A recurrence relation is a functional relation between the independent variable x, dependent variable f(x) and the differences of various order of f (x). A recurrence relation is also called a difference equation, and we will use these two terms interchangeably.
Example1: The equation f (x + 3h) + 3f (x + 2h) + 6f (x + h) + 9f (x) = 0 is a recurrence relation.
It can also be written as
ar+3 + 3ar+2 + 6ar+1 + 9ar = 0 yk+3 + 3yk+2 + 6yk+1 + 9yk = 0
Example2: The Fibonacci sequence is defined by the recurrence relation ar = ar-2 + ar-1, r≥2,with the initial conditions a0=1 and a1=1.
Order of the Recurrence Relation:
The order of the recurrence relation or difference equation is defined to be the difference between the highest and lowest subscripts of f(x) or ar=yk.
Example1: The equation 13ar+20ar-1=0 is a first order recurrence relation.
Example2: The equation 8f (x) + 4f (x + 1) + 8f (x+2) = k (x)
Degree of the Difference Equation:
The degree of a difference equation is defined to be the highest power of f (x) or ar=yk
Example1: The equation y3k+3+2y2k+2+2yk+1=0 has the degree 3, as the highest power of yk is 3.
Example2: The equation a4r+3a3r-1+6a2r-2+4ar-3 =0 has the degree 4, as the highest power of ar is 4.
Example3: The equation yk+3 +2yk+2 +4yk+1+2yk= k(x) has the degree 1, because the highest power of yk is 1 and its order is 3.
Example4: The equation f (x+2h) - 4f(x+h) +2f(x) = 0 has the degree1 and its order is 2.