Three-term recurrence

Three-term recurrence
Name	Three-term recurrence
Field	Mathematics
Related	Linear algebra; Numerical analysis; Orthogonal polynomials
Notation	an = bnan−1 + cnan−2

Three-term recurrence. Three-term recurrence relations appear across mathematics and applied science, connecting sequences in combinatorics, numerical analysis, and mathematical physics. They underlie computational algorithms used in approximation theory, spectral methods, and quantum mechanics, and appear in classical works by Euler, Gauss, and Sturm. Prominent occurrences include recurrence relations for Chebyshev, Legendre, and Hermite polynomials, and implementations in Lanczos and Golub methods for eigenproblems.

Definition and examples

A three-term recurrence is an identity relating three successive terms of a sequence, typically written with coefficients that may depend on the index. Classic examples include the recurrence for Fibonacci numbers in combinatorics, the recurrence for Legendre polynomials in special function theory, and the three-term relations satisfied by Hermite polynomials and Chebyshev polynomials of the first kind. Historical instances arise in the work of Leonhard Euler, Carl Friedrich Gauss, and Jacques Sturm, and practical examples appear in algorithms by Cornelius Lanczos and Gene H. Golub. In physics, three-term recurrences occur in the radial equations studied by Erwin Schrödinger and in continued-fraction expansions used by Lord Rayleigh.

Linear three-term recurrence relations

Linear three-term recurrences have the form an = αn an−1 + βn an−2 with scalar or matrix coefficients and arise in linear difference equations and matrix eigenvalue problems. They are central to the Lanczos tridiagonalization procedure associated with John von Neumann’s spectral investigations and with the Golub–Kahan bidiagonalization related to James H. Wilkinson’s numerical linear algebra. Matrix interpretations connect to tridiagonal matrices studied by Issai Schur and to Jacobi matrices appearing in the theory developed by Carl Gustav Jacobi and Tikhonov-regularization contexts influenced by Andrey Tikhonov.

Solutions and characteristic equations

When coefficients are constant, solutions follow from a quadratic characteristic equation whose roots determine closed-form expressions via linear combinations, a method dating to Augustin-Louis Cauchy and refined by Gaspard Monge. For variable coefficients, solution techniques invoke continued fractions as in the works of Adrien-Marie Legendre and Niels Henrik Abel, spectral theory of David Hilbert and John von Neumann, and asymptotic methods from S. R. Srinivasa Varadhan and Harold Jeffreys. Connections to resolvent operators and Green’s functions echo developments by George B. Arfken and Morse and Feshbach in mathematical physics.

Orthogonal polynomials and applications

Orthogonal polynomial systems satisfy three-term recurrences with recurrence coefficients determined by moment functionals and measures studied by Pafnuty Chebyshev, S. N. Bernstein, and Marcel Riesz. The Favard theorem, associated with Jean Favard and furthered by M. G. Kreĭn, characterizes such sequences via Jacobi matrices linked to spectral theorems of Frigyes Riesz and John von Neumann. Applications span Gaussian quadrature developed by Carl Friedrich Gauss and matrix approximation techniques used in computational physics by Richard Feynman and Enrico Fermi; specific families include Laguerre polynomials, Gegenbauer polynomials, and Jacobi polynomials used in models by Pierre-Simon Laplace and J. Willard Gibbs.

Computational methods and numerical stability

Numerical treatment of three-term recurrences appears in algorithms by Gene H. Golub, Cornelius Lanczos, and James Demmel for eigenvalue computation and in quadrature routines used in software from the Numerical Algorithms Group and libraries arising from Netlib. Stability analysis leverages backward error techniques advanced by Higham and perturbation theory from T. Kato, while forward error bounds reflect work by Alston Householder and Wilkinson. Practical concerns include underflow and overflow in floating-point arithmetic standardized by IEEE 754 and iterative refinement methods inspired by Richard Brent and H. Rutishauser.

Generalizations and higher-order recurrences

Generalizations extend to k-term recurrences, nonlinear recurrences, and matrix-valued recurrences studied by Issai Schur and contemporary researchers in integrable systems such as Mikhail S. Adler and Pierre van Moerbeke. Higher-order linear recurrences connect to companion matrices explored by Eugène Catalan and control theory frameworks influenced by Rudolf E. Kálmán and Norbert Wiener. Nonlinear discrete analogues appear in the theory of discrete Painlevé equations and in dynamical systems investigated by Henri Poincaré and Mitchell Feigenbaum.

Category:Recurrence relations