combinatorics, linear algebra, and dynamical systems), a linear recurrence with constant coefficients: ch. 17 : ch. 10  (also known as a linear recurrence relation...

the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients, because the coefficients of the...

true for a linear equation of order one, with non-constant coefficients. An equation of order two or higher with non-constant coefficients cannot, in...

r^{n}-b_{1}r^{n-1}-\cdots -b_{n}=0,} discussed in more detail at Linear recurrence with constant coefficients. The characteristic roots (roots of the characteristic...

numbers, or complex numbers). The equation is called a linear recurrence with constant coefficients of order d. The order of the sequence is the smallest...

Édouard Lucas. Like every sequence defined by a homogenous linear recurrence with constant coefficients, the Fibonacci numbers have a closed-form expression...

{\text{otherwise}},\end{cases}}} with initial term a 0 = 0. {\displaystyle a_{0}=0.} A linear recurrence with constant coefficients is a recurrence relation of the form...

sequence satisfying a linear recurrence with constant coefficients. This theorem states that, if such a sequence has zeros, then with finitely many exceptions...

the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by...

of polynomials satisfies a linear recurrence with constant coefficients; these coefficients are identical to the coefficients of the fraction denominator...

the discussion of linearity.) If the ai are constants (independent of x and y) then the PDE is called linear with constant coefficients. If f is zero everywhere...

zero), the coefficients of all series involved in second linearly independent solutions can be calculated straightforwardly from tandem recurrence relations...

Method of undetermined coefficients Recurrence relation Dennis G. Zill (15 March 2012). A First Course in Differential Equations with Modeling Applications...

Tausworthe, Robert C. (April 1965). "Random Numbers Generated by Linear Recurrence Modulo Two" (PDF). Mathematics of Computation. 19 (90): 201–209. doi:10...

_{k},\;k=0,1,\ldots } is a sequence of functions that satisfy the linear recurrence relation ϕ k + 1 ( x ) = α k ( x ) ϕ k ( x ) + β k ( x ) ϕ k − 1 (...

satisfies a linear recurrence determines a rational function when used as the coefficients of a Taylor series. This is useful in solving such recurrences, since...

[0,2^{w}-1]} . The Mersenne Twister algorithm is based on a matrix linear recurrence over a finite binary field F 2 {\displaystyle {\textbf {F}}_{2}} ...

undetermined coefficients is an approach to finding a particular solution to certain nonhomogeneous ordinary differential equations and recurrence relations...

applied to this differential equation with step size h yields a linear recurrence relation with characteristic polynomial π ( z ; h λ ) = ( 1 − h λ β s ) z...

other terms, which are assumed to be known, are usually called constants, coefficients or parameters. An example of an equation involving x and y as unknowns...

satisfies a linear homogeneous recurrence relation with polynomial coefficients, or equivalently a linear homogeneous difference equation with polynomial...

coefficients, there is a negative binomial distribution in which the multiset coefficients occur. Multiset coefficients should not be confused with the...

constant coefficient ordinary differential equation: d u d x = c u + x 2 . {\displaystyle {\frac {du}{dx}}=cu+x^{2}.} Homogeneous second-order linear...

for some purposes such as linear recurrence relations (see below). C ( p ) {\displaystyle C(p)} is defined from the coefficients of p ( x ) {\displaystyle...

the coefficients of their Taylor series at any point satisfy a linear recurrence relation with polynomial coefficients, and that this recurrence relation...

linear recurrence equations (or linear recurrence relations or linear difference equations) with polynomial coefficients. These equations play an important...

Bott, L. Garding, "Lacunas for hyperbolic differential operators with constant coefficients I", Acta Mathematica, 124 (1970), 109–189. M. F. Atiyah, R. Bott...

Q ) {\displaystyle V_{n}(P,Q)} are certain constant-recursive integer sequences that satisfy the recurrence relation x n = P ⋅ x n − 1 − Q ⋅ x n − 2 {\displaystyle...

that is, not the root of a non-zero polynomial with integer (or, equivalently, rational) coefficients. The best-known transcendental numbers are π and...

provided a systematic and effective method for solving linear difference equations with constant coefficients, which are ubiquitous in the analysis of discrete-time...