On Bicomplex (<i>p,q</i>)-Fibonacci Quaternions

Abstract

Here, we describe the bicomplex p,q-Fibonacci numbers and the bicomplex p,q-Fibonacci quaternions based on these numbers to show that bicomplex numbers are not defined the same as bicomplex quaternions. Then, we give some of their equations, including the Binet formula, generating function, Catalan, Cassini, and d’Ocagne’s identities, and summation formulas for both. We also create a matrix for bicomplex p,q-Fibonacci quaternions, and we obtain the determinant of a special matrix that gives the terms of that quaternion. With this study, we get a general form of the second-order bicomplex number sequences and the second-order bicomplex quaternions. In addition, we show that these two concepts, defined as the same in many studies, are different.

Keywords

• (<i>p</i>,<i>q</i>)-Fibonacci number
• (<i>p</i>,<i>q</i>)-Fibonacci quaternion
• bicomplex Fibonacci number
• generating function
• Catalan identity