Some general formulas associated with the second-order homogeneous polynomial line-sequences (Q2765402)

The author investigates the construction of a class of linear second-order recurrence sequences \(G_{u_0,u_1}(c,b): u_{n+2}= bu_{n+1}+ cu_n\), where \(n\in \mathbb{Z}\) and \(u_0,u_1,b,c\in \mathbb{R}\). For example, he considers the translation operation \(T: TG_{1,0}(c,b)= cG_{0,1}(c,b)\), the decomposition \(G_{i,j}(c,b)= iG_{1,0}(c,b)+ jG_{0,1}(c,b)\), and establishes some formulae for terms in these sequences. The polynomial case for \(b=2x\), \(c=1\) is also discussed.