| Summary: | Here, we describe the bicomplex <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfenced separators="|"><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></mfenced></mrow></semantics></math></inline-formula>-Fibonacci numbers and the bicomplex <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfenced separators="|"><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></mfenced></mrow></semantics></math></inline-formula>-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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfenced separators="|"><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></mfenced></mrow></semantics></math></inline-formula>-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.
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