To find constant g 0, we perform summation over equation (A 9): ∑ k = 0 N − 1 g k = ∑ k = 0 N − 1 f k − ∑ k = 0 N − 1 f k + 1 = f 0 − f 1 + f 1 − f 2 + ⋯ + f N − 1 − f N = f 0 − f N = 1.
To proceed further, we substitute (A 3) into (A 2): ( s n + λ + ϕ ) ( f n − s b n ) = ϕ ( f n − 1 − s b n − 1 ) + λ ( f n + 1 − s b n + 1 ).