One more about the generalized Fibonacci numbers
So called generalized Fibonacci numbers resulting from Pascal Triangle play the important role in our Harmony Mathematics. For the given non-negative integer p (p = 0, 1, 2, 3, ...) the generalized Fibonacci numbers called the p-Fibonacci numbers are given with the following recurrent formula:
Note that the p-Fibonacci numbers given with (1), (2) are the wide generalization of the classical Fibonacci numbers being the partial case of the p-Fibonacci numbers for p = 1.
The recurrent formula (1) at the initial conditions (2) generates the infinite number of the numerical series, which members are called the p-Fibonacci numbers (see Table 1).
For the case p = 0 the series of the p-Fibonacci numbers is reduced to the classical "binary" series. Let's consider now the p-Fibonacci numbers for the cases ð > 0 and extend them to the negative values of the argument of n. For that we will use the recurrent relation (1) and the initial condition (2) for calculation of the p-Fibonacci numbers Fp(0), Fp(-1), Fp(-2), ..., Fp(-p), ..., Fp(-2p + 1). Representing the p-Fibonacci number of Fp(p + 1) in the form of (1) we get:
Since according to (2) Fp(p) = Fp(p + 1) = 1 it follows from (3) that Fp(0) = 0.
Continuing this process, that is, representing the p-Fibonacci numbers Fp(p), Fp(p - 1), ..., Fp(2) in the form (1) we get:
Let's represent the number Fp(1) in the form:
Since Fp(1) = 1 and Fp(0) = 0 it follows from (5) that
Representing the p-Fibonacci numbers Fp(0), Fp(-1), ..., Fp(-p + 1) in the form (1) we can find that
Continuing this process we can get all values of the p-Fibonacci numbers Fp(n) for the negative values of n (see Table 2).
And now we have all reasons to introduce one more complicated concept, the concept of the generalized Fibonacci matrix called Qp-matrix. Follows us!