id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejpam-6218	S. Gafel, Hanan; Altamimi, Haya	Behavior and Solution Representations of Fourth-Order Rational Systems of Difference Equations	2025	28	.pdf	application/pdf	8748	322	80	Math, 18 (3) (2025), 6218 20 of 28 =  (−1)n−1ηnλn−1σn−1µn−1τn−1κn−1[ ∏n−2 i=0 ((2i + 2)η − τ)(λ + (2i + 1)µ)((2i + 1)λ − κ)(σ + (2i + 2)τ) ((2i + 2)σ − δ)(ζ + (2i + 1)κ) ]  (−1)nηn−1λn−1σnµnτnκn−1[ (σ + τ)(σ − δ) ∏n−2 i=0 ((2i + 1)η − τ)(λ + (2i + 2)µ)((2i + 2)λ − κ) (σ + (2i + 3)τ)((2i + 3)σ − δ)(ζ + (2i + 2)κ) ]  −  (−1)n−1ηnλn−1σn−1µn−1τn−1κn−1[ ∏n−2 i=0 ((2i + 2)η − τ)(λ + (2i + 1)µ)((2i + 1)λ − κ)(σ + (2i + 2)τ) ((2i + 2)σ − δ)(ζ + (2i + 1)κ) ]  +  (−1)n−1ηn−1λn−1σn−1µn−1τnκn−1[ ∏n−2 i=0 ((2i)η − τ)(λ + (2i + 1)µ)((2i + 1)λ − κ)(σ + (2i + 2)τ) ((2i + 2)σ − δ)(ζ + (2i + 1)κ) ]  =  (−1)n−1ηnλn−1σnµnτnκn−1[ (σ + τ)(σ − δ) ∏n−2 i=0 ((2i + 2)η − τ) ∏n−2 i=0 ((2i + 1)η − τ)(λ + (2i + 2)µ) ((2i + 2)λ − κ)(σ + (2i + 3)τ)((2i + 3)σ − δ)(ζ + (2i + 2)κ) ]  [ η∏n−2 i=0 ((2i+2)η−τ) ] − [ τ∏n−2 i=0 ((2i)η−τ) ] =  (−1)n−1ηnλn−1σnµnτnκn−1[ (σ + τ)(σ − δ) ∏n−2 i=0 ((2i + 1)η − τ)(λ + (2i + 2)µ)((2i + 2)λ − κ) (σ + (2i + 3)τ)((2i + 3)σ − δ)(ζ + (2i + 2)κ) ]  η − [ τ ∏n−2 i=0 ((2i+2)η−τ)∏n−2 i=0 ((2i)η−τ) ] =  (−1)n−1ηnλn−1σnµnτnκn−1[ (σ + τ)(σ − δ) ∏n−2 i=0 ((2i + 1)η − τ)(λ + (2i + 2)µ)((2i + 2)λ − κ) (σ + (2i + 3)τ)((2i + 3)σ − δ)(ζ + (2i + 2)κ) ]  η + ((2n − 2)η − τ) H. S. Gafel, H. A. Altamimi / Eur. = (−1)nηn+1λnσnµnτnκn[ ∏n−1 i=0 ((2i + 2)η − τ)(λ + (2i + 1)µ)((2i + 1)λ − κ)(σ + (2i + 2)τ) ((2i + 2)σ − δ)(ζ + (2i + 1)κ) ] , Θ6n+1 = (−1)n+1ηnλnσn+1µn+1τnκn[ (σ − δ) ∏n−1 i=0 ((2i + 1)η − τ)(λ + (2i + 2)µ)((2i + 2)λ − κ) (σ + (2i + 1)τ)((2i + 3)σ − δ)(ζ + (2i + 2)κ) ] , Θ6n+2 = (−1)n+1ηn+1λn+1σnµnτnκn+1[ (λ − κ)(ζ + κ) ∏n−1 i=0 ((2i + 2)η − τ)(λ + (2i + 1)µ)((2i + 3)λ − κ) (σ + (2i + 2)τ)((2i + 2)σ − δ)(ζ + (2i + 3)κ) ] , Ω6n−3 = (−1)nηnλnσnµnτnκnδ[ ∏n−1 i=0 ((2i)η − τ)(λ + (2i + 1)µ)((2i + 1)λ − κ)(σ + (2i)τ) ((2i)σ − δ)(ζ + (2i + 1)κ) ] , Ω6n−2 = (−1)nηnλnσnµnτnκn+1[ ∏n−1 i=0 ((2i + 1)η − τ)(λ + (2i)µ)((2i)λ − κ)(σ + (2i + 1)τ) ((2i + 1)σ − δ)(ζ + (2i + 2)κ) ] , Ω6n−1 = (−1)nηnλnσnµnτn+1κn[ ∏n−1 i=0 ((2i)η − τ)(λ + (2i + 1)µ)((2i + 1)λ − κ)(σ + (2i + 2)τ) ((2i + 2)σ − δ)(ζ + (2i + 1)κ) ] , Ω6n = (−1)nηnλnσnµn+1τnκn[ ∏n−1 i=0 ((2i + 1)η − τ)(λ + (2i + 2)µ)((2i + 2)λ − κ)(σ + (2i + 1)τ) ((2i + 1)σ − δ)(ζ + (2i + 2)κ) ] , Ω6n+1 = (−1)nηn+1λnσnµnτnκn+1[ (ζ + κ) ∏n−1 i=0 ((2i + 2)η − τ)(λ + (2i + 1)µ)((2i + 1)λ − κ) (σ + (2i + 2)τ)((2i + 2)σ − δ)(ζ + (2i + 3)κ) ] , Ω6n+2 = (−1)n+1ηnλnσn+1µn+1τn+1κn[ (σ + τ)(σ − δ) ∏n−1 i=0 ((2i + 1)η − τ)(λ + (2i + 2)µ)((2i + 2)λ − κ) (σ + (2i + 3)τ)((2i + 3)σ − δ)(ζ + (2i + 2)κ) ] , where Θ−3 = ζ, Θ−2 = σ, Θ−1 = λ, Θ0 = η, Ω−3 = δ, Ω−2 = κ, Ω−1 = τ and Ω0 = µ. Proof.	cache/ejpam-6218.pdf	txt/ejpam-6218.txt