Cassini-Like Formula for Generalized Hyper-Fibonacci Numbers

Lyes Ait-Amrane^{1}

Ecole nationale Supérieure d’Informatique (ESI)

BP 68M, Oued Smar, 16309

El Harrach, Algiers

Algeria

lyesait@gmail.com

Djilali Behloul Faculty of Computer Science

University of Sciences and Technology Houari Boumediene (USTHB) BP 32, El Alia, 16111 Bab Ezzouar, Algiers

Algeria

dbehloul@yahoo.fr

Abstract

In this paper, we give some properties of generalized hyper-Fibonacci numbers in order to obtain a Cassini-like formula for them.

1 Introduction

In this paper, we consider the generalized Fibonacci sequence (w_{n})_{n≥0} defined by

\begin{cases}w_{0} = a, w_{1} = b; \\ w_{n+2} = αw_{n+1} + βw_{n}, & (n ≥ 0) \end{cases}

1Corresponding author.

where a, b, α, and β are integers. Several authors [3, 7, 8, 12] studied properties of these numbers. If α^{2} + 4β ≠ 0, then we have

w_{n} = as_{1}^{n} + (b − as_{1}) (\frac{s_{1}^{n} − s_{2}^{n}}{s_{1} − s_{2}} ) ,  n ≥ 0,  (1)

where s_{1} = (α + \sqrt{α + 4β})/2 and s_{2} = (α − \sqrt{α + 4β})/2 are the roots of x^{2} − αx − β = 0. If A = b − as_{2} and B = b − as_{1}, then the identity (1) is equivalent to the well-known Binet-like formula

w_{n} = \frac{As_{1}^{n} − Bs_{2}^{n}}{s_{1} − s_{2}} , n ≥ 0.

If β ≠ 0 and α^{2} + 4β ≠ 0, then the generalized Fibonacci numbers for negative subscripts are defined, using Identity (1), by w_{−n} = a(s_{1}^{−n} + s_{2}^{−n}) − (−β)^{−n}w_{n} for n ≥ 0. If β ≠ 0 and α^{2} + 4β = 0, then α = 2t and β = −t^{2}, where t is the double root of x^{2} − αx − β = 0. Thus, w_{n} = ((2bα^{−1} − a)n + a)t^{n} and w_{−n} = ((a − 2bα^{−1})n + a)t^{−n}, for n ≥ 0.

We introduce the generalized hyper-Fibonacci numbers associated with the sequence (w_{n})_{n≥0} as follows:

w_{n}^{(r+1)} = ∑_{k=0}^{n} α^{n−k}w_{k}^{(r)}, w_{n}^{(0)} = w_{n}, w_{0}^{(r)} = a, w_{1}^{(r)} = αar + b,  (2)

where r is a nonnegative integer.

The aim of this paper is to extend the well-known Cassini formula [5,10,14]

F_{n}F_{n+2} − F_{n+1}^{2} = (−1)^{n+1}  (3)

to the generalized hyper-Fibonacci numbers (2), where (F_{n})_{n} denotes the classical Fibonacci sequence. The identity (3) can be written in a determinant form as

|\begin{matrix}F_{n} & F_{n+1} \\ F_{n+1} & F_{n+2} \end{matrix}| = (−1)^{n+1}.  (4)

In [11], Stakhov generalized the Cassini formula (3) to the p-Fibonacci numbers and developed a new coding theory based on the Q_{p}-matrices. By analogy, one can use the compagnon matrices given bellow in the formula (19) instead of the Q_{p}-matrices. Halici [6] established the Cassini formula for the Fibonacci quaternions. Martinjak and Urbiha [9] extended the Cassini formula (4) to the hyper-Fibonacci numbers defined in [2,4] as

F_{n}^{(r+1)} = ∑_{k=0}^{n} F_{k}^{(r)}, F_{n}^{(0)} = F_{n}, F_{0}^{(r)} = 0, F_{1}^{(r)} = 1,

where r is a nonnegative integer. The number F_{n}^{(r)} is called the nth hyper-Fibonacci number of the rth generation. The hyper-Fibonacci numbers satisfy many interesting numbertheoretical and combinatorial properties, e.g., [2]. Martinjak and Urbiha [9] defined the matrix

A_{r,n} = (\begin{matrix}F_{n}^{(r)} & F_{n+1}^{(r)} & ⋯ & F_{n+r+1}^{(r)} \\ F_{n+1}^{(r)} & F_{n+2}^{(r)} & ⋯ & F_{n+r+2}^{(r)} \\ ⋮ & ⋮ & ... & ⋮ \\ F_{n+r+1}^{(r)} & F_{n+r+2}^{(r)} & ⋯ & F_{n+2r+2}^{(r)} \end{matrix})

and proved that

det(A_{r,n}) = (−1)^{n+⌊(r+3)/2⌋},  (5)

where n and r are nonnegative integers. It is clear that for r = 0, we obtain (4). Recently, in [1], the authors generalized the identity (5) to the (a, b)-hyper-Fibonacci numbers.

If (b_{n})_{n≥0} and (c_{n})_{n≥0} are two sequences satisfying the relation a_{n+2} = αa_{n+1} + βa_{n}, then we have the identity [13]

b_{n}c_{n−1} − b_{n−1}c_{n} = (−β)^{n−1}(b_{1}c_{0} − b_{0}c_{1}).  (6)

If we take b_{n} = F_{n+2} and c_{n} = F_{n+1}, then the identity (6) reduces to (3). For b_{n} = w_{n+2} and c_{n} = w_{n+1}, the identity (6) reduces to

w_{n}w_{n+2} − w_{n+1}^{2} = (−β)^{n}(βa^{2} + αab − b^{2}).  (7)

Cassini formula (7) can also be expressed as a determinant in the following way

|\begin{matrix}w_{n} & w_{n+1} \\ w_{n+1} & w_{n+2} \end{matrix}| = (−β)^{n−1}(βb^{2} − αβab − β^{2}a^{2}).

For n, r ∈ ℤ such that n ≥ 0 and r ≥ 0, let us define the (r + 2) × (r + 2) matrix

W_{r,n} = (\begin{matrix}w_{n}^{(r)} & w_{n+1}^{(r)} & ⋯ & w_{n+r+1}^{(r)} \\ w_{n+1}^{(r)} & w_{n+2}^{(r)} & ⋯ & w_{n+r+2}^{(r)} \\ ⋮ & ⋮ & ... & ⋮ \\ w_{n+r+1}^{(r)} & w_{n+r+2}^{(r)} & ⋯ & w_{n+2r+2}^{(r)} \end{matrix}) .

In Section 2, we establish some combinatorial properties involving the generalized hyper-Fibonacci sequence (w_{n}^{(r)})_{n≥0}. In Section 3, we evaluate the determinant of the matrix W_{r,n}, which gives the identity (5) for a = 0 and b = α = β = 1. For a = 2 and b = α = β = 1, we deduce the determinant of the matrix

(\begin{matrix}L_{n}^{(r)} & L_{n+1}^{(r)} & ⋯ & L_{n+r+1}^{(r)} \\ L_{n+1}^{(r)} & L_{n+2}^{(r)} & ⋯ & L_{n+r+2}^{(r)} \\ ⋮ & ⋮ & ... & ⋮ \\ L_{n+r+1}^{(r)} & L_{n+r+2}^{(r)} & ⋯ & L_{n+2r+2}^{(r)} \end{matrix})

involving the hyper-Lucas numbers given by [2,4]

L_{n}^{(r+1)} = ∑_{k=0}^{n} L_{k}^{(r)}, L_{n}^{(0)} = L_{n}, L_{0}^{(r)} = 2, L_{1}^{(r)} = 2r + 1.

We let \binom{n}{k} denote the binomial coefficient which is defined for a nonnegative integer n and an integer k by

\binom{n}{k} = \begin{cases}\frac{n!}{k!(n − k)!}, & if 0 ≤ k ≤ n; \\ 0, & otherwise, \end{cases}

and for a negative integer n and an integer k by

\binom{n}{k} = \begin{cases}(−1)^{k}\binom{−n + k − 1}{k}, & if k ≥ 0; \\ (−1)^{n−k}\binom{−k − 1}{n − k}, & if k ≤ n; \\ 0, & otherwise. \end{cases}

In Sections 2 and 3 we assume that β ≠ 0.

2 Properties of the sequences (w_{n}^{(r)})_{n≥0}

In this section, we give some properties satisfied by the generalized hyper-Fibonacci sequence (w_{n}^{(r)})_{n≥0} given by the formula (2).

Lemma 1. Let n ≥ 0 be an integer. Then

w_{n}^{(1)} = \frac{1}{β} w_{n+2} − \frac{α^{n+1}}{β} b.  (8)

Proof. We prove the lemma by induction on n ≥ 0. For n = 0, the identity (8) is trivially checked. Now assume that (8) holds for an integer n ≥ 0. Then

w_{n+1}^{(1)} = ∑_{k=0}^{n+1} α^{n+1−k}w_{k}

= α ∑_{k=0}^{n} α^{n−k}w_{k} + w_{n+1}

= αw_{n}^{(1)} + w_{n+1}

= α (\frac{1}{β} w_{n+2} − \frac{α^{n+1}}{β} b) + w_{n+1}

= \frac{1}{β} w_{n+3} − \frac{α^{n+2}}{β} b.

We conclude that (8) holds for all n ≥ 0.

The following proposition expresses a generalized hyper-Fibonacci number of any generation r ≥ 1 in terms of the 0th generation.

Proposition 2. Let r ≥ 1 be an integer. Then

w_{n}^{(r)} = \frac{1}{β^{r}} w_{n+2r} − α^{n+1} ∑_{l=0}^{r−1} \binom{n + r − l − 1}{r − l − 1}\frac{w_{2l+1}}{β^{l+1}} , n ≥ 0.  (9)

Proof. We prove the proposition by induction on r ≥ 1. We deduce from Lemma 1 that (9) holds for r = 1. Now assume that (9) holds for an integer r ≥ 1. Then

w_{n}^{(r+1)} = ∑_{k=0}^{n} α^{n−k}w_{k}^{(r)}

= ∑_{k=0}^{n} α^{n−k} (\frac{1}{β^{r}} w_{k+2r} − α^{k+1} ∑_{l=0}^{r−1} \binom{k + r − l − 1}{r − l − 1}\frac{w_{2l+1}}{β^{l+1}} )

= \frac{1}{β^{r}} ∑_{k=0}^{n} α^{n−k}w_{k+2r} − ∑_{k=0}^{n} ∑_{l=0}^{r−1} \binom{k + r − l − 1}{r − l − 1}α^{n+1} \frac{w_{2l+1}}{β^{l+1}}

= \frac{1}{β^{r}} ∑_{l=0}^{n+2r} α^{n+2r−l}w_{l} − \frac{α^{n+1}}{β^{r}} ∑_{l=0}^{2r−1} α^{2r−1−l}w_{l} − α^{n+1} ∑_{l=0}^{r−1} \frac{w_{2l+1}}{β^{l+1}} ∑_{k=0}^{n} \binom{k + r − l − 1}{r − l − 1}

= \frac{1}{β^{r}} w_{n+2r}^{(1)} − \frac{α^{n+1}}{β^{r}} w_{2r−1}^{(1)} − α^{n+1} ∑_{l=0}^{r−1} \binom{n + r − l}{r − l}\frac{w_{2l+1}}{β^{l+1}}

= \frac{1}{β^{r+1}} w_{n+2r+2} − \frac{α^{n+1}}{β^{r+1}} w_{2r+1} − α^{n+1} ∑_{l=0}^{r−1} \binom{n + r − l}{r − l}\frac{w_{2l+1}}{β^{l+1}}

= \frac{1}{β^{r+1}} w_{n+2r+2} − α^{n+1} ∑_{l=0}^{r} \binom{n + r − l}{r − l}\frac{w_{2l+1}}{β^{l+1}} .

We deduce that (9) holds for all r ≥ 1.

To go further, we need to define the generalized hyper-Fibonacci numbers of negative subscripts. To do this, we first study the particular case a = 0 and b = 1 of the sequence (w_{n})_{n} ≥ 0. Thus, let (u_{n})_{n} denote this particular sequence. The definition given by (2) reduces to

u_{n}^{(r+1)} = ∑_{k=0}^{n} α^{n−k}u_{k}^{(r)}, u_{n}^{(0)} = u_{n}, u_{0}^{(r)} = 0, u_{1}^{(r)} = 1.  (10)

Proposition 3. Let r ≥ 0 be an integer. Then

u_{n}^{(r+1)} = \frac{1}{β} u_{n+2}^{(r)} − \binom{n + r + 1}{r}\frac{α^{n+1}}{β} , n ≥ 0.  (11)

Proof. We prove the proposition by induction on r ≥ 0. We deduce from Lemma 1 that (11) holds for r = 0. Now assume that (11) holds for an integer r ≥ 0. Then

u_{n}^{(r+2)} = ∑_{k=0}^{n} α^{n−k}u_{k}^{(r+1)}

= ∑_{k=0}^{n} α^{n−k} (\frac{1}{β} u_{k+2}^{(r)} − \binom{k + r + 1}{r}\frac{α^{k+1}}{β} )

= \frac{1}{β} ∑_{k=0}^{n} α^{n−k}u_{k+2}^{(r)} − \frac{1}{β} ∑_{k=0}^{n} \binom{k + r + 1}{r}α^{n+1}

= \frac{1}{β} ∑_{l=2}^{n+2} α^{n+2−l}u_{l}^{(r)} − \frac{1}{β} ∑_{l=2}^{n+2} \binom{l + r − 1}{r}α^{n+1}

= \frac{1}{β} (∑_{l=0}^{n+2} α^{n+2−l}u_{l}^{(r)} − α^{n+1}) − \frac{1}{β} (∑_{l=1}^{n+2} \binom{l + r − 1}{r}α^{n+1} − α^{n+1})

= \frac{1}{β} u_{n+2}^{(r+1)} − \frac{1}{β} ∑_{l=1}^{n+2} \binom{l + r − 1}{r}α^{n+1}

= \frac{1}{β} u_{n+2}^{(r+1)} − \binom{n + r + 2}{r + 1}\frac{α^{n+1}}{β} .

We deduce that (11) holds for all r ≥ 0.

The following corollary allows us to extend the sequence (u_{n})_{n≥0} to negative subscripts.

Corollary 4. Let r ≥ 0 and n ≥ 0 be integers. Then

u_{n+2}^{(r+1)} = αu_{n+1}^{(r+1)} + βu_{n}^{(r+1)} + \binom{n + r + 1}{r}α^{n+1}.  (12)

Proof. According to (10), we have

u_{n+1}^{(r)} = u_{n+1}^{(r+1)} − αu_{n}^{(r+1)},  (13)

where n ≥ 0 and r ≥ 0. From (11) and (13) we get

u_{n}^{(r+1)} = \frac{1}{β} u_{n+2}^{(r+1)} − \frac{α}{β}u_{n+1}^{(r+1)} − \binom{n + r + 1}{r}\frac{α^{n+1}}{β} .

We deduce that

u_{n+2}^{(r+1)} = αu_{n+1}^{(r+1)} + βu_{n}^{(r+1)} + \binom{n + r + 1}{r}α^{n+1}.

Now we give a linear relation between the sequences (w_{n})_{n≥0} and (u_{n})_{n≥0}. Lemma 5. Let n ≥ 0 be an integer. Then

w_{n+1} = bu_{n+1} + βau_{n}.  (14)

Proof. We prove the lemma by induction on n ≥ 0. For n = 0, 1 and 2, the identity (14) is trivially checked. Now assume that (14) holds up to some order n ≥ 1. Then

w_{n+2} = αw_{n+1} + βw_{n}
= α (bu_{n+1} + βau_{n}) + β (bu_{n} + βau_{n−1})
= b (αu_{n+1} + βu_{n}) + βa (αu_{n} + βu_{n−1})
= bu_{n+2} + βau_{n+1}.

We conclude that (14) holds for all n ≥ 0.

In the following lemma, we give a relation between the sequences (w_{n}^{(1)})_{n≥0} and (u_{n}^{(1)})_{n≥0}. Lemma 6. Let n ≥ 0 be an integer. Then

w_{n}^{(1)} = bu_{n}^{(1)} + βau_{n−1}^{(1)} + aα^{n}.  (15)

Proof. We prove the lemma by induction on n ≥ 0. Since u_{0}^{(1)} = u_{−1}^{(1)} = 0, the identity (15) holds for n = 0. Now assume that (15) holds for an integer n ≥ 0. Then

w_{n+1}^{(1)} = ∑_{k=0}^{n+1} α^{n+1−k}w_{k}

= w_{n+1} + α ∑_{k=0}^{n} α^{n−k}w_{k}

= w_{n+1} + αw_{n}^{(1)}

= bu_{n+1} + βau_{n} + α(bu_{n}^{(1)} + βau_{n−1}^{(1)} + aα^{n}) = b(u_{n+1} + αu_{n}^{(1)}) + βa(u_{n} + αu_{n−1}^{(1)}) + aα^{n+1} = bu_{n+1}^{(1)} + βau_{n}^{(1)} + aα^{n+1}.

We conclude that (15) holds for all n ≥ 0.

Now we derive a relation between the sequences (w_{n}^{(r)})_{n≥0} and (u_{n}^{(r)})_{n≥0}, for any generation r ≥ 1. Proposition 7. Let r ≥ 1 be an integer. Then

w_{n}^{(r)} = bu_{n}^{(r)} + βau_{n−1}^{(r)} + a\binom{n + r − 1}{r − 1}α^{n},  n ≥ 0.  (16)

Proof. We prove the proposition by induction on n ≥ 0. We deduce from Lemma 6 that Identity (16) holds for r = 1. Now assume that (16) holds for an integer r ≥ 1. Then

w_{n}^{(r+1)} = ∑_{k=0}^{n} α^{n−k}w_{k}

= ∑_{k=0}^{n} α^{n−k} (bu_{k}^{(r)} + βau_{k−1}^{(r)} + a\binom{k + r − 1}{r − 1}α^{k})

= ∑_{k=0}^{n} α^{n−k}u_{k}^{(r)} + βa ∑_{k=0}^{n} α^{n−k}u_{k−1}^{(r)} + aα^{n} ∑_{k=0}^{n} \binom{k + r − 1}{r − 1}

= bu_{n}^{(r+1)} + βa ∑_{l=0}^{n−1} α^{(n−1)−l}u_{l}^{(r)} + \binom{n + r}{r}aα^{n}

= bu_{n}^{(r+1)} + βau_{n−1}^{(r+1)} + \binom{n + r}{r}aα^{n}.

We conclude that (16) holds for all n ≥ 0.

Remark 8. Using Identity (12), we extend, for r ≥ 1 and α ≠ 0, the sequence (u_{n})_{n≥0} to negative subscripts as follows:

u_{−n}^{(r)} = \frac{1}{β} u_{−n+2}^{(r)} − \frac{α}{β}u_{−n+1}^{(r)} − \binom{−n + r}{r − 1}\frac{α^{−n+1}}{β} , n ≥ 0.

It is easy to see that

u_{−n}^{(r)} = 0, for 0 ≤ n ≤ r and u_{−r−1}^{(r)} = \frac{1}{β} (\frac{−1}{α})^{r} ≠ 0.

Now, using Identity (16), we define, for r ≥ 1 and α ≠ 0, the generalized hyper-Fibonacci numbers for negative subscripts by

w_{−n}^{(r)} = bu_{−n}^{(r)} + βau_{−n+1}^{(r)} + a\binom{−n + r − 1}{r − 1}α^{−n}, n ≥ 0.

We deduce that

w_{−n}^{(r)} = 0 for 1 ≤ n ≤ r.

If α = 0, we have respectively from (2) and (10) that w_{n}^{(r)} = w_{n} and u_{n}^{(r)} = u_{n} for n ≥ 0. Thus,

w_{−n}^{(r)} = w_{−n} and u_{−n}^{(r)} = u_{−n}, n ≥ 0.

The following theorem is the key assertion behind the computation of the generalized Cassini formula.

Theorem 9. Assume that α ≠ 0 and let r ≥ 0 be an integer. Then

w_{n+r+2}^{(r)} = ∑_{k=0}^{r+1}(−1)^{r+1−k}α^{r−k} (α^{2}\binom{r + 1}{k − 1} − β\binom{r}{k}) w_{n+k}^{(r)},  n ≥ −r.  (17)

Proof. Let us proceed by induction on r. For r = 0, the identity (17) holds by definition of the sequence (w_{n})_{n}. Now assume that (17) holds for an integer r ≥ 0. Since n+1 ≥ n ≥ −r, we have

w_{n+r+3}^{(r)} = ∑_{k=0}^{r+1}(−1)^{r+1−k}α^{r−k} (α^{2}\binom{r + 1}{k − 1} − β\binom{r}{k}) w_{n+k+1}^{(r)}.  (18)

Since n + r + 3 ≥ n + r + 2 ≥ 0, we have w_{n+r+3}^{(r)} = w_{n+r+3}^{(r+1)} − αw_{n+r+2}^{(r+1)}. For k = 0, 1 …, r + 1, we have w_{n+k+1}^{(r)} = w_{n+k+1}^{(r+1)} − αw_{n+k}^{(r+1)} because

• If n + k + 1 < 0 then we obtain 0 = 0 − 0.

• If n + k + 1 ≥ 0 and n + k < 0 then n + k = −1, we obtain a = a − 0.

• If n + k ≥ 0 then n + k + 1 > 0 and we obtain w_{n+k+1}^{(r)} = w_{n+k+1}^{(r+1)} − αw_{n+k}^{(r+1)}.

Thus, from (18) we get

w_{n+r+3}^{(r+1)} − αw_{n+r+2}^{(r+1)} = ∑_{k=0}^{r+1}(−1)^{r+1−k}α^{r−k} (α^{2}\binom{r + 1}{k − 1} − β\binom{r}{k}) (w_{n+k+1}^{(r+1)} − αw_{n+k}^{(r+1)}).

We deduce that

w_{n+r+3}^{(r+1)} = αw_{n+r+2}^{(r+1)} + ∑_{k=0}^{r+1}(−1)^{r+1−k}α^{r−k} (α^{2}\binom{r + 1}{k − 1} − β\binom{r}{k}) w_{n+k+1}^{(r+1)}

+ ∑_{k=0}^{r+1}(−1)^{r−k}α^{r+1−k} (α^{2}\binom{r + 1}{k − 1} − β\binom{r}{k}) w_{n+k}^{(r+1)}

= (αr + 2α)w_{n+r+2}^{(r+1)} + ∑_{k=0}^{r}(−1)^{r+1−k}α^{r−k} (α^{2}\binom{r + 1}{k − 1} − β\binom{r}{k}) w_{n+k+1}^{(r+1)}

+ (−1)^{r+1}α^{r+1}βw_{n}^{(r+1)} + ∑_{k=1}^{r+1}(−1)^{r−k}α^{r+1−k} (α^{2}\binom{r + 1}{k − 1} − β\binom{r}{k}) w_{n+k}^{(r+1)}

= (αr + 2α)w_{n+r+2}^{(r+1)} + ∑_{l=1}^{r+1}(−1)^{r−l}α^{r+1−l} (α^{2}\binom{r + 1}{l − 2} − β\binom{r}{l − 1}) w_{n+l}^{(r+1)}

+ (−1)^{r+1}α^{r+1}βw_{n}^{(r+1)} + ∑_{k=1}^{r+1}(−1)^{r−k}α^{r+1−k} (α^{2}\binom{r + 1}{k − 1} − β\binom{r}{k}) w_{n+k}^{(r+1)}

= ∑_{k=0}^{r+2}(−1)^{r−k}α^{r+1−k} (α^{2}\binom{r + 1}{k − 2} + α^{2}\binom{r + 1}{k − 1} − β\binom{r}{k − 1} − β\binom{r}{k}) w_{n+k}^{(r+1)}

= ∑_{k=0}^{r+2}(−1)^{r−k}α^{r+1−k} (α^{2}\binom{r + 2}{k − 1} − β\binom{r + 1}{k}) w_{n+k}^{(r+1)}.

We conclude that (17) holds for all r ≥ 0.

3 A generalization of the Cassini formula

Cassini formula (7) can be expressed as a 2 × 2 determinant in the following way

|\begin{matrix}w_{n} & w_{n+1} \\ w_{n+1} & w_{n+2} \end{matrix}| = (−β)^{n−1}(βb^{2} − αβab − β^{2}a^{2}).

For n, r ∈ ℤ such that n ≥ 0 and r ≥ 0, let us define the (r + 2) × (r + 2) matrix

W_{r,n} = (\begin{matrix}w_{n}^{(r)} & w_{n+1}^{(r)} & ⋯ & w_{n+r+1}^{(r)} \\ w_{n+1}^{(r)} & w_{n+2}^{(r)} & ⋯ & w_{n+r+2}^{(r)} \\ ⋮ & ⋮ & ... & ⋮ \\ w_{n+r+1}^{(r)} & w_{n+r+2}^{(r)} & ⋯ & w_{n+2r+2}^{(r)} \end{matrix}) .

Our purpose is to evaluate the determinant of the matrix W_{r,n}. Note that

W_{0,n} = (\begin{matrix}w_{n} & w_{n+1} \\ w_{n+1} & w_{n+2} \end{matrix}) .

From Theorem 9, we can write

w_{n+r+2}^{(r)} = ∑_{k=0}^{r+1} q_{k}w_{n+k}^{(r)}, n ≥ −r,

where

q_{k} = (−1)^{r+1−k}α^{r−k} (α^{2}\binom{r + 1}{k − 1} − β\binom{r}{k}) , 0 ≤ k ≤ r + 1.

Let

V_{r+2} = (\begin{matrix}0 & 1 & 0 & ⋯ & 0 & 0 \\ 0 & 0 & 1 & ⋯ & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & 0 & ⋯ & 1 & 0 \\ 0 & 0 & 0 & ⋯ & 0 & 1 \\ q_{0} & q_{1} & q_{2} & ⋯ & q_{r} & q_{r+1} \end{matrix})  (19)

denote the companion matrix of the generalized hyper-Fibonacci sequence (w_{n}^{(r)})_{n}. We

deduce from Theorem 9 that the generalized hyper-Fibonacci sequence (w_{n}^{(r)})_{n} can be defined by the vector recurrence relation

( w_{n+1}^{(r)} ; w_{n+2}^{(r)} ; ⋮ ; w_{n+r+2}^{(r)}) = V_{r+2} ( w_{n}^{(r)} ; w_{n+1}^{(r)} ; ⋮ ; w_{n+r+1}^{(r)}) ,  (20)

where n + r ≥ 0.

Lemma 10. Let n ≥ 0 and r ≥ 0 be integers. Then

W_{r,n} = V_{r+2}^{n}W_{r,0}.

Proof. From the relation (20) we can write W_{r,n} = V_{r+2}W_{r,n−1}. It follows that

W_{r,n} = V_{r+2}W_{r,n−1} = V_{r+2}^{2}W_{r,n−2} = ⋯ = V_{r+2}^{n}W_{r,0}.

Lemma 11. Let r ≥ 0 be an integer. Then

det(V_{r+2}) = −α^{r}β.

Proof. It is clear that

det(V_{r+2}) = (−1)^{r+3}q_{0} = (−1)^{r+3}(−1)^{r+2}α^{r}β = −α^{r}β.

Theorem 12. Let n ≥ 0 and r ≥ 0 be integers. Then

det(W_{r,n}) = (−1)^{n+⌊(r+3)/2⌋}α^{nr+r^{2}}β^{n+r}b^{r}(b^{2} − αab − βa^{2}).  (21)

Proof. For r = 0, the result follows from Identity (7). Assume that r ≥ 1. For α ≠ 0, we deduce from (20) that multiplication by V_{r+2}^{−1} decreases by 1 the subscript of each component, i.e.,

V_{r+2}^{−1}W_{r,0} = (\begin{matrix}w_{−1}^{(r)} & w_{0}^{(r)} & ⋯ & w_{r}^{(r)} \\ w_{0}^{(r)} & w_{1}^{(r)} & ⋯ & w_{r+1}^{(r)} \\ ⋮ & ⋮ & ... & ⋮ \\ w_{r}^{(r)} & w_{r+1}^{(r)} & ⋯ & w_{2r+1}^{(r)} \end{matrix}) .

Thus,

V_{r+2}^{−r}W_{r,0} = (\begin{matrix}w_{−r}^{(r)} & w_{1−r}^{(r)} & ⋯ & w_{1}^{(r)} \\ w_{1−r}^{(r)} & w_{2−r}^{(r)} & ⋯ & w_{2}^{(r)} \\ ⋮ & ⋮ & ... & ⋮ \\ w_{1}^{(r)} & w_{2}^{(r)} & ⋯ & w_{r+2}^{(r)} \end{matrix}) .

Since w_{−n}^{(r)} = 0 for 1 ≤ n ≤ r, then

V_{r+2}^{−r}W_{r,0} = (\begin{matrix}0 & 0 & ⋯ & 0 & w_{0}^{(r)} & w_{1}^{(r)} \\ 0 & 0 & ⋯ & w_{0}^{(r)} & w_{1}^{(r)} & w_{2}^{(r)} \\ ⋮ & ⋮ & ... & ⋮ & ⋮ & ⋮ \\ 0 & w_{0}^{(r)} & ⋯ & w_{r−2}^{(r)} & w_{r−1}^{(r)} & w_{r}^{(r)} \\ w_{0}^{(r)} & w_{1}^{(r)} & ⋯ & w_{r−1}^{(r)} & w_{r}^{(r)} & w_{r+1}^{(r)} \\ w_{1}^{(r)} & w_{2}^{(r)} & ⋯ & w_{r}^{(r)} & w_{r+1}^{(r)} & w_{r+2}^{(r)} \end{matrix}) .

Thus,

det(W_{r,0}) = det(V_{r+2})^{r} · Δ,  (22)

where

Δ = |\begin{matrix}0 & 0 & ⋯ & 0 & w_{0}^{(r)} & w_{1}^{(r)} \\ 0 & 0 & ⋯ & w_{0}^{(r)} & w_{1}^{(r)} & w_{2}^{(r)} \\ ⋮ & ⋮ & ... & ⋮ & ⋮ & ⋮ \\ 0 & w_{0}^{(r)} & ⋯ & w_{r−2}^{(r)} & w_{r−1}^{(r)} & w_{r}^{(r)} \\ w_{0}^{(r)} & w_{1}^{(r)} & ⋯ & w_{r−1}^{(r)} & w_{r}^{(r)} & w_{r+1}^{(r)} \\ w_{1}^{(r)} & w_{2}^{(r)} & ⋯ & w_{r}^{(r)} & w_{r+1}^{(r)} & w_{r+2}^{(r)} \end{matrix}| .

Let L_{j} denote the jth line of Δ, where j = 1, 2, …, r+2. First, we replace L_{i+1} by L_{i+1}−αL_{i} for i = r + 1, r, …, 1. Since w_{i+1}^{(r−1)} = w_{i+1}^{(r)} − αw_{i}^{(r)}, we get

Δ = |\begin{matrix}0 & 0 & ⋯ & w_{0}^{(r−1)} & w_{1}^{(r−1)} + 2α \\ 0 & 0 & ⋯ & w_{1}^{(r−1)} & w_{2}^{(r−1)} \\ ⋮ & ⋮ & ... & ⋮ & ⋮ \\ w_{0}^{(r−1)} & w_{1}^{(r−1)} & ⋯ & w_{r}^{(r−1)} & w_{r+1}^{(r−1)} \\ w_{1}^{(r−1)} & w_{2}^{(r−1)} & ⋯ & w_{r+1}^{(r−1)} & w_{r+2}^{(r−1)} \end{matrix}| .

Using the same method again (r − 1) times, we obtain

Δ = |\begin{matrix}0 & 0 & ⋯ & 0 & w_{0} & w_{1} + d_{1} \\ 0 & 0 & ⋯ & w_{0} & w_{1} & w_{2} + d_{2} \\ ⋮ & ⋮ & ... & ⋮ & ⋮ & ⋮ \\ 0 & w_{0} & ⋯ & w_{r−2} & w_{r−1} & w_{r} + d_{r} \\ w_{0} & w_{1} & ⋯ & w_{r−1} & w_{r} & w_{r+1} \\ w_{1} & w_{2} & ⋯ & w_{r} & w_{r+1} & w_{r+2} \end{matrix}| ,

where d_{i} = a(−1)^{i−1}\binom{r}{i}α^{i} for 1 ≤ i ≤ r. Now let C_{j} denote the jth column of this last determinant, where j = 1, …, r + 2. Replacing the column C_{i} by C_{i} − (αC_{i−1} + βC_{i−2}) for i = r + 2, r + 1, …, 3 and using the fact that w_{i} = αw_{i−1} + βw_{i−2} we get

Δ = |\begin{matrix}0 & 0 & 0 & ⋯ & 0 & w_{0} & w_{1} − αw_{0} + d_{1} \\ 0 & 0 & 0 & ⋯ & w_{0} & w_{1} − αw_{0} & d_{2} \\ 0 & 0 & 0 & ⋯ & w_{1} − αw_{0} & 0 & d_{3} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ & ⋮ \\ 0 & w_{0} & w_{1} − αw_{0} & ⋯ & 0 & 0 & d_{r} \\ w_{0} & w_{1} & 0 & ⋯ & 0 & 0 & 0 \\ w_{1} & w_{2} & 0 & ⋯ & 0 & 0 & 0 \end{matrix}| .

Now we permute the column C_{i} with column C_{r+3−i} for 1 ≤ i ≤ ⌊(r + 2)/2⌋, and obtain

Δ = (−1)^{⌊ \frac{r+2}{2} ⌋} |\begin{matrix}w_{1} − αw_{0} + d_{1} & w_{0} & 0 & ⋯ & 0 & 0 & 0 \\ d_{2} & w_{1} − αw_{0} & w_{0} & ⋯ & 0 & 0 & 0 \\ d_{3} & 0 & w_{1} − αw_{0} & ⋯ & 0 & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ & ⋮ \\ d_{r} & 0 & 0 & ⋯ & w_{1} − αw_{0} & w_{0} & 0 \\ 0 & 0 & 0 & ⋯ & 0 & w_{1} & w_{0} \\ 0 & 0 & 0 & ⋯ & 0 & w_{2} & w_{1} \end{matrix}| .

We deduce that

Δ = (−1)^{⌊ \frac{r+2}{2} ⌋}Δ′ |\begin{matrix}w_{1} & w_{0} \\ w_{2} & w_{1} \end{matrix}| ,  (23)

where

Δ′ = |\begin{matrix}d_{1} − (αa − b) & a & 0 & ⋯ & 0 & 0 \\ d_{2} & −(αa − b) & a & ⋯ & 0 & 0 \\ d_{3} & 0 & −(αa − b) & ⋯ & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ \\ d_{r−1} & 0 & 0 & ⋯ & −(αa − b) & a \\ d_{r} & 0 & 0 & ⋯ & 0 & −(αa − b) \end{matrix}| .

We distinguish two cases for the calculation of Δ′.

• If αa = b ≠ 0, the expansion with respect to the first column gives

Δ′ = |\begin{matrix}d_{1} & a & 0 & ⋯ & 0 & 0 \\ d_{2} & 0 & a & ⋯ & 0 & 0 \\ d_{3} & 0 & 0 & ⋯ & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ \\ d_{r−1} & 0 & 0 & ⋯ & 0 & a \\ d_{r} & 0 & 0 & ⋯ & 0 & 0 \end{matrix}| = (−1)^{r+1}d_{r}a^{r−1} = (αa)^{r}.

• If b − αa ≠ 0, let L_{j} denote the jth line of Δ′, where j = 1, …, r. Replacing the line

L_{i} by L_{i} + \frac{a}{αa − b}L_{i+1} for i = r − 1, …, 1 gives

Δ′ = |\begin{matrix}(b − αa) + ∑_{i=1}^{r} (\frac{a}{αa − b})^{i−1} d_{i} & 0 & ⋯ & 0 & 0 \\ & b − αa & ⋯ & 0 & 0 \\ ⋮ & ⋮ & ... & ⋮ & ⋮ \\ d_{r−1} + \frac{a}{αa − b}d_{r} & 0 & ⋯ & b − αa & 0 \\ d_{r} & 0 & ⋯ & 0 & b − αa \end{matrix}| .

We deduce that

Δ′ = (b − αa)^{r−1} ((b − αa) + ∑_{i=1}^{r} (\frac{a}{αa − b})^{i−1} d_{i})

= (b − αa)^{r−1} ((b − αa) + a ∑_{i=1}^{r} (\frac{a}{b − αa})^{i−1} \binom{r}{i}α^{i})

= (b − αa)^{r−1} ((b − αa) + (b − αa) ∑_{i=1}^{r} (\frac{αa}{b − αa})^{i} \binom{r}{i})

= (b − αa)^{r−1} ((b − αa) ∑_{i=0}^{r} (\frac{αa}{b − αa})^{i} \binom{r}{i})

= (b − αa)^{r} (\frac{αa}{b − αa} + 1)^{r}

= (b − αa)^{r} (\frac{b}{b − αa})^{r}

= b^{r},

which coincides with the case αa = b ≠ 0. Since |\begin{matrix}w_{1} & w_{0} \\ w_{2} & w_{1} \end{matrix}| = b^{2} − αab − βa^{2}, we deduce from Identities (22), (23) and Lemmas 10, 11 that

det (W_{r,n}) = (−1)^{n+⌊(r+3)/2⌋}α^{nr+r^{2}}β^{n+r}b^{r}(b^{2} − αab − βa^{2}).

Corollary 13. Let n ≥ 0 and r ≥ 0 be integers. Then

|\begin{matrix}F_{n}^{(r)} & F_{n+1}^{(r)} & ⋯ & F_{n+r+1}^{(r)} \\ F_{n+1}^{(r)} & F_{n+2}^{(r)} & ⋯ & F_{n+r+2}^{(r)} \\ ⋮ & ⋮ & ... & ⋮ \\ F_{n+r+1}^{(r)} & F_{n+r+2}^{(r)} & ⋯ & F_{n+2r+2}^{(r)} \end{matrix}| = (−1)^{n+⌊(r+3)/2⌋}.

Proof. Follows from Identity (21) for a = 0 and b = α = β = 1. Corollary 14. Let n ≥ 0 and r ≥ 0 be integers. Then

|\begin{matrix}L_{n}^{(r)} & L_{n+1}^{(r)} & ⋯ & L_{n+r+1}^{(r)} \\ L_{n+1}^{(r)} & L_{n+2}^{(r)} & ⋯ & L_{n+r+2}^{(r)} \\ ⋮ & ⋮ & ... & ⋮ \\ L_{n+r+1}^{(r)} & L_{n+r+2}^{(r)} & ⋯ & L_{n+2r+2}^{(r)} \end{matrix}| = 5(−1)^{n+⌊(r+1)/2⌋}.

Proof. Follows from Identity (21) for a = 2 and b = α = β = 1.

References

[1] L. Ait-Amrane and D. Behloul, Cassini determinant involving the (a, b)-hyper-Fibonacci numbers, Malaya J. Mat. 8 (2020), 939–944.

[2] N. N. Cao and F. Z. Zhao, Some properties of hyperfibonacci and hyperlucas numbers, J. Integer Sequences 13 (2010), Article 10.8.8.

[3] G. Cerda-Morales, On the properties of new families of generalized Fibonacci numbers, Konuralp J. Math. 5 (2017), 96–101.

[4] A. Dil and I. Mező, A symmetric algorithm for hyperharmonic and Fibonacci numbers, Appl. Math. Comput. 206 (2008), 942–951.

[5] R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, Addison-Wesley, 1994.

[6] S. Halici, On Fibonacci quaternions, Adv. Appl. Clifford Algebr. 22 (2012), 321–327.

[7] A. F. Horadam, Basic properties of a certain generalized sequence of numbers, Fibonacci Quart. 3 (1965), 161–176.

[8] R. Keskin and Z. Şiar, Some new identities concerning the Horadam sequence and its companion sequence, Commun. Korean Math. Soc. 34 (2019), 1–16.

[9] I. Martinjak and I. Urbiha, A new generalized Cassini determinant, Colloq. Math. 145 (2016), 209–218.

[10] E. P. Miles, Jr., Generalized Fibonacci numbers and associated matrices, Amer. Math. Monthly 67 (1960), 745–752.

[11] A. P. Stakhov, Fibonacci matrices, a generalization of the “Cassini formula”, and a new coding theory, Chaos Solitons Fractals 30 (2006), 56–66.

[12] A. Tagiuri, Di alcune successioni ricorrenti a termini interi e positivi, Periodico di Mat. 3 (1900), 1–12.

[13] N. G. Voll, The Cassini identity and its relatives, Fibonacci Quart. 48 (2010), 197–201.

[14] M. Werman and D. Zeilberger, A bijective proof of Cassini’s Fibonacci identity, Discrete Math. 58 (1986), 109.

2010 Mathematics Subject Classification: Primary 11B39; Secondary 11B37, 20G20, 15B36. Keywords: Fibonacci number, generalized Fibonacci number, hyper-Fibonacci number, generalized hyper-Fibonacci number, Cassini formula.

(Concerned with sequences \underline{A000032}, \underline{A000045}, \underline{A000071}, \underline{A001610}, \underline{A001924}, and \underline{A023548}.)

Received November 27 2021; revised versions received March 8 2022. Published in Journal of Integer Sequences, March 10 2022.

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