Recurrence RelationsA 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 a_{r+3} + 3a_{r+2} + 6a_{r+1} + 9a_{r} = 0 y_{k+3} + 3y_{k+2} + 6y_{k+1} + 9y_{k} = 0 Example2: The Fibonacci sequence is defined by the recurrence relation a_{r} = a_{r-2} + a_{r-1}, r≥2,with the initial conditions a_{0}=1 and a_{1}=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 a_{r}=y_{k}. Example1: The equation 13a_{r}+20a_{r-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 a_{r}=y_{k} Example1: The equation y^{3}_{k+3}+2y^{2}_{k+2}+2y_{k+1}=0 has the degree 3, as the highest power of y_{k} is 3. Example2: The equation a^{4}_{r}+3a^{3}_{r-1}+6a^{2}_{r-2}+4a_{r-3} =0 has the degree 4, as the highest power of a_{r} is 4. Example3: The equation y_{k+3} +2y_{k+2} +4y_{k+1}+2y_{k}= k(x) has the degree 1, because the highest power of y_{k} 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. |
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