A Tribonacci identity (Q2746564)

Let \(\{V_n\}\) be defined by \(V_n= rV_{n-1}+ sV_{n-2}+ tV_{n-3}\), where \(V_0\), \(V_1\), \(V_2\) are arbitrary complex numbers and \(r\), \(s\), \(t\) are arbitrary integers with \(t\neq 0\). It can be extended to negative subscripts by NEWLINE\[NEWLINEV_{-n}= -\tfrac{r}{t} V_{-(n-1)}- \tfrac{s}{t} V_{-(n-2)}+ \tfrac{1}{t} V_{-(n-3)} \quad\text{for } n=1,2,3\dots\;.NEWLINE\]NEWLINE With the notation \(V_n= V_n(V_0,V_1,V_2;r,s,t)\) the special sequence \(J_n= V_n(3,r,r^2+2s;r,s,t)\) is defined. Then the main result is the identity \(V_{n+2m}= J_m V_{n+m}- t^m J_{-m}V_n+ t^m V_{n-m}\), where \(n,m\) are arbitrary integers.