Some new identities concerning the Horadam sequence and its companion sequence

Abstract

Let a,b,P, and Q be real numbers with PQ 6≠ 0 and (a,b) 6≠ (0,0). The Horadam sequence (W n ) is defined by W 0 = a, W 1 = b and Wn = PW n-1 + QW n-2 for n ≥ 2. Let the sequence (X n ) be defined by Xn = W n+1 + QW n-1 . In this study, we obtain some new identities between the Horadam sequence (X n ) and the sequence (Xn). By the help of these identities, we show that Diophantine equations such as x 2 - Pxy - y 2 = ±(b 2 - Pab - a 2 )(P 2 + 4), x 2 - Pxy + y 2 = -(b 2 - Pab + a 2 )(P 2 - 4), x 2 - (P 2 + 4)y 2 = ±4(b 2 - Pab - a 2 ), and x 2 - (P 2 - 4)y 2 = 4(b 2 - Pab + a 2 ) have infinitely many integer solutions x and y, where a, b, and P are integers. Lastly, we make an application of the sequences (X n ) and (X n ) to trigonometric functions and get some new angle addition formulas such as sin rθ sin(m + n + r)θ = sin(m + r)θ sin(n + r)θ - sinmθ sin nθ; cos rθ cos(m + n + r)θ = cos(m + r)θ cos(n + r)θ - sinmθ sin nθ; and cos rθ sin(m + n)θ = cos(n + r)θ sinmθ + cos(m - r)θ sin nθ. © 2019 Korean Mathematical Society.