Total Positivity of Toeplitz Matrices Involving Generalized Hyper-Fibonacci Numbers

Abdelhak Taane University of Kasdi Merbah Department of Mathematics

Laboratory of Applied Mathematics

Ouargla, 30000

Algeria

abdelhak.taane@gmail.com

Ihab-Eddine Djellas

CERIST Scientific and Technical Information Research Center Algiers Algeria and

University of Sciences and Technology Houari Boumediene (USTHB) Faculty of Mathematics RECITS Laboratory

P. O. Box 32, El Alia, 16111

Bab Ezzouar, Algiers

Algeria

ihabeddinedjellas@gmail.com

Mohammed Mekkaoui École Normale Supérieure Department of Mathematics

EDPNLHM Laboratory B. P. 92, Vieux Kouba, 16050

Kouba, Algiers

Algeria

mohammed.mekkaoui@g.ens-kouba.dz
mohammed91mekkaoui@gmail.com

Abstract We study the total positivity of Toeplitz matrices built from generalized hyper-Fibonacci sequences of a fixed generation. Using this approach, we prove that each sequence becomes log-concave beyond a certain point for every generation. We also present several notable special cases that illustrate the scope of our results.

1 Introduction

Let M be an infinite real matrix. If all minors of M have positive determinants, we say that M is totally positive, or simply TP. If all minors of M of every order ≤ r for some integer r are positive, we say that M is totally positive of order r, or simply TP_{r}.

Several studies in the literature have addressed the total positivity of matrices associated with various classes of recurrent sequences. In particular, many combinatorial triangular matrices have been shown to possess the total positivity property. For example, Wang and Wang [30] investigated the total positivity of the Catalan triangle. Additionally, Ahmia and Belbachir [1] proved that the generalized Pascal triangle is TP_{2}. For additional results concerning triangular matrices and Riordan arrays, we refer the reader to [8, 9, 10].

A Toeplitz matrix T = [t_{i,j}] is a (finite or infinite) matrix whose entries satisfy t_{i+1,j+1} = t_{i,j}. In the finite case, that is

T = (\begin{matrix}t_{0} & t_{1} & ⋯ & t_{n} \\ t_{−1} & t_{0} & ⋯ & t_{n−1} \\ ⋮ & ⋮ & ... & ⋮ \\ t_{−n} & t_{−n+1} & ⋯ & t_{0} \end{matrix}) .

A Hankel matrix H = [h_{i,j}] is a (finite or infinite) matrix whose entries satisfy t_{i+1,j−1} = t_{i,j}. In the finite case, that is

H = (\begin{matrix}h_{0} & h_{1} & ⋯ & h_{n} \\ h_{1} & h_{2} & ⋯ & h_{n+1} \\ ⋮ & ⋮ & ... & ⋮ \\ h_{n} & h_{n+1} & ⋯ & h_{2n} \end{matrix}) .

Recall that a sequence (b_{n})_{n≥0} of positive numbers is log-concave (resp., log-convex) if b_{i+1}^{2} ≥ b_{i}b_{i+2} for all i ≥ 0 (resp., b_{i+1}^{2} ≤ b_{i}b_{i+2} for all i ≥ 0). The TP_{2} property of Hankel and Toeplitz matrices is closely connected to the log-convexity and log-concavity, respectively, of the associated sequences. The properties of log-concavity and log-convexity have been widely studied in the literature due to their importance in combinatorics. Recent advances have introduced novel techniques for establishing such properties; see, for instance, [12, 20, 22, 30]. These expansions complement the classical results given by Stanley [27].

The total positivity of Toeplitz and Hankel matrices is among the important properties of these matrices; such properties can find applications in a number of fields, including chemistry, electrical networks, game theory, differential equations, stochastic processes, orthogonal polynomials, combinatorics, quantum groups, algebraic geometry, symmetric functions, and representation theory; see [8, 16, 17, 23, 24] for some applications.

Let p ≥ 1, a ≥ 0, and b ≥ 0 be integers. We consider the generalized Fibonacci sequence (W_{p,n})_{n≥0} defined recursively as follows:

W_{p,n} = pW_{p,n−1} + W_{p,n−2} (n ≥ 2), W_{p,0} = a, W_{p,1} = b.  (1)

Remark 1. Some particular cases of the above sequence are

1. For (a, b) = (0, 1), we obtain the p-Fibonacci numbers F_{p,n} defined by Falcón [15].

2. For (a, b) = (2, p), we obtain the p-Lucas numbers L_{p,n} defined by Falcón and Plaza [14].

Furthermore, for p = 1, the sequences (F_{1,n})_{n≥0} = (F_{n})_{n≥0} and (L_{1,n})_{n≥0} = (L_{n})_{n≥0} correspond to the classical Fibonacci and Lucas sequences, respectively. We refer to [14, 15, 19, 21, 28, 29] for some properties and identities involving the p-Fibonacci and p-Lucas numbers.

Let r ≥ 0 be a fixed integer. Dil and Mező [11] introduced the hyper-Fibonacci numbers F_{n}^{(r)} and the hyper-Lucas numbers L_{n}^{(r)} as follows:

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

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

Let p ≥ 1 and r ≥ 1 be two integers. The generalized hyper-Fibonacci numbers W_{p,n}^{(r)} associated with the generalized Fibonacci numbers W_{p,n} are defined as follows [3, 6]:

W_{p,n}^{(r)} = ∑_{k=0}^{n} p^{n−k}W_{p,k}^{(r−1)}, W_{p,n}^{(0)} = W_{p,n}, W_{p,0}^{(r)} = a, W_{p,1}^{(r)} = par + b.  (2)

Remark 2. Some particular cases of the generalized hyper-Fibonacci numbers are

1. For (a, b) = (0, 1), we get the hyper p-Fibonacci numbers F_{p,n}^{(r)}; see [3, 4, 7].

2. For (a, b) = (2, p), we get the hyper p-Lucas numbers L_{p,n}^{(r)}; see [5].

Furthermore, for p = 1, we have F_{1,n}^{(r)} = F_{n}^{(r)} and L_{1,n}^{(r)} = L_{n}^{(r)}. Also, for p = 2, the numbers F_{2,n}^{(r)} = P_{n}^{(r)} and L_{2,n}^{(r)} = Q_{n}^{(r)} correspond to the n^{th} hyper-Pell and hyper-Pell-Lucas numbers, respectively; see [2].

Došlić et al. [13] studied the total positivity of a class of Toeplitz matrices composed of the hyper-Fibonacci sequence (F_{n}^{(r)})_{n≥0}. A similar study but for the hyper-Lucas sequence (L_{n}^{(r)})_{n≥0} has been studied by Rezig and Ahmia [25]. The aim of our work is to extend these results to the generalized hyper-Fibonacci numbers (W_{p,n}^{(r)})_{n≥0}. To do so, we define the following m × m Toeplitz matrix:

T_{m,n}^{(r)} := (\begin{matrix}W_{p,n}^{(r)} & W_{p,n−1}^{(r)} & ⋯ & W_{p,n−m+1}^{(r)} \\ W_{p,n+1}^{(r)} & W_{p,n}^{(r)} & ⋯ & W_{p,n−m+2}^{(r)} \\ ⋮ & ⋮ & ... & ⋮ \\ W_{p,n+m−1}^{(r)} & W_{p,n+m−2}^{(r)} & ⋯ & W_{p,n}^{(r)} \end{matrix}) .

In Section 2, we present several identities and auxiliary results that we use in this work. In Section 3, we establish the positivity of the determinant det(T_{m,n}^{(r)}) under the condition m ≤ r +1. This, in turn, implies the log-concavity of the sequences (W_{p,n}^{(r)})_{n≥0} for all positive integers r and sufficiently large n. Finally, in Section 4, we examine the total positivity of the Toeplitz matrix T_{r+2,n}^{(r)}.

Throughout this paper, we assume that p ≥ 1, a ≥ 0, and b > 0 are integers.

2 Preliminary results

In this section, we present some of the results we need for the rest of the paper.

Initial minors play a fundamental role in the study of total positivity of matrices. Let I and J denote the row set and column set, respectively. A minor b_{I,J} is called initial if both I and J consist of consecutive indices and I ∪ J contain 1. Consequently, each matrix entry is the lower-right corner of exactly one initial minor. The result established by Gasca and Peña [18] provides a useful method for testing the total positivity of matrices:

Theorem 3. [18] A matrix is totally positive if and only if all its initial minors are positive.

Numerous identities involving generalized Fibonacci numbers have been established in

the literature. Let Δ = a^{2} + pab − b^{2}, α = ^{\frac{p+\sqrt{p^{2}+4}}{2}}, and β = ^{\frac{p−\sqrt{p^{2}+4}}{2}}. Then we have

Lemma 4. Let m ≥ 0 and n ≥ 0 be two integers. Then the following identities hold.

W_{p,n} = Aα^{n} + Bβ^{n}, where A = \frac{b − aβ}{\sqrt{p^{2} + 4}}, B = \frac{aα − b}{\sqrt{p^{2} + 4}},  (3)
F_{p,m−2}W_{p,n−2} − F_{p,m−3}W_{p,n−1} = (−1)^{m−1}W_{n−m+1},  (4)
W_{p,n}W_{p,n+2} − W_{n+1}^{2} = (−1)^{n}Δ,  (5)
lim_{n→+∞} \frac{W_{p,n+1}}{W_{p,n}} = α.  (6)

Lemma 5. [3] Let r ≥ 1 be an integer. The following identities hold

W_{p,n}^{(r)} = pW_{p,n−1}^{(r)} + W_{p,n}^{(r−1)} (n ≥ 1),  (7)
W_{p,n−r}^{(r)} = W_{p,n+r} − S_{r} (n ≥ r),  (8)

where S_{r} = p^{n} ∑_{k=0}^{r−1} \binom{n−k−1}{r−k−1} \frac{W_{p,2k+1}}{p^{r−1}} .

Remark 6. In (8), the numbers S_{r} can be expressed as S_{r} = p^{n}f_{r}(n), where f_{r}(n) is a polynomial in n.

Let r ≥ 0 and n ≥ 0 be two integers. We consider the following Hankel matrix of order r + 2:

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

Proposition 7. [3] Let r ≥ 0 and n ≥ 0 be integers. We have

det(H_{r,n}) = (−1)^{n+1+⌊ \frac{r+3}{2} ⌋}p^{nr+r^{2}}b^{r}Δ.  (9)

3 Positivity of Toeplitz determinants

In this section, we discuss the positivity of determinants of some classes of Toeplitz matrices where the entries are generalized hyper-Fibonacci numbers of the r^{th} generation.

Let T_{m,n}^{(r)} denote the Toeplitz matrix of order m consisting of generalized hyper-Fibonacci numbers of the r^{th} generation

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

with the condition m ≤ r + 1.

Theorem 8. Let m ≥ 1 be an integer. There exists a positive integer n_{m} such that det(T_{m,n}^{(m−1)}) > 0 for all n ≥ n_{m}.

Proof. For m = 2, using (7), we obtain

det(T_{2,n}^{(1)}) := |\begin{matrix}W_{p,n}^{(1)} & W_{p,n−1}^{(1)} \\ W_{p,n+1}^{(1)} & W_{p,n}^{(1)} \end{matrix}| = |\begin{matrix}W_{p,n} & W_{p,n−1}^{(1)} \\ W_{p,n+1} & W_{p,n}^{(1)} \end{matrix}| = |\begin{matrix}W_{p,n} & W_{p,n−1}^{(1)} \\ W_{p,n−1} & W_{p,n} \end{matrix}| .

Thus, by using (8), we have

det(T_{2,n}^{(1)}) = |\begin{matrix}W_{p,n} & W_{p,n+1} − S_{1} \\ W_{p,n−1} & W_{p,n} \end{matrix}| .

According to (5), the last determinant is positive starting from a certain value n_{2}.

In the rest of the proof, we assume that m ≥ 3. Using (7) and some elementary transformations on columns, we get

det(T_{m,n}^{(m−1)}) := |\begin{matrix}W_{p,n}^{(m−1)} & W_{p,n−1}^{(m−1)} & ⋯ & W_{p,n−m+1}^{(m−1)} \\ W_{p,n+1}^{(m−1)} & W_{p,n}^{(m−1)} & ⋯ & W_{p,n−m+2}^{(m−1)} \\ ⋮ & ⋮ & ... & ⋮ \\ W_{p,n+m−1}^{(m−1)} & W_{p,n+m−2}^{(m−1)} & ⋯ & W_{p,n}^{(m−1)} \end{matrix}|

= |\begin{matrix}W_{p,n} & W_{p,n−1}^{(1)} & W_{p,n−2}^{(2)} & ⋯ & W_{p,n−m+1}^{(m−1)} \\ W_{p,n+1} & W_{p,n}^{(1)} & W_{p,n−1}^{(2)} & ⋯ & W_{p,n−m+2}^{(m−1)} \\ ⋮ & ⋮ & ⋮ & & ⋮ \\ W_{p,n+m−1} & W_{p,n+m−2}^{(1)} & W_{p,n+m−3}^{(2)} & ⋯ & W_{p,n}^{(m−1)} \end{matrix}| .

Again, using (7) and some elementary transformations on rows, we get

det(T_{m,n}^{(m−1)}) = |\begin{matrix}W_{p,n} & W_{p,n−1}^{(1)} & W_{p,n−2}^{(2)} & ⋯ & W_{p,n−m+1}^{(m−1)} \\ W_{p,n−1} & W_{p,n} & W_{p,n−1}^{(1)} & ⋯ & W_{p,n−m+2}^{(m−2)} \\ ⋮ & ⋮ & ⋮ & & ⋮ \\ W_{p,n−m+2} & W_{p,n−m+3} & W_{p,n−m+4} & ⋯ & W_{n−1}^{(1)} \\ W_{p,n−m+1} & W_{p,n−m+2} & W_{p,n−m+3} & ⋯ & W_{p,n} \end{matrix}| .

Now, using (8), we write

det(T_{m,n}^{(m−1)}) = |\begin{matrix}W_{p,n} & W_{p,n+1} − S_{1} & W_{p,n+2} − S_{2} & ⋯ & W_{p,n+m−1} − S_{m−1} \\ W_{p,n−1} & W_{p,n} & W_{p,n+1} − S_{1} & ⋯ & W_{p,n+m−2} − S_{m−2} \\ ⋮ & ⋮ & ⋮ & & ⋮ \\ W_{p,n−m+2} & W_{p,n+m+3} & W_{p,n−m+3} & ⋯ & W_{p,n+1} − S_{1} \\ W_{p,n−m+1} & W_{p,n−m+2} & W_{p,n−m+3} & ⋯ & W_{p,n} \end{matrix}| .

Using the recurrence relation (1) on columns, we obtain

det(T_{m,n}^{(m−1)}) = |\begin{matrix}S_{2} − pS_{1} & S_{3} − pS_{2} − S_{1} & ⋯ & W_{p,n+m−2} − S_{m−2} & W_{p,n+m−1} − S_{m−1} \\ S_{1} & S_{2} − pS_{1} & ⋯ & W_{p,n+m−3} − S_{m−3} & W_{p,n+m−2} − S_{m−2} \\ 0 & S_{1} & ⋯ & W_{p,n+m−4} − S_{m−4} & W_{p,n+m−3} − S_{m−3} \\ ⋮ & ⋮ & & ⋮ & ⋮ \\ 0 & 0 & ⋯ & W_{p,n} & W_{p,n+1} − S_{1} \\ 0 & 0 & ⋯ & W_{p,n−1} & W_{p,n} \end{matrix}| .

Again, using (1) on rows, we obtain

\begin{aligned}det(T_{m,n}^{(m−1)}) & = |\begin{matrix}S_{2} − 2pS_{1} & S_{3} − 2pS_{2} + (p^{2} − 2)S_{1} & ⋯ & −S_{m−2} + pS_{m−3} + S_{m−4} & −S_{m−1} + pS_{m−2} + S_{m−3} \\ S_{1} & S_{2} − 2pS_{1} & ⋯ & −S_{m−3} + pS_{m−4} + S_{m−5} & −S_{m−2} + pS_{m−3} + S_{m−4} \\ 0 & S_{1} & ⋯ & −S_{m−4} + pS_{m−5} + S_{m−6} & −S_{m−3} + pS_{m−4} + S_{m−5} \\ ⋮ & ⋮ & & ⋮ & ⋮ \\ 0 & 0 & ⋯ & W_{p,n} & W_{p,n+1} − S_{1} \\ 0 & 0 & ⋯ & W_{p,n−1} & W_{p,n} \end{matrix}| \\ & = det(Ω). \end{aligned}

Let a_{ij} denote the elements of the first m−2 column of Ω and let b_{i,j} denote the elements

of the last two columns, i.e., Ω = {a_{i,j}, b_{i,m−1}, b_{i,m}} 1≤i≤m_{1≤j≤m−2}.

The elements a_{i,j} of Ω satisfy a_{i,j} = a_{i+1,j+1}, (1 ≤ i ≤ m − 1, 1 ≤ j ≤ m − 3), where

\{ \begin{matrix}a_{1,1} = S_{2} − 2pS_{1}, \\ a_{1,2} = S_{3} − 2pS_{2} + (p^{2} − 2)S_{1}, \\ a_{1,3} = S_{4} − 2pS_{3} + (p^{2} − 2)S_{2} + 2pS_{1}, \\ a_{1,j} = S_{j+1} − 2pS_{j} + (p^{2} − 2)S_{j−1} + 2pS_{j−2} + S_{j−3}, 4 ≤ j ≤ m − 2; \\ a_{2,1} = S_{1}, \\ a_{i,1} = 0, 3 ≤ i ≤ m. \end{matrix}

The elements b_{i,j} of Ω satisfy b_{i,m−1} = b_{i+1,m}, (1 ≤ i ≤ m − 3), where

\{ \begin{matrix}b_{i,m−1} = −S_{m−i−1} + pS_{m−i−2} + S_{m−i−3}, 1 ≤ i ≤ m − 4; \\ b_{m−3,m−1} = −S_{2} + pS_{1}, \\ b_{m−2,m−1} = −S_{1}, \\ b_{m−1,m−1} = W_{p,n}, \\ b_{m,m−1} = W_{p,n−1}, \\ b_{1,m} = −S_{m−1} + pS_{m−2} + S_{m−3}, \\ b_{m−1,m} = W_{p,n+1} − S_{1}, \\ b_{m,m} = W_{p,n}. \end{matrix}

Now we use the following column transformations in the last determinant:

b_{i,m−1} ⟶ b′_{i,m−1} := b_{i,m−1} + ∑_{j=1}^{m−3} a_{i,j}, (1 ≤ i ≤ m),

b_{i,m} ⟶ b′_{i,m} := b_{i,m} + ∑_{j=1}^{m−2} a_{i,j}, (1 ≤ i ≤ m),

we obtain

det(T_{m,n}^{(m−1)}) = |\begin{matrix}a_{1,1} & a_{1,2} & ⋯ & a_{1,m−2} & b′_{1,m−1} & b′_{1,m} \\ a_{2,1} & a_{2,2} & ⋯ & a_{2,m−2} & b′_{2,m−1} & b′_{2,m} \\ ⋮ & ⋮ & & ⋮ & ⋮ & ⋮ \\ 0 & 0 & ⋯ & a_{n−2,n−2} & b′_{m−2,m−1} & b′_{m−2,m} \\ 0 & 0 & ⋯ & S_{1} & W_{p,n} & W_{p,n+1} \\ 0 & 0 & ⋯ & 0 & W_{p,n−1} & W_{p,n} \end{matrix}| ,  (10)

where the last two columns are

(\begin{matrix}(p^{2} − 1)S_{1} + p^{2} ∑_{j=2}^{m−6} S_{j} + (p^{2} − 1)S_{m−5} + (p^{2} − 2p)S_{m−4} + (1 − p)S_{m−3} & (p^{2} − 1)S_{1} + p^{2} ∑_{j=2}^{m−5} S_{j} + (p^{2} − 1)S_{m−4} + (p^{2} − 2p)S_{m−3} + (1 − p)S_{m−2} \\ p^{2} ∑_{j=1}^{m−7} S_{j} + (p^{2} − 1)S_{m−6} + (p^{2} − 2p)S_{m−5} + (1 − p)S_{m−4} & p^{2} ∑_{j=1}^{m−6} S_{j} + (p^{2} − 1)S_{m−5} + (p^{2} − 2p)S_{m−4} + (1 − p)S_{m−3} \\ ⋮ & ⋮ \\ p^{2}(S_{1} + S_{2}) + (p^{2} − 1)S_{3} + (p^{2} − 2p)S_{4} + (1 − p)S_{5} & p^{2}(S_{1} + S_{2} + S_{3}) + (p^{2} − 1)S_{4} + (p^{2} − 2p)S_{5} + (1 − p)S_{6} \\ p^{2}S_{1} + (p^{2} − 1)S_{2} + (p^{2} − 2p)S_{3} + (1 − p)S_{4} & p^{2}(S_{1} + S_{2}) + (p^{2} − 1)S_{3} + (p^{2} − 2p)S_{4} + (1 − p)S_{5} \\ (p^{2} − 1)S_{1} + (p^{2} − 2p)S_{2} + (1 − p)S_{3} & p^{2}S_{1} + (p^{2} − 1)S_{2} + (p^{2} − 2p)S_{3} + (1 − p)S_{4} \\ (p^{2} − 2p)S_{1} + (1 − p)S_{2} & (p^{2} − 1)S_{1} + (p^{2} − 2p)S_{2} + (1 − p)S_{3} \\ (1 − p)S_{1} & (p^{2} − 2p)S_{1} + (1 − p)S_{2} \\ 0 & (1 − p)S_{1} \\ W_{p,n} & W_{p,n+1} \\ W_{p,n−1} & W_{p,n} \end{matrix}).

Furthermore, we perform the following column transformations on (10):

b′_{i,m−1} ⟶ b′′_{i,m−1} := b′_{i,m−1} + ∑_{j=1}^{m−4}(F_{p,m−j−2} − 1)a_{i,j}, (1 ≤ i ≤ m),

b′_{i,m} ⟶ b′′_{i,m} := b′_{i,m} + ∑_{j=1}^{m−3}(F_{p,m−j−1} − 1)a_{i,j}, (1 ≤ i ≤ m),

where F_{p,n} is the n^{th} p-Fibonacci number, we get

det(T_{m,n}^{(m−1)}) = |\begin{matrix}a_{1,1} & a_{1,2} & ⋯ & a_{1,m−2} & −S_{1}F_{p,m−2} & −S_{1}F_{p,m−1} \\ a_{2,1} & a_{2,2} & ⋯ & a_{2,m−2} & 0 & 0 \\ ⋮ & ⋮ & & ⋮ & ⋮ & ⋮ \\ 0 & 0 & ⋯ & a_{m−2,m−2} & 0 & 0 \\ 0 & 0 & ⋯ & S_{1} & W_{p,n} & W_{p,n+1} \\ 0 & 0 & ⋯ & 0 & W_{p,n−1} & W_{p,n} \end{matrix}| .

Next, we apply (5) with the following computational techniques on two last rows:

det(T_{m,n}^{(m−1)}) = \frac{1}{W_{p,n−1}} | \begin{matrix}& ⋮ & ⋮ & ⋮ \\ ⋯ & S_{1} & W_{p,n} & W_{p,n+1} \\ ⋯ & 0 & W_{p,n−1}^{2} & W_{p,n}W_{p,n−1} \end{matrix}|

= \frac{1}{W_{p,n−1}} | \begin{matrix}& ⋮ & ⋮ & ⋮ \\ ⋯ & S_{1} & W_{p,n} & W_{p,n+1} \\ ⋯ & −S_{1}W_{p,n−2} & (−1)^{n−1}Δ & p(−1)^{n−1}Δ \end{matrix}|

= \frac{(−1)^{n−1}}{ΔW_{p,n−1}} |\begin{matrix}& ⋮ & ⋮ & ⋮ \\ ⋯ & (−1)^{n−1}ΔS_{1} & (−1)^{n−1}ΔW_{p,n} & (−1)^{n−1}ΔW_{p,n+1} \\ ⋯ & −S_{1}W_{p,n−2} & (−1)^{n−1}Δ & p(−1)^{n−1}Δ \end{matrix}|

= \frac{(−1)^{n−1}}{Δ} |\begin{matrix}a_{1,1} & ⋯ & a_{1,m−2} & −S_{1}F_{p,m−2} & −S_{1}F_{p,m−1} \\ a_{2,1} & ⋯ & a_{2,m−2} & 0 & 0 \\ ⋮ & & ⋮ & ⋮ & ⋮ \\ 0 & ⋯ & S_{1}W_{p,n−1} & 0 & (−1)^{n−1}Δ \\ 0 & ⋯ & −S_{1}W_{p,n−2} & (−1)^{n−1}Δ & (−1)^{n−1}pΔ \end{matrix}| .

Thus, we have

det(T_{m,n}^{(m−1)}) = \frac{(−1)^{n}}{Δ} |\begin{matrix}a_{1,1} & ⋯ & a_{1,m−2} & −S_{1}F_{p,m−3} & −S_{1}F_{p,m−2} \\ a_{2,1} & ⋯ & a_{2,m−2} & 0 & 0 \\ ⋮ & & ⋮ & ⋮ & ⋮ \\ 0 & ⋯ & S_{1}W_{p,n−1} & (−1)^{n−1}Δ & 0 \\ 0 & ⋯ & −S_{1}W_{p,n−2} & 0 & (−1)^{n−1}Δ \end{matrix}|

= \frac{(−1)^{n}}{Δ} |\begin{matrix}a_{1,1} & ⋯ & A & −S_{1}F_{p,m−3} & −S_{1}F_{p,m−2} \\ a_{2,1} & ⋯ & a_{2,m−2} & 0 & 0 \\ ⋮ & & ⋮ & ⋮ & ⋮ \\ 0 & ⋯ & 0 & (−1)^{n−1}Δ & 0 \\ 0 & ⋯ & 0 & 0 & (−1)^{n−1}Δ \end{matrix}|

= (−1)^{n}Δ |\begin{matrix}a_{1,1} & a_{1,2} & ⋯ & a_{1,m−3} & A \\ S_{1} & a_{2,2} & ⋯ & a_{2,m−3} & a_{2,m−2} \\ 0 & S_{1} & ⋯ & a_{3,m−3} & a_{3,m−2} \\ ⋮ & ⋮ & & ⋮ \\ 0 & 0 & ⋯ & S_{1} & a_{m−2,m−2} \end{matrix}| ,

where

A = a_{1,m−2} + \frac{(−1)^{n}S_{1}^{2}}{Δ} (F_{p,m−2}W_{p,n−2} − F_{p,m−3}W_{p,n−1})
= a_{1,m−2} + \frac{(−1)^{n}S_{1}^{2}}{Δ} (−1)^{m−1}W_{p,n−m+1} (using (4)).  (11)

The last determinant can be rewritten as follows:

det(T_{m,n}^{(m−1)}) = (−1)^{n}Δ ((−1)^{m−1}S_{1}^{m−3}A + ∑_{j=1}^{m−3}(−1)^{i+1}a_{1,j}M_{1,j}) ,  (12)

where M_{1,j} denotes the determinant obtained from det(T_{m,n}^{(m−1)}) by omitting the first row and j^{th} column (1 ≤ j ≤ m − 3). Using Remark 6, we get a_{i,j} = p^{n}f_{i,j}(n) where f_{i,j}(n) is a polynomial in n. This implies that M_{1,j} = p^{n(m−3)}f_{j}(n) where f_{j}(n) is a polynomial on n (1 ≤ j ≤ m − 3). Therefore, we have

det(T_{m,n}^{(m−1)}) = (−1)^{n}Δ((−1)^{m−1}S_{1}^{m−3}A + p^{n(m−2)}f(n)) (S_{1} := bp^{n})
= p^{n(m−2)}b^{m−1}(p^{n}W_{p,n−m+1} + g(n)) (using (11)),

where f(n) and g(n) are polynomials in n. The existence of a positive integer n_{m} such that W_{p,n−m+1} > P (n) for all n > n_{m}, where P (n) is a polynomial of any degree, completes the proof.

Corollary 9. Let r ≥ 1 be an integer. For every integer m such that 1 ≤ m ≤ r + 1, there exists a positive integer n_{m} such that det(T_{m,n}^{(r)}) > 0 for all n ≥ n_{m}.

Proof. We use induction on r. The case r = m − 1 is given in Theorem 8. Assume that the claim is true for all m − 1 ≤ t ≤ r − 1; we prove that

det(T_{m,n}^{(r)}) = |\begin{matrix}W_{p,n}^{(r)} & W_{p,n−1}^{(r)} & ⋯ & W_{p,n−m+1}^{(r)} \\ W_{p,n+1}^{(r)} & W_{p,n}^{(r)} & ⋯ & W_{p,n−m+2}^{(r)} \\ ⋮ & ⋮ & ... & ⋮ \\ W_{p,n+m−1}^{(r)} & W_{p,n+m−2}^{(r)} & ⋯ & W_{p,n}^{(r)} \end{matrix}| > 0.

By performing row transformations using relation (7), we obtain

det(T_{m,n}^{(r)}) = |\begin{matrix}W_{p,n}^{(r)} & W_{p,n−1}^{(r)} & ⋯ & W_{p,n−m+1}^{(r)} \\ W_{p,n+1}^{(r−1)} & W_{p,n}^{(r−1)} & ⋯ & W_{p,n−m+2}^{(r−1)} \\ ⋮ & ⋮ & ... & ⋮ \\ W_{p,n+m−1}^{(r−1)} & W_{p,n+m−2}^{(r−1)} & ⋯ & W_{p,n}^{(r−1)} \end{matrix}|

= ∑_{j=1}^{m}(−1)^{j+1} \frac{W_{p,n−j+1}^{(r)}}{W_{p,n−j+1}^{(r−1)}} W_{p,n−j+1}^{(r−1)}M_{1,j},

where M_{1,j} denotes the determinant obtained from det(T_{m,n}^{(r)}) by omitting the first row and j^{th} column (1 ≤ j ≤ m). We define a function f : ℝ^{n} ⟶ ℝ by

f(x_{1}, x_{2}, …, x_{n}) = ∑_{j=1}^{m}(−1)^{j+1}x_{j}W_{p,n−j+1}^{(r−1)}M_{1,j}.

Let ϕ := 1+pα, where α = ^{\frac{p+\sqrt{p^{2}+4}}{2}}. We have f(ϕ, ϕ, …, ϕ) = ϕ ∑_{j=1}^{m}(−1)^{j+1}W_{p,n−j+1}^{(r−1)}M_{1,j} = ϕ det(T_{m,n}^{(r−1)}) > 0. Since f is continuous, there is a neighborhood

I = (ϕ − ϵ_{1}, ϕ + ϵ_{1}) × (ϕ − ϵ_{2}, ϕ + ϵ_{2}) × ⋯ × (ϕ − ϵ_{n}, ϕ + ϵ_{n})

of the point (ϕ, ϕ, …, ϕ) such that f is positive on I. Using (8), by dividing both sides by W_{n}^{(r−1)}, and passing to limit n ⟶ ∞, one readily obtains

lim_{n→∞} \frac{W_{p,n}^{(r)}}{W_{p,n}^{(r−1)}} = lim_{n→∞} \frac{W_{p,n+2r}}{W_{p,n+2r−2}} = ϕ.

This implies that, for sufficiently large n, the coefficient x_{j} = \frac{W_{p,n−j+1}^{(r)}}{W_{p,n−j+1}^{(r−1)}} falls into (ϕ−ϵ_{j}, ϕ+ϵ_{j})

for all 1 ≤ j ≤ n. Therefore, there exists a positive integer n_{m} such that

det(T_{m,n}^{(r)}) = f (\frac{W_{p,n}^{(r)}}{W_{p,n}^{(r−1)}} , \frac{W_{p,n−1}^{(r)}}{W_{p,n−1}^{(r−1)}} , …, \frac{W_{p,n−m+1}^{(r)}}{W_{p,n−m+1}^{(r−1)}} ) > 0

for all n ≥ n_{m}. The proof is thus completed.

If we take m = 2 in Corollary 9, we obtain

Corollary 10. Let r ≥ 1 be an integer. The sequence (W_{p,n}^{(r)})_{n} is log-concave for large enough n.

Corollary 11. Let r ≥ 1 be an integer. The sequences (F_{p,n}^{(r)})_{n} and (L_{p,n}^{(r)})_{n} are log-concave for large enough n.

For p = 1, we obtain the log-concavity of the sequences (F_{n}^{(r)})_{n} and (L_{n}^{(r)})_{n}. This recovers the main result established by Zheng et al. [31].

4 Total positivity of Toeplitz matrices

Let T_{r+2,n}^{(r)} denote the Toeplitz matrix of order r+2 consisting of generalized hyper-Fibonacci numbers of the r^{th} generation; that is

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

Let Δ := a^{2} + pab − b^{2}. Then we have the following main theorem.

Theorem 12. Let r ≥ 1 be an integer.

1. If Δ > 0, then there exists a positive integer n_{r} such that the matrix T_{r+2,2n}^{(r)} is totally positive for every n ≥ n_{r}.

2. If Δ < 0, then there exists a positive integer n′_{r} such that the matrix T_{r+2,2n+1}^{(r)} is totally

positive for every n ≥ n′_{r}. Proof. The 2r + 1 initial minors of order two of T_{r+2,n}^{(r)} have the form T_{2,n_{2}}^{(r)}, where n_{2} ≥ n − r. Using Corollary 9, there exists a positive integer q_{2} such that these initial minors are positive for n_{2} ≥ q_{2}. Similarly, according to Corollary 9, there exist positive integers q_{3}, q_{4}, …, q_{r+1} such that the initial minors of order 3, 4, …, r + 1, which are of the form T_{3,n_{3}}^{(r)},T_{4,n_{4}}^{(r)},…, T_{r+1,n_{r+1}}^{(r)}, are positive for n_{3} ≥ q_{3}, n_{4} ≥ q_{4}, …, n_{r+1} ≥ q_{r+1}. It remains to discuss the positivity of det(T_{r+2,n}^{(r)}). By reversing the order of columns of T_{r+2,n}^{(r)}, we obtain

det(T_{r+2,n}^{(r)}) = (−1)^{⌊ \frac{r+2}{2} ⌋} |\begin{matrix}W_{p,n−r−1}^{(r)} & W_{p,n−r}^{(r)} & ⋯ & W_{p,n}^{(r)} \\ W_{p,n−r}^{(r)} & W_{p,n−r+1}^{(r)} & ⋯ & W_{p,n+1}^{(r)} \\ ⋮ & ⋮ & ... & ⋮ \\ W_{p,n}^{(r)} & W_{p,n+1}^{(r)} & ⋯ & W_{p,n+r+1}^{(r)} \end{matrix}|
= (−1)^{⌊ \frac{r+2}{2} ⌋} det(H_{r,n−r−1}) (using Proposition 7)
= (−1)^{n}p^{(n−1)r}b^{r}Δ.

The following corollary follows directly from Theorem 12. Corollary 13. Let r ≥ 1 be a positive integer. There exists a positive integer n_{r} such that the matrix T_{r+2,n}^{(r)} is TP_{r+1} for all n ≥ n_{r}.

Furthermore, the following corollaries follow directly from Theorem 12 and Corollary 13. Corollary 14. Let r ≥ 1 be a positive integer. There exist positive integers n_{r} and n′_{r} such that

1. The matrix

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

is totally positive for every n ≥ n_{r}.

2. The matrix

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

is TP_{r+1} for every n ≥ n′_{r}. For p = 1, we obtain the main results given by Došlić et al. [13].

Corollary 15. Let r ≥ 1 be an integer. There exist positive integers n_{r} and n′_{r} such that: 1. The matrix

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

is totally positive for every n ≥ n_{r}.

2. The matrix

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

is TP_{r+1} for every n ≥ n′_{r}.

For p = 1, we obtain the main results given by Rezig and Ahmia [25].

5 Final remarks

This work relies on specific choices of coefficients, initial conditions, and parity assumptions. A complete characterization of total positivity without these constraints remains open and will likely require new techniques.

6 Acknowledgments

We thank the reviewer and the editor for their valuable remarks and suggestions, which have improved the quality of the paper.

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2020 Mathematics Subject Classification: Primary 15B36; Secondary 11B39, 15A45. Keywords: hyper-Fibonacci number, hyper-Lucas number, Toeplitz matrix, total positivity, log-concavity.

(Concerned with sequences \underline{A000032}, \underline{A000045}, \underline{A000129}, and \underline{A001333}.)

Received May 12 2025; revised versions received May 14 2025; February 26 2026. Published in Journal of Integer Sequences, April 22 2026.

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