MathDB

a.) Let

a,b

be real numbers. Define sequence

x_k

and

y_k

such that x_0 = 1, y_0 = 0, x_{k+1} = a \cdot x_k - b \cdot y_l, y_{k+1} = x_k - a \cdot y_k \text{ for } k = 0,1,2, \ldots Prove that

x_k = \sum^{[k/2]}_{l=0} (-1)^l \cdot a^{k - 2 \cdot l} \cdot \left(a^2 + b \right)^l \cdot \lambda_{k,l}

where

\lambda_{k,l} = \sum^{[k/2]}_{m=l} \binom{k}{2 \cdot m} \cdot \binom{m}{l}

b.) Let

u_k = \sum^{[k/2]}_{l=0} \lambda_{k,l}

. For positive integer

m,

denote the remainder of

u_k

divided by

2^m

z_{m,k}

. Prove that

z_{m,k},

k = 0,1,2, \ldots

is a periodic function, and find the smallest period.

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