# difference equation

## difference equation

[′dif·rəns i′kwā·zhən] (mathematics)

An equation expressing a functional relationship of one or more independent variables, one or more functions dependent on these variables, and successive differences of these functions.

McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

## 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.

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.

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