On the Study of Bifurcating Right Fibonomial Numbers and Rb-Trinomial Numbers

Fontene ́ once introduced a generalized form of binomial coefficients by substituting natural numbers with terms from an arbitrary sequence {A_n} of real or complex numbers, which he referred to as Fibonomial coefficients. Since then, significant interest has developed around Fibonomial numbers which is two dimensional in which n is divided into two parts, particularly when the sequence {A_n} is chosen as {F_n}, the well-known Fibonacci sequence. More recently, researchers have explored a further extension by considering {A_n }={F_n^R}, the sequence of right Fibonacci numbers. In this paper, we take this generalization a step further by defining Fibonomial coefficients based on the sequence {A_n }={F_n^(R(a,b))}, known as the right Bifurcating Fibonacci numbers. Also, there were a new generalization was established for three-dimensional Fibonomial numbers which is the extension of n divided into three parts, known as F-trinomial numbers. In this paper, we choose right bifurcating Fibonacci sequence and introduced RB-trinomial numbers. Then, we derive several identities associated with both of them. Additionally, we examine some of their bounds for both numbers.