Complex factorization of some two-periodic linear recurrence systems (Q2786865)

The two-periodic second-order linear recurrence system \(\{ v_n \}\) is defined for \(n \geq 1\) by: NEWLINE\[NEWLINEv_{2n} := a_0v_{2n-1} + b_0v_{2n-2}NEWLINE\]NEWLINE NEWLINE\[NEWLINEv_{2n+1} := 1_0v_{2n} + b_1v_{2n-1},NEWLINE\]NEWLINE where \(v_0 := 0, v_1\) is a real number. Let \(A := a_0a_1 + b_0 + b_1, B := b_0b_1\) and let us assume that \(A^2 - 4B \neq 0.\) It is proved that for \(n \geq 2\): NEWLINE\[NEWLINEv_{2n} = a_0v_1 \prod_{k=1}^{n-1} \left (\frac{a_0+a_1}{2} \pm \sqrt{\left (\frac{a_0-a_1}{2} \right)^2 - (b_0+b_1) + 2\sqrt{b_0b_1} \cos \frac{k\pi}{n}} \right)NEWLINE\]NEWLINE and for \(b_1 = 0\): NEWLINE\[NEWLINEv_{2n+1} = a_0a_1v_1(a_0a_1+b_0)^{n-1}.NEWLINE\]NEWLINE The following special cases are discussed, respectively for Fibonacci, Pell, Jacobsthal and Mersenne numbers: NEWLINE\[NEWLINEF_{2n}= \prod_{k=1}^{n-1} \left (3 - 2\cos \frac{k\pi}{n} \right ),NEWLINE\]NEWLINE NEWLINE\[NEWLINEP_{2n}= 2^n \prod_{k=1}^{n-1} \left (3 - \cos \frac{k\pi}{n} \right ),NEWLINE\]NEWLINE NEWLINE\[NEWLINEJ_{2n}= \prod_{k=1}^{n-1} \left (5 - 4\cos \frac{k\pi}{n} \right ),NEWLINE\]NEWLINE NEWLINE\[NEWLINEM_{2n}= 3 \prod_{k=1}^{n-1} \left (5 - 4\cos \frac{k\pi}{n} \right ).NEWLINE\]