Pell and Pell–Lucas Numbers with Applications by Thomas Koshy

Author:Thomas Koshy
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

9.8 Lockwood’s Identity

Let x and y be arbitrary real numbers. Then, by the binomial theorem, we have

In each case, the expression x n + y n is expressed as a sum of terms in xy and x + y.

More generally, we have the following identity, developed by E.H. Lockwood in 1967 [15]:

where n ≥ 1. This identity can be confirmed using strong induction, Pascal’s identity, and a lot of algebra:

Proof. When n = 1:

When n = 2:

So the identity is true when n = 1 and n = 2.

Now assume that it is true for all positive integers ≤ n, where n is an arbitrary integer ≥ 2; that is, assume that

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