id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejpam-6492	Demirtürk, Bahar; Topal, Nazim	Product Difference Fibonacci Identities Revisited: Quaternionic Generalizations of Everman and Koshy	2025	22	.pdf	application/pdf	7317	460	79	Using the Binet formula and the relation αβ = −q, we have Un−1Un+1 − U2 n = (αn−1 − βn−1)(αn+1 − βn+1) (α − β)2 − (αn − βn)2 (α − β)2 = α2n + β2n − αn−1βn+1 − αn+1βn−1 (α − β)2 − α2n + β2n − 2αnβn (α − β)2 = −αn+1βn−1 − αn−1βn+1 + 2αnβn (α − β)2 = −(αβ)n−1 α2 + β2 − 2αβ (α − β)2 Using the Binet formula and αβ = −q, we have Un−rUn+r − U2 n = (αn−r − βn−r)(αn+r − βn+r) (α − β)2 − (αn − βn)2 (α − β)2 = α2n + β2n − αn−rβn+r − αn+rβn−r (α − β)2 − α2n + β2n − 2αnβn (α − β)2 = −αn+rβn−r − αn−rβn+r + 2αnβn (α − β)2 = −(αβ)n−r α2r + β2r − 2αrβr (α − β)2 = −(αβ)n−r ( αr − βr α − β )2 = −(αβ)n−r(Ur)2 = −(−q)n−rU2 r .	cache/ejpam-6492.pdf	txt/ejpam-6492.txt