On the Pseudo-Fibonacci and Pseudo-Lucas Quaternions

Ferns introduced pseudo-Fibonacci and pseudo-Lucas sequences in 1968 as novel generalizations of the Fibonacci and Lucas sequences as follows:\, consider two recurrence relations{align}{d1}Φₙ₊₁=Φₙ+Ψₙ, {align}{align}{d2}Ψₙ₊₁=Φₙ₊₁+γΦₙ {align}with initial conditions Φ₁=1 and Ψ₁=1 in which γ is a positive integer. Φ and Ψ are pseudo-Fibonacci and pseudo-Lucas numbers, respectively (see {Ferns}). Actually, by eliminating first the Φs and then the Ψs, from ({d1}) and ({d2}), the following pseudo-Fibonacci and pseudo-Lucas sequences are obtained{align}{d3}Φₙ₊₂=2Φₙ₊₁+γΦₙ, {align}{align}{d4}Ψₙ₊₂=2Ψₙ₊₁+γΨₙ {align}with initial conditions Φ₀=0, Φ₁=1 and Ψ₀=1, Ψ₁=1, respectively. There are a lot of quaternion numbers that are related to the Fibonacci and Lucas numbers or their generalizations have been described and extensively explored. The coefficients of these quaternions have been chosen from terms of Fibonacci and Lucas numbers. In this study, we define two new quaternions that are pseudo-Fibonacci and pseudo-Lucas quaternions. Then, we give their Binet-like formula, generating functions, certain binomial sums and Honsberg-like, d'Ocagne-like, Catalan-like and Cassini-like identities.