomial (in {{mvar|n}})
:<math>n</math>
as its only coefficient.

===Logistic map===
An example of a recurrence relation is the [[logistic map]] defined by

:<math>x_{n+1} = r x_n (1 - x_n),</math>

for a given constant <math>r.</math> The behavior of the sequence depends dramatically on <math>r,</math> but is stable when the initial condition <math>x_0</math> varies.

===Fibonacci numbers===
The recurrence of order two satisfied by the [[Fibonacci number]]s is the canonical example of a homogeneous [[linear recurrence]] relation with constant coefficients (see below).  The Fibonacci sequence is defined using the recurrence

:<math>F_n = F_{n-1}+F_{n-2}</math>

with [[initial condition]]s

:<math>F_0 = 0</math>
:<math>F_1 = 1.</math>

Explicitly, the recurrence yields the equations
:<math>F_2 = F_1 + F_0</math>
:<math>F_3 = F_2 + F_1</math>
:<math>F_4 = F_3 + F_2</math>
etc.

We obtain the sequence of Fibonacci numbers, which begins
:0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

The recurrence can be solved by methods described below yielding [[Binet's formula]], which involves powers of the two roots of the characteristic polynomial <math>t^2 = t + 1</math>; the [[generating function]] of the sequence is the [[rational function]]
:  <math>\frac{t}{1-t-t^2}.</math>

===Binomial coefficients===
A simple example of a multidimensional recurrence relation is given by the [[binomial coefficient]]s <math>\tbinom{n}{k}</math>, which count the ways of selecting <math>k</math> elements out of a set of <math>n</math> elements.
They can be computed by the recurrence relation
:<math>\binom{n}{k}=\binom{n-1}{k-1}+\binom{n-1}{k},</math>
with the base cases <math>\tbinom{n}{0}=\tbinom{n}{n}=1</math>. Using this formula to compute the values of all binomial coefficients generates an infinite array called [[Pascal's triangle]]. The same values can also be computed directly by a different formula that is not a recurrence, but uses [[factorial]]s, multiplication and division, not just additions:
:<math>\binom{n}{k}=\frac{n!}{k!(n-k)!}.</math>

The binomial coefficients can also be computed with a uni-dimensional recurrence:
:<math>\binom n k = \binom n{k-1}(n-k+1)/k,</math>
with the initial value <math display = inline>\binom n 0 =1</math> (The division is not displayed as a fraction for emphasizing that it must be computed after the multiplication, for not introducing fractional numbers).
This recurrence is widely used in computers because it does not require to build a table as does the bi-dimensional recurrence, and does involve very large integers as does the formula with factorials (if one uses <math display = inline>\binom nk= \binom n{n-k}, </math> all involved integers are smaller than the final result).

== Difference operator and difference equations {{anchor|Relationship to difference equations narrowly defined}} ==

The '''{{vanchor|difference operator}}''' is an [[operator (mathematics)|operator]] that maps [[sequence]]s to sequences, and, more generally, [[function (mathematics)|functions]] to functions. It is commonly denoted <math>\Delta,</math> and is defined, in [[functional notation]], as 
:<math>(\Delta f)(x)=f(x+1)-f(x).</math>
It is thus a special case of [[finite difference]].

When using the index notation for sequences, the definition becomes
:<math>(\Delta a)_n= a_{n+1} - a_n.</math>
The parentheses around <math>\Delta f</math> and <math>\Delta a</math> are generally omitted, and <math>\Delta a_n</math> must be understood as the term of index {{mvar|n}} in the sequence <math>\Delta a,</math> and not <math>\Delta</math> applied to the element <math>a_n.</math>

Given [[sequence]] <math>a=(a_n)_{n\in \N},</math> the '''{{vanchor|first difference}}''' of {{mvar|a}} is <math>\Delta a.</math>

The '''{{vanchor|second difference}}''' is 
<math>\Delta^2 a=(\Delta\circ\Delta)a= \Delta(\Delta a).</math> A simple computation shows that
:<math>\Delta^2 a_n= a_{n+2} - 2a_{n+1} + a_n.</math>

More generally: the  {{mvar|k}}''th difference'' is defined recursively as <math>\Delta^k=\Delta\circ \Delta^{k-1},</math> and one has
:<math>\Delta^k a_n = \sum_{t=0}^k (-1)^t \binom{k}{t} a_{n+k-t}.</math>

This relation can be inverted, giving
:<math>a_{n+k} = a_n + {k\choose 1} \Delta a_n  + \cdots + {k\choose k} \Delta^k(a_n).</math>

A '''{{vanchor|difference equation}}''' of order {{mvar|k}} is an equation that involves the {{mvar|k}} first differences of a sequence or a function, in the same way as a [[ordinary differential equation|differential equation]] of order {{mvar|k}} relates the {{mvar|k}} first [[derivative]]s of a function.

The two above relations allow transforming a recurrence relation of order {{mvar|k}} into a difference equation of order {{mvar|k}}, and, conversely, a difference equation of order {{mvar|k}} into recurrence relation of order {{mvar|k}}. Each transformation is the [[inverse function|inverse]] of the other, and the sequences that are solution of the difference equation are exactly those that satisfies the recurrence relation.

For example, the difference equation
:<math>3\Delta^2 a_n + 2\Delta a_n + 7a_n = 0</math>
is equivalent to the recurrence relation
:<math>3a_{n+2} = 4a_{n+1} - 8a_n,</math>
in the sense that the two equations are satisfied by the same sequences.

As it is equivalent for a sequence to satisfy a recurrence relation or to be the solution of a difference equation, the two terms "recurrence relation" and "difference equation" are sometimes used interchangeably. See [[Rational difference equation]] and [[Matrix difference equation]] for example of uses of "difference equation" instead of "recurrence relation"

Difference equations resemble differential equations, and this resemblance is often used to mimic methods for solving differentiable equations to apply to solving difference equations, and therefore recurrence relations.

[[Summation equation]]s relate to difference equations as [[integral equation]]s relate to differential equations. See [[time scale calculus]] for a unification of the theory of difference equations with that of differential equations.

===From sequences to grids===
Single-variable or one-dimensional recurrence relations are about sequences (i.e. functions defined on one-dimensional grids). Multi-variable or n-dimensional recurrence relations are about <math>n</math>-dimensional grids. Functions defined on <math>n</math>-grids can also be studied with partial difference equations.<ref>[https://books.google.com/books?id=1klnDGelHGEC Partial difference equations], Sui Sun Cheng, CRC Press, 2003, {{isbn|978-0-415-29884-1}}</ref>

== Solving ==

=== Solving linear recurrence relations with constant coefficients ===
{{main|Linear recurrence with constant coefficients}}

===Solving first-order non-homogeneous recurrence relations with variable coefficients===
Moreover, for the general first-order non-homogeneous linear recurrence relation with variable coefficients:

:<math>a_{n+1} = f_n a_n + g_n, \qquad f_n \neq 0,</math>
there is also a nice method to solve it:<ref>{{cite web |url=http://faculty.pccu.edu.tw/%7Emeng/Math15.pdf |title=Archived copy |access-date=2010-10-19 |url-status=live |archive-url=https://web.archive.org/web/20100705023731/http://faculty.pccu.edu.tw/~meng/Math15.pdf |archive-date=2010-07-05 }}</ref>
:<math>a_{n+1} - f_n a_n = g_n</math>
:<math>\frac{a_{n+1}}{\prod_{k=0}^n f_k} - \frac{f_n a_n}{\prod_{k=0}^n f_k} = \frac{g_n}{\prod_{k=0}^n f_k}</math>

:<math>\frac{a_{n+1}}{\prod_{k=0}^n f_k} - \frac{a_n}{\prod_{k=0}^{n-1} f_k} = \frac{g_n}{\prod_{k=0}^n f_k}</math>

Let 
:<math>A_n = \frac{a_n}{\prod_{k=0}^{n-1} f_k},</math>
Then 
:<math>A_{n+1} - A_n = \frac{g_n}{\prod_{k=0}^n f_k}</math>
:<math>\sum_{m=0}^{n-1}(A_{m+1} - A_m) = A_n - A_0 = \sum_{m=0}^{n-1}\frac{g_m}{\prod_{k=0}^m f_k}</math>
:<math>\frac{a_n}{\prod_{k=0}^{n-1} f_k} = A_0 + \sum_{m=0}^{n-1}\frac{g_m}{\prod_{k=0}^m f_k}</math>
:<math>a_n = \left(\prod_{k=0}^{n-1} f_k \right) \left(A_0 + \sum_{m=0}^{n-1}\frac{g_m}{\prod_{k=0}^m f_k}\right)</math>

If we apply the formula to <math>a_{n+1} = (1 + h f_{nh}) a_n + hg_{nh}</math> and take the limit <math>h \to 0</math>, we get the formula for first order [[linear differential equation]]s with variable coefficients; the sum becomes an integral, and the product becomes the exponential function of an integral.

===Solving general homogeneous linear recurrence relations===
Many homogeneous linear recurrence relations may be solved by means of the [[generalized hypergeometric series]]. Special cases of these lead to recurrence relations for the [[orthogonal polynomials]], and many [[special function]]s. For example, the solution to

:<math>J_{n+1}=\frac{2n}{z}J_n-J_{n-1}</math>

is given by

:<math>J_n=J_n(z), </math>

the [[Bessel function]], while

:<math>(b-n)M_{n-1} +(2n-b+z)M_n - nM_{n+1}=0 </math>

is solved by

:<math>M_n=M(n,b;z) </math>

the [[confluent hypergeometric series]]. Sequences which are the solutions of [[P-recursive equation|linear difference equations with polynomial coefficients]] are called [[Holonomic function|P-recursive]]. For these specific recurrence equations algorithms are known which find [[Polynomial solutions of P-recursive equations|polynomial]], [[Abramov's algorithm|rational]] or [[Petkovšek's algorithm|hypergeometric]] solutions.

===Solving general non-homogeneous linear recurrence relations with constant coefficients===
Furthermore, for the general non-homogeneous linear recurrence relation with constant coefficients, one can solve it based on variation of parameter.<ref>[https://www.techrxiv.org/doi/full/10.36227/techrxiv.174439127.75795694/v1 Solution of Nonhomogeneous Linear Recurrence Relations with Constant Coefficient based on Variation of Parameter], Haoran Han, 2025</ref>


===Solving first-order rational difference equations===
{{Main|Rational difference equation}}

A first order rational difference equation has the form <math>w_{t+1} = \tfrac{aw_t+b}{cw_t+d}</math>. Such an equation can be solved by writing <math>w_t</math> as a nonlinear transformation of another variable <math>x_t</math> which itself evolves linearly.  Then standard methods can be used to solve the linear difference equation in <math>x_t</math>.

==Stability==

===Stability of linear higher-order recurrences===
The linear recurrence of order <math>d</math>,

:<math>a_n = c_1a_{n-1} + c_2a_{n-2}+\cdots+c_da_{n-d},  </math>

has the [[Characteristic polynomial|characteristic equation]]

:<math>\lambda^d - c_1 \lambda^{d-1} - c_2 \lambda^{d-2} - \cdots - c_d \lambda^0 =0. </math>

The recurrence is [[Stability theory|stable]], meaning that the iterates converge asymptotically to a fixed value, if and only if the [[eigenvalues]] (i.e., the roots of the characteristic equation), whether real or complex, are all less than [[unity (mathematics)|unity]] in absolute value.

===Stability of linear first-order matrix recurrences===
{{Main|Matrix difference equation}}

In the first-order matrix difference equation

:<math>[x_t - x^*] = A[x_{t-1}-x^*]</math>

with state vector <math>x</math> and transition matrix <math>A</math>, <math>x</math> converges asymptotically to the steady state vector <math>x^*</math> if and only if all eigenvalues of the transition matrix <math>A</math> (whether real or complex) have an [[absolute value]] which is less than&nbsp;1.

===Stability of nonlinear first-order recurrences===
Consider the nonlinear first-order recurrence

:<math>x_n=f(x_{n-1}).</math>

This recurrence is [[stability theory|locally stable]], meaning that it [[limit of a sequence|converges]] to a fixed point <math>x^*</math> from points sufficiently close to <math>x^*</math>, if the slope of <math>f</math>  in the neighborhood of <math>x^*</math> is smaller than [[unity (mathematics)|unity]] in absolute value:  that is,

: <math>| f' (x^*) | < 1. </math>

A nonlinear recurrence could have multiple fixed points, in which case some fixed points may be locally stable and others locally unstable; for continuous ''f''  two adjacent fixed points cannot both be locally stable.

A nonlinear recurrence relation could also have a cycle of period <math>k</math> for <math>k > 1</math>.  Such a cycle is stable, meaning that it attracts a set of initial conditions of positive measure, if the composite function

:<math>g(x) := f \circ f \circ \cdots \circ f(x)</math>

with <math>f</math> appearing <math>k</math> times is locally stable according to the same criterion:

: <math>| g' (x^*) | < 1,</math>

where <math>x^*</math> is any point on the cycle.

In a [[chaos theory|chaotic]] recurrence relation, the variable <math>x</math> stays in a bounded region but never converges to a fixed point or an attracting cycle; any fixed points or cycles of the equation are unstable.  See also [[logistic map]], [[dyadic transformation]], and [[tent map]].

== Relationship to differential equations ==
When solving an [[ordinary differential equation]] [[numerical ordinary differential equations|numerically]], one typically encounters a recurrence relation. For example, when solving the [[initial value problem]]

:<math>y'(t) = f(t,y(t)), \ \ y(t_0)=y_0,</math>

with [[Euler's method]] and a step size <math>h</math>, one calculates the values

:<math>y_0=y(t_0), \ \ y_1=y(t_0+h), \ \ y_2=y(t_0+2h), \ \dots</math>

by the recurrence

:<math>\, y_{n+1} = y_n + hf(t_n,y_n), t_n = t_0 + nh </math>

Systems of linear first order differential equations can be discretized exactly analytically using the methods shown in the [[discretization]] article.

== Applications ==

=== Mathematical biology ===
Some of the best-known difference equations have their origins in the attempt to model [[population dynamics]]. For example, the [[Fibonacci number]]s were once used as a model for the growth of a rabbit population.

The [[logistic map]] is used either directly to model population growth, or as a starting point for more detailed models of population dynamics. In this context, coupled difference equations are often used to model the interaction of two or more [[population]]s. For example, the [[Nicholson–Bailey model]] for a host-[[parasite]] interaction is given by

:<math>N_{t+1} = \lambda N_t e^{-aP_t} </math>
:<math>P_{t+1} = N_t(1-e^{-aP_t}), </math>

with <math>N_t</math> representing the hosts, and <math>P_t</math> the parasites, at time <math>t</math>.

[[Integrodifference equation]]s are a form of recurrence relation important to spatial [[ecology]]. These and other difference equations are particularly suited to modeling [[voltinism|univoltine]] populations.

===Computer science===
Recurrence relations are also of fundamental importance in [[analysis of algorithms]].<ref>Cormen, T. et al, ''Introduction to Algorithms'', MIT Press, 2009</ref><ref>R. Sedgewick, F. Flajolet, ''An Introduction to the Analysis of Algorithms'', Addison-Wesley, 2013</ref> If an [[algorithm]] is designed so that it will break a problem into smaller subproblems ([[Divide and conquer algorithm|divide and conquer]]), its running time is described by a recurrence relation.

A simple example is the time an algorithm takes to find an element in an ordered vector with <math>n</math> elements,  in the worst case.

A naive algorithm will search from left to right, one element at a time. The worst possible scenario is when the required element is the last, so the number of comparisons is <math>n</math>.

A better algorithm is called [[Binary search algorithm|binary search]]. However, it requires a sorted vector. It will first check if the element is at the middle of the vector. If not, then it will check if the middle element is greater or lesser than the sought element. At this point, half of the vector can be discarded, and the algorithm can be run again on the other half. The number of comparisons will be given by

:<math>c_1=1</math>
:<math>c_n=1+c_{n/2}</math>

the [[time complexity]] of which will be <math>O(\log_2(n))</math>.

=== Digital signal processing ===
In [[digital signal processing]], recurrence relations can model feedback in a system, where outputs at one time become inputs for future time. They thus arise in [[infinite impulse response]] (IIR) [[digital filter]]s.

For example, the equation for a "feedforward" IIR [[comb filter]] of delay <math>T</math> is:

:<math>y_t = (1 - \alpha) x_t + \alpha y_{t - T},</math>

where <math>x_t</math> is the input at time <math>t</math>, <math>y_t</math> is the output at time <math>t</math>, and <math>\alpha</math> controls how much of the delayed signal is fed back into the output.  From this we can see that

:<math>y_t = (1 - \alpha) x_t + \alpha ((1-\alpha) x_{t-T} + \alpha y_{t - 2T})</math>
:<math>y_t = (1 - \alpha) x_t + (\alpha-\alpha^2) x_{t-T}  + \alpha^2 y_{t - 2T}</math>

etc.

===Economics===
{{see also|time series analysis|simultaneous equations model}}
Recurrence relations, especially linear recurrence relations, are used extensively in both theoretical and empirical economics.<ref>{{cite book |first1=Nancy L. |last1=Stokey |author-link=Nancy Stokey | first2=Robert E. Jr. |last2=Lucas |author-link2=Robert Lucas, Jr. |first3=Edward C. |last3=Prescott |author-link3=Edward C. Prescott |title=Recursive Methods in Economic Dynamics |location=Cambridge |publisher=Harvard University Press |year=1989 |isbn=0-674-75096-9 |url=https://books.google.com/books?id=BgQ3AwAAQBAJ }}</ref><ref>{{cite book |last2=Sargent |first2=Thomas J. |author-link2=Thomas J. Sargent |first1=Lars |last1=Ljungqvist |author-link=Lars Ljungqvist |title=Recursive Macroeconomic Theory |location=Cambridge |publisher=MIT Press |edition=Second |year=2004 |isbn=0-262-12274-X |url=https://archive.org/details/recursivemacroec02edljun |url-access=registration }}</ref>  In particular, in macroeconomics one might develop a model of various broad sectors of the economy (the financial sector, the goods sector, the labor market, etc.) in which some agents' actions depend on lagged variables.  The model would then be solved for current values of key variables ([[interest rate]], real [[GDP]], etc.) in terms of past and current values of other variables.

==See also==
{{colbegin}}

* [[Circle points segments proof]]
* [[Combinatorial principles]]
* [[Continued fraction]]
* [[Holonomic function|Holonomic sequences]]
* [[Infinite impulse response]]
* [[Integration by reduction formulae]]
* [[Iterated function]]
* [[Lagged Fibonacci generator]]
* [[Master theorem (analysis of algorithms)]]
* [[Mathematical induction]]
* [[Orthogonal polynomials]]
* [[Recursion]]
* [[Recursion (computer science)]]
* [[Time scale calculus]]
{{colend}}

== References ==
=== Footnotes ===
{{reflist}}

=== Bibliography ===
* {{cite book|first1=Paul M. | last1=Batchelder
|title=An introduction to linear difference equations
|year=1967
|publisher=Dover Publications
}}
* {{cite book|first1=Kenneth S. | last1=Miller
|title=Linear difference equations
|year=1968 | publisher=W. A. Benjamin
}}
* {{cite news|first1=Jay P. | last1=Fillmore | first2=Morris L. | last2=Marx
|title=Linear recursive sequences | journal= SIAM Rev. | volume=10|number=3
|pages=324–353 | jstor=2027658| year=1968}}
* {{cite book| first1=Alfred | last1=Brousseau
|title=Linear Recursion and Fibonacci Sequences
|year=1971
|publisher=Fibonacci Association
|url=http://www.fq.math.ca/linear.html
}}
* [[Thomas H. Cormen]], [[Charles E. Leiserson]], [[Ronald L. Rivest]], and [[Clifford Stein]]. ''[[Introduction to Algorithms]]'', Second Edition. MIT Press and McGraw-Hill, 1990. {{isbn|0-262-03293-7}}. Chapter 4: Recurrences, pp.&nbsp;62–90.
*{{cite book| first1=Ronald L. | last1=Graham |
first2=Donald E. | last2=Knuth
|first3=Oren | last3=Patashnik
|title=Concrete Mathematics: A Foundation for Computer Science
|edition=2
|publisher=Addison-Wesley
|year=1994 |isbn=0-201-55802-5
}}
*{{cite book
 |year        = 2010
 |url         = http://eu.wiley.com/WileyCDA/WileyTitle/productCd-EHEP000338.html
 |first1      = Walter
 |last1       = Enders
 |title       = Applied Econometric Times Series
 |edition     = 3
 |url-status     = dead
 |archive-url  = https://web.archive.org/web/20141110221941/http://eu.wiley.com/WileyCDA/WileyTitle/productCd-EHEP000338.html
 |archive-date = 2014-11-10
}}
*{{cite book| first1=Paul | last1=Cull | first2=Mary | last2=Flahive|author2-link= Mary Flahive
|first3=Robbie | last3=Robson
|title=Difference Equations: From Rabbits to Chaos
|title-link= Difference Equations: From Rabbits to Chaos
|publisher=Springer | year=2005 | isbn=0-387-23234-6 }} chapter 7.
* {{cite book| first1=Ian | last1=Jacques
|title=Mathematics for Economics and Business
| url=https://archive.org/details/mathematicsforec00jacq | url-access=limited |edition=Fifth
|publisher=Prentice Hall
|year=2006
|isbn=0-273-70195-9
|pages=[https://archive.org/details/mathematicsforec00jacq/page/n561 551]–568
}} Chapter 9.1: Difference Equations.
* {{cite web | first1=Tang | last1=Minh | first2=Tan | last2=Van To | title=Using generating functions to solve linear inhomogeneous recurrence equations | pages=399–404 | url=http://www.wseas.us/e-library/conferences/2006lisbon/papers/517-481.pdf | year=2006 | work=Proc. Int. Conf. Simulation, Modelling and Optimization, SMO'06 | access-date=2014-08-07 | archive-date=2016-03-04 | archive-url=https://web.archive.org/web/20160304035925/http://www.wseas.us/e-library/conferences/2006lisbon/papers/517-481.pdf | url-status=dead }}
* {{cite web| first1=Andrei D. | last1=Polyanin
|title=Difference and Functional Equations: Exact Solutions
|url=http://eqworld.ipmnet.ru/en/solutions/fe.htm}} at EqWorld - The World of Mathematical Equations.
* {{cite web| first1=Andrei D. | last1=Polyanin
|title=Difference and Functional Equations: Methods
|url=http://eqworld.ipmnet.ru/en/education/edu-fe.htm}} at EqWorld - The World of Mathematical Equations.
*{{cite journal | first1= Xiang-Sheng | last1=Wang
| first2=Roderick | last2=Wong
|title= Asymptotics of orthogonal polynomials via recurrence relations
|journal =  Anal. Appl. | year=2012 | volume=10 | number=2
|pages=215–235 | doi=10.1142/S0219530512500108 
|arxiv=1101.4371| s2cid=28828175
}}

== External links ==
* {{springer|title=Recurrence relation|id=p/r080150}}
* {{MathWorld | urlname= RecurrenceEquation | title= Recurrence Equation}}
* {{cite web |title= OEIS Index Rec|url=http://oeis.org/wiki/Index_to_OEIS:_Section_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)

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[[Category:Algebra]]
[[Category:Recurrence relations| ]]
[[Category:Combinatorics]]