If the \{a_n\} and \{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 a_n=b_n=1.
Orthogonal polynomials Pn all have a TTRR with respect to degree
n, : P_n(x) = (A_n x + B_n) P_{n-1}(x) + C_n P_{n-2}(x) 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 :J_{n+1}=\frac{2n}{z}J_n-J_{n-1} is given by the
Bessel function 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 ==