Three-term recurrence relation

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

${\displaystyle y_{n+1}=a_{n}y_{n}+b_{n}y_{n-1}}$ for ${\displaystyle n=1,2,...,}$

where the sequences ${\displaystyle \{a_{n}\}}$ and ${\displaystyle \{b_{n}\}}$, together with the initial values ${\displaystyle y_{0},y_{1}}$ govern the evolution of the sequence ${\displaystyle \{y_{n}\}}$.

Applications

If the ${\displaystyle \{a_{n}\}}$ and ${\displaystyle \{b_{n}\}}$ 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 ${\displaystyle a_{n}=b_{n}=1}$.

Orthogonal polynomials P_n all have a TTRR with respect to degree n,

${\displaystyle P_{n}(x)=(A_{n}x+B_{n})P_{n-1}(x)+C_{n}P_{n-2}(x)}$

where A_n 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

${\displaystyle J_{n+1}={\frac {2n}{z}}J_{n}-J_{n-1}}$

is given by the Bessel function ${\displaystyle J_{n}=J_{n}(z)}$. TTRRs are an important tool for the numeric computation of special functions.

TTRRs are closely related to continued fractions.

Solution

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 ${\displaystyle y_{0},y_{1}}$.^[2]

See also

• Miller's recurrence algorithm

Literature

• Walter Gautschi. Computational Aspects of Three-Term Recurrence Relations. SIAM Review, 9:24–80 (1967).
• Walter Gautschi. Minimal Solutions of Three-Term Recurrence Relation and Orthogonal Polynomials. Mathematics of Computation, 36:547–554 (1981).
• Amparo Gil, Javier Segura, and Nico M. Temme. Numerical Methods for Special Functions. siam (2007)
• J. Wimp, Computation with recurrence relations, London: Pitman (1984)

References

1. Gi, Segura, Temme (2007), Chapter 4.1
2. Gi, Segura, Temme (2007), Chapter 4.1