<mathematics> 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) = 0can 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) = 0and vice versa. a, b, p, q are constants.
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