Difference Equations

Motivation: arises from discretization of ODEs/PDEs.

Discrete Calculus

Difference

\[ \begin{aligned} D a_n &= a_{n+1} - a_n := a_n^{(1)} \\ D^2 a_n &= D a_{n+1} - D a_n = a_{n+2} - 2 a_{n+1} + a_n := a_n^{(2)} \\ \vdots & \\ D^k a_n &= D a_{n}^{(k-1)} = \sum_{j=0}^k (-1)^j \binom{k}{j} a_{n+k-j} \end{aligned} \]

Discrete Integral

\[ b_n = \sum_{j = n_0}^{n} a_j + c_0 \]

• \(D(f_n + g_n) = D f_n + D g_n\)
• \(D(f_n g_n) = f_{n+1} D g_n + g_n D f_n\)
• \(D\left(\frac{f_n}{g_n}\right) = \frac{g_n D f_n - f_n D g_n}{g_n g_{n+1}}\)

Continuous	Discrete
\(y = x^k\)	\(f_n = n(n+1)\ldots(n+k-1)\)
\((x^k)' = k x^{k-1}\)	\(\begin{aligned} D f_n &= (n+1)\ldots (n+k) - n(n+1)\ldots (n+k-1) \\ &= k \underset{k-1 \text{ terms}}{\underbrace{(n+1)\ldots (n+k-1)}} \end{aligned}\)
\(\int x^k \mathrm{d}x = \frac{x^{k+1}}{k+1} + c\)	\(\sum_{j=n_0}^n j(j+1)\ldots (j+k-1)\)

Difference Equations

\(N\)-th order difference equation:

\[ D^N a_n = F(n, a_n, D a_n, \ldots, D^{N-1} a_n) \]

• \(D^k a_n\) depends on \(a_n, a_{n+1}, \ldots, a_{n+k}\)

We can omit the \(D\) operator and rewrite it in algebraic form:

\[ a_{n+N} = G(n, a_n, a_{n+1}, \ldots, a_{n+N-1}) \]

• \(a_{n+1} = D a_n + a_n\)
• \(a_{n+2} = D^2 a_n + 2 D a_n + a_n\)

similar to ODEs, we have

• General solutions: $$ a_n = a(n; \underset{\text{general const.}}{\underbrace{c_1, c_2, \ldots, c_N}}) $$

Factorial

\[ a_{n+1} = n a_n \]

general solution:

\[ a_n = c_1 (n-1)! \]

Gamma function

\[ \Gamma(n) = (n-1)! \]

Definition:

\[ \Gamma(z) = \int_0^\infty t^{z-1} e^{-t} \, \mathrm{d}t, \quad \text{Re}(z) > 0 \]

Integration by parts gives

\[ \Gamma(z+1) = z \Gamma(z) \]

extends to \(\mathbb{C}\) except at \(z = 0, -1, -2, \ldots\)

First Order

Homogeneous

\[ a_{n+1} = p(n) a_n \]

take natural log on both sides:

\[ \ln a_{n+1} - \ln a_n = \ln p(n) \]

\[ D(\ln a_n) = \ln p(n) \]

thus,

\[ \ln a_n = \sum_{j=n_0}^{n} \ln p(j) + c_0 \]

the general solution is

\[ a_n = e^{c_0} \prod_{j=1}^{n} p(j) \]

Inhomogeneous

\[ a_{n+1} = p(n) a_n + q(n) \]

For ODEs like

\[ y' = p(x) y + q(x) \]

we have the multiplying factor \(e^{\int p(x) \mathrm{d}x}\).

• summing factors

\[ \left(\prod_{j=1}^n p(j)\right)^{-1} \]

\[ a_n = \prod_{j=1}^{n} p(j) \left(\sum_{k=1}^{n} \frac{q(k)}{\prod_{j=1}^{k} p(j)} + c_0\right) \]

Higher Order

\[ a_{n+N} + p_{N-1}(n) a_{n+N-1} + \ldots + p_1(n) a_{n+1} + p_0(n) a_n = 0 \]

Linear dependency of \(N\) solutions by Wronskian

\[ \begin{aligned} W_n &= W(n) = W\left[a^1, a^2, \ldots, a^N\right] \\[1ex] &= \det \begin{vmatrix} a^1(n) & a^2(n) & \ldots & a^N(n) \\[1ex] a^1(n+1) & a^2(n+1) & \ldots & a^N(n+1) \\ \vdots & \vdots & \ddots & \vdots \\[1ex] a^1(n+N-1) & a^2(n+N-1) & \ldots & a^N(n+N-1) \end{vmatrix} \end{aligned} \]

Abel's formula

\[ W_n = W_{n_0} \prod_{j=n_0}^{n-1} p_0(j), \quad n > n_0 \]

• IVP:

\[ \begin{array}{c} a_{n_0}, D a_{n_0}, D^2 a_{n_0}, \ldots, D^{N-1} a_{n_0} \\ a_{n_0}, a_{n_0+1}, a_{n_0+2}, \ldots, a_{n_0+N-1} \end{array} \]

• BVP: \(a_n\) at \(N\) different \(n\) values

Constant coeff. Linear Homogeneous

Recall for linear homogeneous ODEs with constant coeff., we use \(y = e^{r x}\) ansatz.

For difference equations, we use the ansatz

\[ a_n = r^n. \]

• for single root \(r\): \(a_n = r^n\)
• for multiple root \(r\) of multiplicity \(m\): \(a_n = n^k r^n, k = 0, 1, \ldots, m-1\)

Euler eq. (equidimensional eq.)

ODE:

\[ x^N y^{(N)} + p_{N-1} x^{N-1} y^{(N-1)} + \ldots + p_1 x y' + p_0 y = 0 \]

\(x^N \leftrightarrow n(n-1)\ldots(n-N+1) = \Gamma(n+N) / \Gamma(n)\)

Difference Euler eq.:

\[ \frac{\Gamma(n+N)}{\Gamma(n)} a_n^{(N)} + p_{N-1} \frac{\Gamma(n+N-1)}{\Gamma(n)} a_n^{(N-1)} + \ldots + p_1 n a_n^{(1)} + p_0 a_n = 0 \]

Try the ansatz \(a_n = \frac{\Gamma(n+r)}{\Gamma(n)}\).

Inhomogeneous Linear Eq.

1. Variation of parameters
2. Reduction of order

Generating Functions

Given \(a_n\) (integer function), we can construct

\[ F(x) = \sum_{n=0}^{\infty} a_n (x - x_0)^n \]

converges near \(x_0\).

e.g.

\[ (n+1)(n+2) a_{n+2} - 2(n+1) a_{n+1} - 3 a_n = 0, \quad \text{i.c. } a_0 = 2, a_1 = 2 \]

Let \(F(x) = \sum_{n=0}^{\infty} a_n x^n\). Then

\[ F'' - 2 F' - 3 F = 0 \]