The wonderful world of two-term recurrences

The Fibonacci and Lucas sequences are elements of R(1, 1), and many of their properties
follow immediately from the recursive rule that each term is the sum of the two
preceding terms. Similarly, it is often easy to establish corresponding properties for elements
ofR(a, b) directly from the fundamental identity (1). For example, in R(1, 1),
the Sum of Squares identity is
F2
1 + F2
2 +・ ・ ・+ F2
n = FnFn+1.
The generalization of this to R(a, b) is
bnF2
0 + bn−1F2
1 +・ ・ ・+bF2
n−1 + F2
n =
FnFn+1
a
. (2)
This can be proved quite easily using (1) and induction.
VOL. 76, NO. 3, JUNE 2003 171
Many of the other famous properties can likewise be established by induction. But
to provide more insight about these properties, we will develop some analytic methods,
organized loosely into three general contexts. First, we can think of R(a, b) as a subset
of R∞, the real vector space of real sequences, and use the machinery of difference
operators. Second, by deriving Binet formulas for elements of R(a, b), we obtain explicit
representations as linear combinations of geometric progressions. Finally, there
is a natural matrix formulation which is tremendously useful.We explore each of these
contexts in turn.
Difference operators We will typically represent elements of R∞ with uppercase
roman letters, in the form
A = A0, A1, A2, . . . .
There are three fundamental linear operators on R∞ to consider. The first is the leftshift,
. For any real sequence A = A0, A1, A2, . . . , the shifted sequence is A = A1, A2, A3, . . . .
This shift operator is a kind of discrete differential operator. Recurrences like (1)
are also called difference equations. Expressed in terms of , (1) becomes
( 2 − a − b)A = 0.
This is analogous to expressing a differential equation in terms of the differential operator,
and there is a theory of difference equations that perfectly mirrors the theory
of differential equations. Here, we have in mind linear constant coefficient differential
and difference equations.
As one fruit of this parallel theory, we see at once that 2 − a − b is a linear
operator on R∞, and that R(a, b) is its null space. This shows that R(a, b) is a subspace
of R∞. We will discuss another aspect of the parallel theories of difference and
differential equation in the succeeding section on Binet formulas.
Note that any polynomial in is a linear operator on R∞, and that all of these operators
commute. For example, the forward difference operator , defined by (A)k = Ak+1 − Ak, is given by = − 1. Similarly, consider the k-term sum, _k, defined
by (_k A)n = An + An+1 + ・ ・ ・ + An+k−1. To illustrate, _2(A) is the sequence A0 + A1, A1 + A2, A2 + A3, . . . . These sum operators can also be viewed as polynomials
in : _k = 1 + + 2 +・ ・ ・+ k−1.
Because these operators commute with , they are operators onR(a, b), as well. In
general, if _ is an operator that commutes with , we observe that _ also commutes
with 2 − a − b. Thus, if A ∈ R(a, b), then ( 2 − a − b)_ A = _( 2 − a − b)A = _0 = 0. This shows that _A ∈ R(a, b). In particular, R(a, b) is closed under
differences and k-term sums.


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