AN EXTENDED FRAMEWORK FOR BIHYPERBOLIC GENERALIZED TRIBONACCI NUMBERS - Gurses, Nurten; Isbilir, Zehra (ANKARA UNIV, FAC SCI, 2024)
The aim of this article is to identify and analyze a new type special number system which is called bihyperbolic generalized Tribonacci numbers ( BGTN for short). For this purpose, we give both classical and several new properties such as; recurrence relation, Binet formula, generating function, exponential generating function, summation formulae, matrix formula, and special determinant equations of BGTN .Also, the system of BGTN is quite a big family and includes several type special cases with respect to initial values and r, s, t values, we give the subfamilies and special cases of it. In addition to these, we construct some numerical algorithms including recurrence relation and special two types determinant equations related to calculating the terms of this new type special number system. Then, we examine several properties by taking two special cases and including some illustrative numerical examples.
Keywords: Generalized Tribonacci numbers; bihyperbolic numbers; bihyperbolic generalized Tribonacci numbers; special sequences; FIBONACCI NUMBERS; PELL; Mathematics