ON A STUDY OF BINOMIAL FORM TO THE NEW (S, T)-JACOBSTHAL SEQUENCE

Abstract

Many (s, t)-type of sequences has been introduced earlier such as (s, t)-Fibonacci sequence, (s, t)-Lucas sequence, (s, t)-Jacobsthal sequence, (s, t)Jacobsthal-Lucas sequence etc . However in this article, we give a new type of (s, t)-Jacobsthal sequence ⟨Un (s, t)⟩n∈N Un = iUn−1 + 2Un−2, n ≥ 2 and U0 = s− 2t, U1 = i (s− t) where i = √ −1 and s, t ∈ Z+. Next we define a binomial form ⟨Xn (s, t)⟩n∈N to the new (s, t)-Jacobsthal sequence and then some fundamental properties for the binomial form ⟨Xn (s, t)⟩n∈N are obtained. Furthermore a new kind of matrix sequence ⟨Zn (s, t)⟩n∈N will be presented for the binomial form ⟨Xn (s, t)⟩n∈N. 2010 Mathematics Subject Classification: 11B37, 11B39.