リュカ数列
V m + n = V m V n − Q n V m − n = D U m U n + Q n V m − n {\displaystyle V_{m+n}=V_{m}V_{n}-Q^{n}V_{m-n}=DU_{m}U_{n}+Q^{n}V_{m-n}} ,
2 n − 1 U n = ( n 1 ) P n − 1 + ( n 3 ) P n − 3 D + ⋯ {\displaystyle 2^{n-1}U_{n}={n \choose 1}P^{n-1}+{n \choose 3}P^{n-3}D+\cdots } ,
2 n − 1 V n = P n + ( n 2 ) P n − 2 D + ( n 4 ) P n − 4 D 2 + ⋯ {\displaystyle 2^{n-1}V_{n}=P^{n}+{n \choose 2}P^{n-2}D+{n \choose 4}P^{n-4}D^{2}+\cdots } .
また n が正のとき
F n = ∏ d ∣ n U n / d μ ( d ) = U n ∏ p ≠ q , p , q ∣ n U n / p q ⋯ ∏ p ∣ n U n / p ⋯ = ∏ gcd ( k , n ) = 1 ( α − ζ n k β ) = β φ ( n ) Φ n ( α / β ) {\displaystyle F_{n}=\prod _{d\mid n}U_{n/d}^{\mu (d)}={\frac {U_{n}\prod _{p\neq q,p,q\mid n}U_{n/pq}\cdots }{\prod _{p\mid n}U_{n/p}\cdots }}=\prod _{\gcd(k,n)=1}(\alpha -\zeta _{n}^{k}\beta )=\beta ^{\varphi (n)}\Phi _{n}(\alpha /\beta )}