The Fibonacci p-numbers and Pascal’s triangle

Abstract: Pascal’s triangle is the most famous of all number arrays full of patterns and surprises. It is well known that the Fibonacci numbers can be read from Pascal’s triangle. In this paper, we consider the Fibonacci p-numbers and derive an explicit formula for these numbers by using some properties of the Pascal’s triangle. We also introduce the companion matrix of the Fibonacci p-numbers and give some identities of the Fibonacci p-numbers by using some properties of our matrix.


Introduction
As is well known, the sequence of Fibonacci {F n } ∞ n=0 is defined by the following recurrent relation The Fibonacci numbers have many interesting properties and applications to almost every fields of science and art (e.g. see Koshy, 2001;Vajda, 1989). For instance, the ratio of two consecutive of these numbers converges to the irrational number = 1+ √ 5 2 called the Golden Proportion (Golden Mean), see Debnart (2011), Vajda (1989).
It is well-known that the Fibonacci number can be derived by the summing of elements on the rising diagonal lines in the Pascal's triangle (see Koshy, 2001, chap. 12).

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The Fibonacci numbers have many interesting properties and applications. Pascal's triangle has been explored for links to the Fibonacci sequence as well as to generalized sequences. In this paper, the author considers the Fibonacci p-numbers and shows that it can be computed in a systematic way from the Pascal's triangle. He also gives the companion matrix of the Fibonacci p-numbers and obtains some identities of the Fibonacci p-numbers by using some properties of his matrix.
In 1960, Charles H. King studied the Q-matrix in his master thesis and computed the nth powers of this matrix He got the Cassini's identity or Simson's formula F n+1 F n−1 − F 2 n = (−1) n .
One of the generalization of Fibonacci numbers is given by Stakhov (2006). The generalization is called the Fibonacci p-numbers F p (n) that are given for any positive integer p by the following relation with the initial values F p (0) = 0, F p (1) = F p (2) = ⋯ = F p (p) = 1.
Recently, the author (Kuhapatanakul, 2015(Kuhapatanakul, , 2016 introduced the Lucas p-matrix and the Lucas p-triangle, and their applications. In this paper, we first introduce the Fibonacci p-triangle by shifting the column of the Pascal's triangle and derive an explicit formula for the Fibonacci p-numbers.
Second, we introduce the companion matrix for the Fibonacci p-numbers, which is different form Q p and give some identities of the Fibonacci p-numbers.

The Fibonacci p-triangle
Consider the Pascal's triangle we now arrange the elements of the Pascal's triangle to form a left-justified triangular array as follows: The Fibonacci numbers can be derived by summing of elements on the rising diagonal lines in the Pascal's triangle. In similar, we will show that the Fibonacci p-numbers can read from the Pascal's triangle.
Definition 1 For fixed p ∈ ℕ, define the Fibonacci p-triangle as follows: where C p (n, 0) = C p (n, n) = 1 and, for 0 ≤ i ≤ n, For p = 1, the Fibonacci 1-triangle becomes the Pascal's triangle, see Figure 1, and C 1 (n, i) = n i which is the binomial coefficient. For example, we give the Fibonacci p-triangles for p = 2, 3 as follows:  We see that each i-th column (i = 0, 1, 2, 3, …) of the Fibonacci 2-triangle is wrote from the same column of the Pascal's triangle by shifting down i places, and each i-th column of the Fibonacci 3-triangle is wrote from the same column of the Pascal's triangle by shifting down 2i places.
In fact, each i-th column (i = 0, 1, 2, 3, …) of the Fibonacci p-triangle is wrote from the same column of the Pascal's triangle by shifting down i(p − 1) places.
Observe that the sum of elements on the rising diagonal lines in the Fibonacci 2-triangle and Fibonacci 3-triangle give the Fibonacci 2-numbers F 2 (n) and Fibonacci 3-numbers F 3 (n), respectively.
The following table provides further information.

Fibonacci 2-triangle Fibonacci 3-triangle
Theorem 1 For fixed p ∈ ℕ and let n be non-negative integer. Then Proof For p = 1, we see that the identity (2.2) becomes the identity (1.1). For p > 1, we will prove this result by induction on n, noting first that Now assume (2.2) holds for n > 1. We will show that this implies the identity holds for n + 1. By the definition of F p (n) and the inductive hypothesis, we get Thus (2.2) holds for every n.
We see that C p (n, i) = 0 for i > ⌊ n p ⌋ . The number of non-zero elements in nth row equal to ⌊ n p ⌋ + 1.
Since each i-th column (i = 0, 1, 2, 3, …) of the Fibonacci p-triangle is wrote from the same column of the Pascal's triangle by shifting down i(p − 1) places, we get that, for 0 ≤ i ≤ ⌊ n p+1 ⌋ , Therefore, we can rewrite (2.2) in terms of binomial coefficient, which is well-known identity, see Tuglu et al. (2011).
Corollary 1 Let n be non-negative integer. Then

Companion matrix for the Fibonacci p-numbers
Define the (p + 1) × (p + 1) matrix H p as follows: and let S p (n) be the sums of the Fibonacci p-numbers from 0 to n, that is, the determinants on both sides of the Equation (3.2) of Theorem 2, we get a generalization of Cassini's identity or Simson formula.