A new method for obtaining Fibonacci identities

For the Lucas sequence {Uk(P,Q)} we discuss the identities like the well-known Fibonacci identities. For example, the generalizations of F2k = F 2 k+1 − F 2 k−1 and F2k+1 = F 2 k+1 + F 2 k are PU2k = U 2 k+1 − QU k−1 and U2k+1 = U k+1 − QU k , respectively. We propose a new simple method for obtaining identities involving any recurrences and use it to obtain new identities involving the Fibonacci numbers such as F 5 k+3−F 5 k−3 + 60F 5 k = 8(F 5 k+2 +F 5 k−2) + 40(F 5 k+1−F 5 k−1). We also present an efficient algorithm for computing any-order linear recurrences.