On The Properties of Generalized k-Pell Like Sequence

The Pell sequence has been generalized in many ways, some by preserving the initial conditions, others by preserving the recurrence relation. In this paper, we define a new generalization {𝑀𝑘,𝑛}_𝑛=1^∞, with initial conditions 𝑀_𝑘,0=2,𝑀_𝑘,1=𝑚+2, which is generated by the recurrence relation 𝑀_𝑘,𝑛+1=2𝑀_𝑘,𝑛+𝑘𝑀_{𝑘,𝑛−1}, for 𝑛≥1, where 𝑘,𝑚 are integer numbers. We produce an extended Binet’s formula for 𝑀𝑘,𝑛 and thereby the identities such as Catalan’s, Simpson’s, d’ Ocagene’s etc.