Recurrence relation

In mathematics and computer science, a recurrence relation is an equation according to which the \(n\)th term of a sequence of numbers is equal to some combination of the previous terms. Often, only \(k\) previous terms of the sequence appear in the equation, for a parameter \(k\) that is independent of \(n\); this number \(k\) is called the order of the relation. If the values of the first \(k\) numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation.

In linear recurrences, the nth term is equated to a linear function of the \(k\) previous terms. A famous example is the recurrence for the Fibonacci numbers, \[F_n=F_{n-1}+F_{n-2}\] where the order \(k\) 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 \(n.\) For these recurrences, one can express the general term of the sequence as a closed-form expression of \(n\). As well, linear recurrences with polynomial coefficients depending on \(n\) 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 \(n\).

The concept of a recurrence relation can be extended to multidimensional arrays, that is, indexed families that are indexed by tuples of natural numbers.

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

\(u_n=\varphi(n, u_{n-1})\quad\text{for}\quad n>0,\)

where

\(\varphi:\mathbb N\times X \to X\)

is a function, where X is a set to which the elements of a sequence must belong. For any \(u_0\in X\), this defines a unique sequence with \(u_0\) as its first element, called the initial value.

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 k has the form

\(u_n=\varphi(n, u_{n-1}, u_{n-2}, \ldots, u_{n-k})\quad\text{for}\quad n\ge k,\)

where \(\varphi: \mathbb N\times X^k \to X\) is a function that involves k consecutive elements of the sequence. In this case, k initial values are needed for defining a sequence.

The factorial is defined by the recurrence relation

\(n!=n\cdot (n-1)!\quad\text{for}\quad n>0,\)

and the initial condition

\(0!=1.\)

This is an example of a linear recurrence with polynomial coefficients of order 1, with the simple polynomial (in n)

as its only coefficient.

An example of a recurrence relation is the logistic map defined by

\(x_{n+1} = r x_n (1 - x_n),\)

for a given constant \(r.\) The behavior of the sequence depends dramatically on \(r,\) but is stable when the initial condition \(x_0\) varies.

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

\(F_n = F_{n-1}+F_{n-2}\)

with initial conditions

\(F_0 = 0\)

\(F_1 = 1.\)

Explicitly, the recurrence yields the equations

\(F_2 = F_1 + F_0\)

\(F_3 = F_2 + F_1\)

\(F_4 = F_3 + F_2\)

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 \(t^2 = t + 1\); the generating function of the sequence is the rational function

\(\frac{t}{1-t-t^2}.\)

A simple example of a multidimensional recurrence relation is given by the binomial coefficients \(\tbinom{n}{k}\), which count the ways of selecting \(k\) elements out of a set of \(n\) elements. They can be computed by the recurrence relation

\(\binom{n}{k}=\binom{n-1}{k-1}+\binom{n-1}{k},\)

with the base cases \(\tbinom{n}{0}=\tbinom{n}{n}=1\). 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 factorials, multiplication and division, not just additions:

\(\binom{n}{k}=\frac{n!}{k!(n-k)!}.\)

The binomial coefficients can also be computed with a uni-dimensional recurrence:

\(\binom n k = \binom n{k-1}(n-k+1)/k,\)

with the initial value \(\binom n 0 =1\) (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 not involve very large integers as does the formula with factorials (if one uses \(\binom nk= \binom n{n-k},\) all involved integers are smaller than the final result).

The difference operator is an operator that maps sequences to sequences, and, more generally, functions to functions. It is commonly denoted \(\Delta,\) and is defined, in functional notation, as

\((\Delta f)(x)=f(x+1)-f(x).\)

It is thus a special case of finite difference.

When using the index notation for sequences, the definition becomes

\((\Delta a)_n= a_{n+1} - a_n.\)

The parentheses around \(\Delta f\) and \(\Delta a\) are generally omitted, and \(\Delta a_n\) must be understood as the term of index n in the sequence \(\Delta a,\) and not \(\Delta\) applied to the element \(a_n.\)

Given sequence \(a=(a_n)_{n\in \N},\) the first difference of a is \(\Delta a.\)

The second difference is \(\Delta^2 a=(\Delta\circ\Delta)a= \Delta(\Delta a).\) A simple computation shows that

\(\Delta^2 a_n= a_{n+2} - 2a_{n+1} + a_n.\)

More generally: the kth difference is defined recursively as \(\Delta^k=\Delta\circ \Delta^{k-1},\) and one has

\(\Delta^k a_n = \sum_{t=0}^k (-1)^t \binom{k}{t} a_{n+k-t}.\)

\(a_{n+k} = a_n + {k\choose 1} \Delta a_n + \cdots + {k\choose k} \Delta^k(a_n).\)

\(3\Delta^2 a_n + 2\Delta a_n + 7a_n = 0\)

\(3a_{n+2} = 4a_{n+1} - 8a_n,\)

Condensed: the full section is in Wikipedia.

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 \(n\)-dimensional grids. Functions defined on \(n\)-grids can also be studied with partial difference equations.

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

\(a_{n+1} = f_n a_n + g_n, \qquad f_n \neq 0,\)

there is also a nice method to solve it:

\(a_{n+1} - f_n a_n = g_n\)

\(\frac{a_{n+1}}{\prod_{k=0}^n f_k} - \frac{f_n a_n}{\prod_{k=0}^n f_k} = \frac{g_n}{\prod_{k=0}^n f_k}\)

\(\frac{a_{n+1}}{\prod_{k=0}^n f_k} - \frac{a_n}{\prod_{k=0}^{n-1} f_k} = \frac{g_n}{\prod_{k=0}^n f_k}\)

Let

\(A_n = \frac{a_n}{\prod_{k=0}^{n-1} f_k},\)

Then

\(A_{n+1} - A_n = \frac{g_n}{\prod_{k=0}^n f_k}\)

\(\sum_{m=0}^{n-1}(A_{m+1} - A_m) = A_n - A_0 = \sum_{m=0}^{n-1}\frac{g_m}{\prod_{k=0}^m f_k}\)

\(\frac{a_n}{\prod_{k=0}^{n-1} f_k} = A_0 + \sum_{m=0}^{n-1}\frac{g_m}{\prod_{k=0}^m f_k}\)

\(a_n = \left(\prod_{k=0}^{n-1} f_k \right) \left(A_0 + \sum_{m=0}^{n-1}\frac{g_m}{\prod_{k=0}^m f_k}\right)\)

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

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 functions. For example, the solution to

\(J_{n+1}=\frac{2n}{z}J_n-J_{n-1}\)

is given by

\(J_n=J_n(z),\)

the Bessel function, while

\((b-n)M_{n-1} +(2n-b+z)M_n - nM_{n+1}=0\)

is solved by

\(M_n=M(n,b;z)\)

the confluent hypergeometric series. Sequences which are the solutions of linear difference equations with polynomial coefficients are called P-recursive. For these specific recurrence equations algorithms are known which find polynomial, rational or hypergeometric solutions.

Furthermore, for the general non-homogeneous linear recurrence relation with constant coefficients, one can solve it based on variation of parameter.

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

The linear recurrence of order \(d\),

\(a_n = c_1a_{n-1} + c_2a_{n-2}+\cdots+c_da_{n-d},\)

has the characteristic equation

\(\lambda^d - c_1 \lambda^{d-1} - c_2 \lambda^{d-2} - \cdots - c_d \lambda^0 =0.\)

The recurrence is 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 in absolute value.

In the first-order matrix difference equation

\([x_t - x^*] = A[x_{t-1}-x^*]\)

with state vector \(x\) and transition matrix \(A\), \(x\) converges asymptotically to the steady state vector \(x^*\) if and only if all eigenvalues of the transition matrix \(A\) (whether real or complex) have an absolute value which is less than 1.