Let
Φ ( x ) = 2 + x + 3 x 2 + 4 x 3 + ⋯ = ∑ n = 0 ∞ L n x n {\displaystyle \Phi (x)=2+x+3x^{2}+4x^{3}+\cdots =\sum _{n=0}^{\infty }L_{n}x^{n}}
be the generating series of the Lucas numbers. By a direct computation,
Φ ( x ) = L 0 + L 1 x + ∑ n = 2 ∞ L n x n = 2 + x + ∑ n = 2 ∞ ( L n − 1 + L n − 2 ) x n = 2 + x + ∑ n = 1 ∞ L n x n + 1 + ∑ n = 0 ∞ L n x n + 2 = 2 + x + x ( Φ ( x ) − 2 ) + x 2 Φ ( x ) {\displaystyle {\begin{aligned}\Phi (x)&=L_{0}+L_{1}x+\sum _{n=2}^{\infty }L_{n}x^{n}\\&=2+x+\sum _{n=2}^{\infty }(L_{n-1}+L_{n-2})x^{n}\\&=2+x+\sum _{n=1}^{\infty }L_{n}x^{n+1}+\sum _{n=0}^{\infty }L_{n}x^{n+2}\\&=2+x+x(\Phi (x)-2)+x^{2}\Phi (x)\end{aligned}}}
which can be rearranged as
Φ ( x ) = 2 − x 1 − x − x 2 . {\displaystyle \Phi (x)={\frac {2-x}{1-x-x^{2}}}.}
The partial fraction decomposition is given by
Φ ( x ) = 1 1 − φ x + 1 1 − ϕ x {\displaystyle \Phi (x)={\frac {1}{1-\varphi x}}+{\frac {1}{1-\phi x}}}
where φ = 1 + 5 2 {\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}} is the golden ratio and ϕ = 1 − 5 2 {\displaystyle \phi ={\frac {1-{\sqrt {5}}}{2}}} is its conjugate.