ABSTRACT
This short chapter offers a gentle introduction to some of the basic ideas in this book. For definitions and additional introductory material see Chapter 2. Consider the difference equation
xn+1 = xn + a
xn − xn−1 , n = 0, 1, 2 . . . (1.1)
where a is a nonzero real number. This equation has order 2, or is second order, because of the difference between the highest index n+1 and the lowest n− 1. Its solution is the sequence of real numbers xn that may be calculated recursively from a pair of unequal real numbers x0, x−1 as the index n increases through the non-negative integers. Can we find this solution or determine its essential properties for all values of n and all unequal initial pairs x0, x−1? Equation (1.1) has a symmetry in its form that is easy to identify when it
is written as xn+1 − xn = a
xn − xn−1 . (1.2)
Setting tn = xn−xn−1 in the right-hand side and tn+1 = xn+1−xn for the left-hand side (to account for the shift in index n) changes Eq. (1.2) to the simpler (and first-order) equation
tn+1 = a
tn . (1.3)
The expression xn − xn−1 is an example of what we call an order-reducing form symmetry. This form symmetry also establishes a link between the second-order (1.1) and the first-order (1.3) in the following sense: Information about each solution {tn} of (1.3) can be translated into information about the corresponding solution of (1.1) using the equation
xn = xn−1 + tn = xn−2 + t1 + t0 = · · · = x0 + n∑
tk. (1.4)
In this particular example, if x0 and x−1 are unequal then each solution of Eq. (1.3) is a sequence of numbers tn taking only two possible values:
tn = t0 = x0 − x−1, if n is even tn =
a
t0 =
a
x0 − x−1 , if n is odd.
