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In mathematics, and especially in numerical analysis, a homogeneous linear three-term recurrence relation (TTRR, the qualifiers "homogeneous linear" are usually taken for granted) [1] is a recurrence relation of the form
where the sequences and , together with the initial values govern the evolution of the sequence .
If the and are constant and independent of the step index n, then the TTRR is a Linear recurrence with constant coefficients of order 2. Arguably the simplest, and most prominent, example for this case is the Fibonacci sequence, which has constant coefficients .
Orthogonal polynomials Pn all have a TTRR with respect to degree n,
where An is not 0. Conversely, Favard's theorem states that a sequence of polynomials satisfying a TTRR is a sequence of orthogonal polynomials.
Also many other special functions have TTRRs. For example, the solution to
is given by the Bessel function . TTRRs are an important tool for the numeric computation of special functions.
TTRRs are closely related to continuous fractions.
Solutions of a TTRR, like those of a linear ordinary differential equation, form a two-dimensional vector space: any solution can be written as the linear combination of any two linear independent solutions. A unique solution is specified through the initial values . [2]
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In mathematics, particularly in computer algebra, Abramov's algorithm computes all rational solutions of a linear recurrence equation with polynomial coefficients. The algorithm was published by Sergei A. Abramov in 1989.
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Miller's recurrence algorithm is a procedure for the backward calculation of a rapidly decreasing solution of a three-term recurrence relation developed by J. C. P. Miller. It was originally developed to compute tables of the modified Bessel function but also applies to Bessel functions of the first kind and has other applications such as computation of the coefficients of Chebyshev expansions of other special functions.
