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I was reading this definition of a linear recurrence, and was wondering what characteristics are required of a linear recurrence for it to be solvable? Meaning, can I find a closed form?

The subject of solving linear recurrence relations was just barely touched on in a Discrete Math course I took a while back, and now I'm trying to pursue the subject further as part of reading Knuth's book, Concrete Mathematics. I've taken a Linear 1 course, but don't know anything about Linear Difference Equations.

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Robert S. Barnes
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Let's write a linear recurrence as $$f_0(n)a_n+f_1(n)a_{n-1}+\cdots+f_r(n)a_{n-r}=g(n)\tag1$$ where the functions $f_0,\dots,f_r$ and $g$ are given.

If the $f_i$ are all constant, and $g$ is identically zero, then the solutions can be expressed in closed form in terms of the roots of the characteristic polynomial, $$f_0x^n+f_1x^{n-1}+\cdots+f_r\tag2$$ Of course, there may be no closed form for the roots of this polynomial.

If the $f_i$ are all constant, and $g$ is of any one of a number of special forms (essentially, polynomials, exponentials, and combinations thereof), then again the solutions of (1) can be expressed in closed form in terms of the roots of (2).

There are other special cases where (1) can be solved in closed form, but I don't think there's any kind of general theory that tells you when it can and when it can't. Click on some of the questions listed to the right under "related" to see examples done.

Gerry Myerson
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  • I'm a little confused. It seems like one of the major cases where I can always find a closed form is when the recurrence is linear, has constant coefficients, and is homogeneous. But then you say, "Of course, there may be no closed form for the roots of this polynomial." Could you maybe explain a little more? Thanks. – Robert S. Barnes Feb 11 '13 at 08:47
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    The roots of the polynomial $x^5-x-1$, for example, cannot be expressed in closed form in terms of the four arithmetic operations, square roots, cube roots, fifth roots, etc., exponentials, logarithms, trig and inverse trig functions --- the usual functions of intro calculus. The solutions of the recurrence $a_n=a_{n-4}+a_{n-5}$ can be expressed in terms of the roots of that polynomial, but the roots of that polynomial.... – Gerry Myerson Feb 11 '13 at 12:01
  • Aren't real? So in addition to being linear, having constant coefficients and being homogeneous, the characteristic equation has to have real roots in order for there to be a closed form? – Robert S. Barnes Feb 11 '13 at 16:05
  • Real/complex has nothing to do with it. $x^2+1=0$ has non-real roots, but $a_n+a_{n-2}=0$ has closed form solutions. There are polynomials of degree $5$ and higher, having all real roots, but no closed form expression for those roots. The question of which polynomial equations can be solved in radicals is the topic of Galois Theory, q.v. – Gerry Myerson Feb 11 '13 at 22:44
  • OK, I looked up Galois Theory I think I get it now ( in a rudimentary sense ). A homogeneous linear recurrence relation with constant coefficients will only have a closed form if it's Galois Group is solvable, which is only always true for polynomials of order 4 or less. Is that more or less correct? – Robert S. Barnes Feb 12 '13 at 09:45
  • Yes.${}{}{}{}{}$ – Gerry Myerson Feb 12 '13 at 11:07
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One of the cases that can always be "solved" is the linear recurrence of first order:

$$ a_{n + 1} = f(n) a_n + g(n) $$

Divide by the summing factor $s_n = \prod_{0 \le k \le n} f(k)$ to get:

$$ \frac{a_{n + 1}}{s_n} - \frac{a_n}{s_{n - 1}} = \frac{g(n)}{s_n} $$

Sum for $n$ from 0 to $n$, the right hand side telescopes:

$$ \frac{a_{n + 1}}{s_n} = a_0 + \sum_{0 \le k \le n} \frac{g(k)}{s_k} $$

This obviously requires $s_k \ne 0$ for the relevant range(i.e., $f(k) \ne 0$), and it qualifies as a "solution" as long as you know the left hand side's sum and $s_n$ has a nice form.

Most linear recurrences of higher orders can't be solved in closed form (just like the case for differential equations, there are uncanny parallels). In some cases defining a generating function (see e.g. Wilf's "generatingfunctionology", the next to last edition is available for free at the link) gets you an algebraic or differential equation that can be solved, in which case you get the terms as the coefficients of the solution. For linear recurrences with constant coefficients generating functions offer a clean, general way to solve them. Some nonlinear recurrences also succumb to this approach.

vonbrand
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Consider a sequence $a_0, a_1, \dotsc$ with a finite linear recurrence with constant coefficients: $a_n = \sum_{i = 1}^mk_ia_{n - i}$ for some constant $m$. Let $v_j = \begin{bmatrix}a_{j - 1} & a_{j - 2} & \cdots & a_{j - m}\end{bmatrix}^\intercal$ for any $j > m$, where $v_m$ is known. Then we have $$ v_{j + 1} = \overbrace{\begin{bmatrix}k_1 & k_2 & \cdots & k_{m - 1} & k_m\\1 & 0 & \cdots & 0 & 0\\0 & 1 & \cdots & 0 & 0\\\vdots & \vdots & \ddots & \vdots & \vdots\\0 & 0 & \cdots & 1 & 0\end{bmatrix}}^Mv_j. $$

In the event that we can diagonalize $M = P^{-1}\operatorname{diag}(d_1, d_2, \dotsc, d_m)P$, $$ (Pv_{m + j}) = \begin{bmatrix}d_1^j & 0 & \cdots & 0\\0 & d_2^j & \cdots & 0\\\vdots & \vdots & \ddots & \vdots\\0 & 0 & \cdots & d_m^j\end{bmatrix}(Pv_m). $$

This should give a closed form for the sequence $a_0, a_1, \dotsc$.

Herng Yi
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