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In mathematics (including combinatorics, linear algebra, and dynamical systems), a linear recurrence with constant coefficients [1]: ch. 17 [2]: ch. 10 (also known as a linear recurrence relation or linear difference equation) sets equal to 0 a polynomial that is linear in the various iterates of a variable—that is, in the values of the elements of a sequence.
The recurrence of order two satisfied by the Fibonacci numbers is the canonical example of a homogeneous linear recurrence relation with constant coefficients (see below). The Fibonacci sequence is defined using the recurrence = + with initial conditions
A homogeneous linear differential equation has constant coefficients if it has the form + ′ + ″ + + = where a 1, ..., a n are (real or complex) numbers. In other words, it has constant coefficients if it is defined by a linear operator with constant coefficients.
A linear differential equation is homogeneous if it is a homogeneous linear equation in the unknown function and its derivatives. It follows that, if φ(x) is a solution, so is cφ(x), for any (non-zero) constant c. In order for this condition to hold, each nonzero term of the linear differential equation must depend on the unknown function or ...
[3] [4] The characteristic equation can only be formed when the differential or difference equation is linear and homogeneous, and has constant coefficients. [1] Such a differential equation, with y as the dependent variable, superscript (n) denoting n th-derivative, and a n, a n − 1, ..., a 1, a 0 as constants,
If the a i are constants (independent of x and y) then the PDE is called linear with constant coefficients. If f is zero everywhere then the linear PDE is homogeneous, otherwise it is inhomogeneous. (This is separate from asymptotic homogenization, which studies the effects of high-frequency oscillations in the coefficients upon solutions to PDEs.)
A non-homogeneous linear recurrence is an equation of the form ... -regular sequence satisfies a linear recurrences with constant coefficients, ...
A polynomial of degree 0 is always homogeneous; it is simply an element of the field or ring of the coefficients, usually called a constant or a scalar. A form of degree 1 is a linear form . [ notes 2 ] A form of degree 2 is a quadratic form .