difference equation
A relation between consecutive elements of a sequence. The
first difference is
D u(n) ◦ u(n:1) - u(n)
where u(n) is the nth element of sequence u. The second
difference is
D2 u(n) ◦ D (D u(n))
◦ (u(n:2) - u(n:1)) - (u(n:1) - u(n))
◦ u(n:2) - 2u(n:1) : u(n)
And so on. A recurrence relation such as
u(n:2) : a u(n:1) : b u(n) ◦ 0
can be converted to a difference equation (in this case, a
second order linear difference equation):
D2 u(n) : p D u(n) : q u(n) ◦ 0
and vice versa. a, b, p, q are constants.
(1995-02-10)