Morgado Identity

An identity satisfied by Generalized Fibonacci Numbers:

$4w_nw_{n+1}w_{n+2}w_{n+4}w_{n+5}w_{n+6}+e^2q^{2n}(w_nU_4U_5-w_{n+1}U_2U_6-w_nU_1U_8)^2$

$= (w_{n+1}w_{n+2}w_{n+6}+w_nw_{n+4}w_{n+5})^2,\quad$ (1)

where

$\displaystyle e$	$\textstyle \equiv$	$\displaystyle pab-qa^2-b^2$	(2)
$\displaystyle U_n$	$\textstyle \equiv$	$\displaystyle w_n(0,1; p,q).$	(3)

See also Generalized Fibonacci Number

References

Morgado, J. ``Note on Some Results of A. F. Horadam and A. G. Shannon Concerning a Catalan's Identity on Fibonacci Numbers.'' Portugaliae Math. 44, 243-252, 1987.

$4w_nw_{n+1}w_{n+2}w_{n+4}w_{n+5}w_{n+6}+e^2q^{2n}(w_nU_4U_5-w_{n+1}U_2U_6-w_nU_1U_8)^2$
$= (w_{n+1}w_{n+2}w_{n+6}+w_nw_{n+4}w_{n+5})^2,\quad$	(1)