A note on hyperbolic (p,q)-Fibonacci quaternions

Abstract

In this paper, we introduce a new quaternion sequence called hyperbolic (p; q)-Fibonacci quaternions. This new quaternion sequence includes hyperbolic Fibonacci, hyperbolic k-Fibonacci, hyperbolic Pell, hyperbolic k-Pell, hyperbolic Jacobsthal, hyperbolic k-Jacobsthal quaternions. We give generating function and Binet's formula for these quaternions. We also obtain some identities such as d'Ocagne's, Catalan's and Cassini's identities involving hyperbolic (p; q)-Fibonacci quaternions.

On the Hyperbolic Leonardo and Hyperbolic Francois Quaternions

“…The hyperbolic Fibonacci and hyperbolic Lucas quaternions and some of their generalizations are given in [19][20][21][22][23].…”

On Bicomplex (p,q)-Fibonacci Quaternions

“…Definition 1. The bicomplex (𝑝, 𝑞) −Fibonacci numbers are introduced by 𝐵𝐹 (𝑝, 𝑞) =𝐹 +𝑖𝐹 + 𝑗𝐹 +𝑖𝑗 𝐹 , (7) where 𝐹 is the 𝑢𝑡ℎ (𝑝, 𝑞) −Fibonacci number and 𝑖, 𝑗 are bicomplex units that provide (5).…”

“…where 𝛽 and 𝜃 are roots of (2) [2]. Furthermore, there are many more articles on (𝑝, 𝑞) −Fibonacci sequence [2][3][4][5][6][7][8].…”

“…Also, the imaginary quaternion units 𝑒 ,𝑒 , and 𝑒 have the rules in (4). There are many more works on quaternions in literature (see, for example, [2,6,7,[10][11][12][13][14][15][16][17][18][19][20]).…”

“…𝑏 For more information on bicomplex numbers, refer to the resources in [7,8,14,18,19,[22][23][24][25][26][27].…”