A New Generalization of Jacobsthal Lucas Numbers (Bi-Periodic Jacobsthal Lucas Sequence)

In this study, we bring into light a new generalization of the Jacobsthal Lucas numbers, whichshall also be called the bi-periodic Jacobsthal Lucas sequence as^cn={b^cn1+ 2^cn2,ifnis evena^cn1+ 2^cn2,ifnis oddn≥2,with initial conditions ^c0= 2,^c1=a. The Binet formula as well as the generating functionfor this sequence are given. The convergence property of the consecutive terms of this sequenceis examined after which the well known Cassini, Catalan and the D'ocagne identities as well assome related summation formulas are also given.