Total positivity

DefinitionMatrices with nonnegative minors#✓fully proved

Let \(\iota\) and \(\kappa\) be linearly ordered types and let \(M\) be a real matrix with rows indexed by \(\iota\) and columns indexed by \(\kappa\). Then \(M\) has nonnegative minors when, for every \(k \in \mathbb{N}\) and all strictly increasing maps \(r \colon \{0, \dots, k-1\} \to \iota\) and \(c \colon \{0, \dots, k-1\} \to \kappa\), the \(k \times k\) submatrix \((M_{r(a), c(b)})_{a, b < k}\) has nonnegative determinant. The case \(k = 0\), the empty minor, is included.

public def Matrix.HasNonnegativeMinors {ι κ : Type*} [LinearOrder ι] [LinearOrder κ]
    (M : Matrix ι κ ℝ) : Prop :=
  ∀ (k : ℕ) (r : Fin k → ι) (c : Fin k → κ),
    StrictMono r → StrictMono c → 0 ≤ (M.submatrix r c).det

Sources

• Ming-Jian Ding and Jiang Zeng, "Some New Results on the Minuscule Polynomials of Type A", Journal of Integer Sequences 28 (2025), Article 25.1.2, "Total positivity of the coefficient matrix" section, definition at lines 638–639 (used by the Schoenberg–Edrei lemma, line 654), https://cs.uwaterloo.ca/journals/JIS/VOL28/Zeng/zeng26.tex.

Used by 6

Used byTotal positivity of Bell-type Riordan arrays from a Pólya frequency series · Bell-type Riordan arrays with a Pólya frequency A-sequence are totally positive · Total positivity of a Riordan array from its coefficient matrix · A total positivity criterion for Riordan arrays · Total positivity of the coefficient matrix of the minuscule polynomials · Aissen–Schoenberg–Whitney–Edrei theorem

DefinitionMatrices with positive minors#✓fully proved

A real matrix \(M\) whose rows and columns are indexed by linearly ordered sets \(R\) and \(C\) has positive minors (is totally positive) when, for every natural number \(k\) and all strictly increasing maps \(r : \{1, \dots, k\} \to R\) and \(c : \{1, \dots, k\} \to C\), the \(k \times k\) submatrix \((M_{r(a), c(b)})_{1 \le a, b \le k}\) has positive determinant. The case \(k = 0\) is the empty minor, whose determinant is \(1\).

public def Matrix.HasPositiveMinors {ι κ : Type*} [LinearOrder ι] [LinearOrder κ]
    (M : Matrix ι κ ℝ) : Prop :=
  ∀ (k : ℕ) (r : Fin k → ι) (c : Fin k → κ),
    StrictMono r → StrictMono c → 0 < (M.submatrix r c).det

Sources

• Abdelhak Taane, Ihab-Eddine Djellas and Mohammed Mekkaoui, Total Positivity of Toeplitz Matrices Involving Generalized Hyper-Fibonacci Numbers, Journal of Integer Sequences 29 (2026), Article 26.2.5, Section 1 (total positivity).

Used by 5

Used byEntries of a matrix with positive minors are positive · Eventual total positivity of hyper-Fibonacci Toeplitz matrices · Gasca–Peña criterion for total positivity · Total positivity of hyper \(p\)-Fibonacci Toeplitz matrices · Total positivity of hyper \(p\)-Lucas Toeplitz matrices

TheoremEntries of a matrix with positive minors are positive#✓fully proved

Let \(M\) be a real matrix whose rows and columns are indexed by linearly ordered sets \(R\) and \(C\). If \(M\) has positive minors, then \(M_{ij} > 0\) for all \(i \in R\) and \(j \in C\); these entries are the \(1 \times 1\) minors of \(M\).

public theorem Matrix.HasPositiveMinors.entry_pos {ι κ : Type*} [LinearOrder ι]
    [LinearOrder κ] {M : Matrix ι κ ℝ} (hM : Matrix.HasPositiveMinors M) (i : ι) (j : κ) :
    0 < M i j

Sources

• Abdelhak Taane, Ihab-Eddine Djellas and Mohammed Mekkaoui, Total Positivity of Toeplitz Matrices Involving Generalized Hyper-Fibonacci Numbers, Journal of Integer Sequences 29 (2026), Article 26.2.5, Section 1 (total positivity).

Uses 1

Statement usesMatrices with positive minors

TheoremGasca–Peña criterion for total positivity#●ready to prove

Let \(A\) be a real \(n \times n\) matrix whose rows and columns are indexed by \(0, \dots, n - 1\). Then \(A\) is totally positive (has positive minors) if and only if all its initial minors are positive: for all natural numbers \(k\), \(i_0\) and \(j_0\) with \(i_0 + k \le n\), \(j_0 + k \le n\) and \(i_0 = 0\) or \(j_0 = 0\), the \(k \times k\) submatrix of \(A\) with rows \(i_0, \dots, i_0 + k - 1\) and columns \(j_0, \dots, j_0 + k - 1\) has positive determinant.

public theorem_wanted hasPositiveMinors_iff_initial_minors_pos {n : ℕ}
    (A : Matrix (Fin n) (Fin n) ℝ) :
    Matrix.HasPositiveMinors A ↔
      ∀ (k i₀ j₀ : ℕ) (hi : i₀ + k ≤ n) (hj : j₀ + k ≤ n), (i₀ = 0 ∨ j₀ = 0) →
        0 < (A.submatrix (fun r : Fin k => (⟨i₀ + r, by omega⟩ : Fin n))
          (fun c : Fin k => (⟨j₀ + c, by omega⟩ : Fin n))).det

Sources

• Abdelhak Taane, Ihab-Eddine Djellas and Mohammed Mekkaoui, Total Positivity of Toeplitz Matrices Involving Generalized Hyper-Fibonacci Numbers, Journal of Integer Sequences 29 (2026), Article 26.2.5, Theorem 3, quoting Gasca and Peña (jis-2026-26.2.5-thm-3).

Uses 1

Statement usesMatrices with positive minors

DefinitionGeneralized \(p\)-Fibonacci sequence#✓fully proved

For integers \(p\), \(a\) and \(b\), the generalized Fibonacci sequence \((W_{p,n})_{n \ge 0}\) of integers is defined by \(W_{p,0} = a\), \(W_{p,1} = b\) and \(W_{p,n} = p W_{p,n-1} + W_{p,n-2}\) for \(n \ge 2\).

public def generalizedPFibonacci (p a b : ℤ) : ℕ → ℤ
  | 0 => a
  | 1 => b
  | n + 2 => p * generalizedPFibonacci p a b (n + 1) + generalizedPFibonacci p a b n

Sources

• Abdelhak Taane, Ihab-Eddine Djellas and Mohammed Mekkaoui, Total Positivity of Toeplitz Matrices Involving Generalized Hyper-Fibonacci Numbers, Journal of Integer Sequences 29 (2026), Article 26.2.5, Section 1, recurrence (1).

Used by 4

Used byGeneralized hyper-Fibonacci numbers · A binomial-sum formula for generalized hyper-Fibonacci numbers · \(p\)-Lucas numbers as a companion Lucas sequence · \(p\)-Fibonacci numbers as a Lucas sequence

Theorem\(p\)-Fibonacci numbers as a Lucas sequence#✓fully proved

For every integer \(p\), the generalized Fibonacci sequence \(W_{p,n}\) with initial values \(a = 0\) and \(b = 1\) (the \(p\)-Fibonacci numbers \(F_{p,n}\)) is the Lucas sequence \(U(p, -1)\): \(W_{p,n} = U_n(p, -1)\) for every natural number \(n\).

public theorem generalizedPFibonacci_zero_one (p : ℤ) (n : ℕ) :
    generalizedPFibonacci p 0 1 n = lucasU ⟨p, -1, by decide⟩ n

Sources

• Abdelhak Taane, Ihab-Eddine Djellas and Mohammed Mekkaoui, Total Positivity of Toeplitz Matrices Involving Generalized Hyper-Fibonacci Numbers, Journal of Integer Sequences 29 (2026), Article 26.2.5, Section 1, remark after recurrence (1) (p-Fibonacci numbers F_{p,n}).

Uses 3

Statement usesParameters of a Lucas sequence · Generalized \(p\)-Fibonacci sequence · Lucas sequence \(U(P, Q)\)

Theorem\(p\)-Lucas numbers as a companion Lucas sequence#✓fully proved

For every integer \(p\), the generalized Fibonacci sequence \(W_{p,n}\) with initial values \(a = 2\) and \(b = p\) (the \(p\)-Lucas numbers \(L_{p,n}\)) is the companion Lucas sequence \(V(p, -1)\): \(W_{p,n} = V_n(p, -1)\) for every natural number \(n\).

public theorem generalizedPFibonacci_two_p (p : ℤ) (n : ℕ) :
    generalizedPFibonacci p 2 p n = lucasV ⟨p, -1, by decide⟩ n

Sources

• Abdelhak Taane, Ihab-Eddine Djellas and Mohammed Mekkaoui, Total Positivity of Toeplitz Matrices Involving Generalized Hyper-Fibonacci Numbers, Journal of Integer Sequences 29 (2026), Article 26.2.5, Section 1, remark after recurrence (1) (p-Lucas numbers L_{p,n}).

Uses 3

Statement usesParameters of a Lucas sequence · Generalized \(p\)-Fibonacci sequence · Companion Lucas sequence \(V(P, Q)\)

DefinitionGeneralized hyper-Fibonacci numbers#✓fully proved

Let \(p, a, b\) be integers and let \(W_{p,n}\) denote the generalized Fibonacci sequence, with \(W_{p,0} = a\), \(W_{p,1} = b\) and \(W_{p,n+2} = p W_{p,n+1} + W_{p,n}\). The generalized hyper-Fibonacci numbers \(W^{(r)}_{p,n}\), for natural numbers \(r\) and \(n\), are the integers defined by \(W^{(0)}_{p,n} = W_{p,n}\) and, for \(r \ge 1\), by \(W^{(r)}_{p,n} = \sum_{k=0}^{n} p^{n-k} W^{(r-1)}_{p,k}\).

public def generalizedHyperFibonacci (p a b : ℤ) : ℕ → ℕ → ℤ
  | 0 => generalizedPFibonacci p a b
  | r + 1 => fun n =>
      ∑ k ∈ Finset.range (n + 1), p ^ (n - k) * generalizedHyperFibonacci p a b r k

Sources

• Abdelhak Taane, Ihab-Eddine Djellas and Mohammed Mekkaoui, Total Positivity of Toeplitz Matrices Involving Generalized Hyper-Fibonacci Numbers, Journal of Integer Sequences 29 (2026), Article 26.2.5, Section 1, definition (2).

Uses 1 · Used by 12

Statement usesGeneralized \(p\)-Fibonacci sequence

Used byEventual log-concavity of generalized hyper-Fibonacci numbers · Hankel determinants of generalized hyper-Fibonacci numbers · A binomial-sum formula for generalized hyper-Fibonacci numbers · Recurrence for generalized hyper-Fibonacci numbers · Eventual positivity of hyper-Fibonacci Toeplitz determinants · Eventual positivity of lower-order hyper-Fibonacci Toeplitz determinants · Eventual total positivity of hyper-Fibonacci Toeplitz matrices · Eventually positive bounded-order minors of hyper-Fibonacci Toeplitz matrices · Initial values of generalized hyper-Fibonacci numbers · Eventual log-concavity of hyper \(p\)-Fibonacci and hyper \(p\)-Lucas numbers · Total positivity of hyper \(p\)-Fibonacci Toeplitz matrices · Total positivity of hyper \(p\)-Lucas Toeplitz matrices

TheoremInitial values of generalized hyper-Fibonacci numbers#✓fully proved

Let \(p, a, b\) be integers and let \(W^{(r)}_{p,n}\) denote the generalized hyper-Fibonacci numbers. For every natural number \(r\), \(W^{(r)}_{p,0} = a\) and \(W^{(r)}_{p,1} = p \cdot a \cdot r + b\).

public theorem generalizedHyperFibonacci_zero_one (p a b : ℤ) (r : ℕ) :
    generalizedHyperFibonacci p a b r 0 = a ∧
      generalizedHyperFibonacci p a b r 1 = p * a * r + b

Sources

• Abdelhak Taane, Ihab-Eddine Djellas and Mohammed Mekkaoui, Total Positivity of Toeplitz Matrices Involving Generalized Hyper-Fibonacci Numbers, Journal of Integer Sequences 29 (2026), Article 26.2.5, Section 1, definition (2) (initial values).

Uses 1

Statement usesGeneralized hyper-Fibonacci numbers

TheoremRecurrence for generalized hyper-Fibonacci numbers#●ready to prove

Let \(p, a, b\) be integers and let \(W^{(r)}_{p,n}\) denote the generalized hyper-Fibonacci numbers. For all natural numbers \(r \ge 1\) and \(n \ge 1\), \(W^{(r)}_{p,n} = p W^{(r)}_{p,n-1} + W^{(r-1)}_{p,n}\).

public theorem_wanted generalizedHyperFibonacci_succ_succ (p a b : ℤ) (r n : ℕ) :
    generalizedHyperFibonacci p a b (r + 1) (n + 1) =
      p * generalizedHyperFibonacci p a b (r + 1) n + generalizedHyperFibonacci p a b r (n + 1)

Sources

• Abdelhak Taane, Ihab-Eddine Djellas and Mohammed Mekkaoui, Total Positivity of Toeplitz Matrices Involving Generalized Hyper-Fibonacci Numbers, Journal of Integer Sequences 29 (2026), Article 26.2.5, Lemma 5, identity (3), quoting Ait-Amrane et al. (jis-2026-26.2.5-lem-5).

Uses 1

Statement usesGeneralized hyper-Fibonacci numbers

TheoremA binomial-sum formula for generalized hyper-Fibonacci numbers#●ready to prove

Let \(p, a, b\) be integers with \(p \ne 0\), let \(W_{p,n}\) denote the generalized Fibonacci sequence and \(W^{(r)}_{p,n}\) the generalized hyper-Fibonacci numbers. For all natural numbers \(r\) and \(n\) with \(1 \le r \le n\), the following identity of rational numbers holds:

\[ W^{(r)}_{p,n-r} = W_{p,n+r} - \frac{p^n}{p^{r-1}} \sum_{k=0}^{r-1} \binom{n-k-1}{r-k-1} W_{p,2k+1}. \]

public theorem_wanted generalizedHyperFibonacci_sub_eq_sub (p a b : ℤ) (hp : p ≠ 0) (r n : ℕ)
    (hr : 1 ≤ r) (hrn : r ≤ n) :
    (generalizedHyperFibonacci p a b r (n - r) : ℚ) =
      generalizedPFibonacci p a b (n + r) -
        (p : ℚ) ^ n * (∑ k ∈ Finset.range r,
          ((n - k - 1).choose (r - k - 1) : ℚ) * generalizedPFibonacci p a b (2 * k + 1)) /
          (p : ℚ) ^ (r - 1)

Sources

• Abdelhak Taane, Ihab-Eddine Djellas and Mohammed Mekkaoui, Total Positivity of Toeplitz Matrices Involving Generalized Hyper-Fibonacci Numbers, Journal of Integer Sequences 29 (2026), Article 26.2.5, Lemma 5, identity (4), quoting Ait-Amrane et al. (jis-2026-26.2.5-lem-5).

Uses 2

Statement usesGeneralized hyper-Fibonacci numbers · Generalized \(p\)-Fibonacci sequence

TheoremHankel determinants of generalized hyper-Fibonacci numbers#●ready to prove

Let \(p, a, b\) be integers, let \(W^{(r)}_{p,n}\) denote the generalized hyper-Fibonacci numbers and put \(\Delta = a^2 + pab - b^2\). For all natural numbers \(r\) and \(n\), the \((r+2) \times (r+2)\) Hankel matrix \(H_{r,n} = (W^{(r)}_{p,n+i+j})_{0 \le i, j \le r+1}\) has determinant \(\det H_{r,n} = (-1)^{n+1+\lfloor (r+3)/2 \rfloor} p^{nr+r^2} b^r \Delta\).

public theorem_wanted generalizedHyperFibonacci_hankel_det (p a b : ℤ) (r n : ℕ) :
    (Matrix.of fun i j : Fin (r + 2) =>
        generalizedHyperFibonacci p a b r (n + i + j)).det =
      (-1) ^ (n + 1 + (r + 3) / 2) * p ^ (n * r + r ^ 2) * b ^ r * (a ^ 2 + p * a * b - b ^ 2)

Sources

• Abdelhak Taane, Ihab-Eddine Djellas and Mohammed Mekkaoui, Total Positivity of Toeplitz Matrices Involving Generalized Hyper-Fibonacci Numbers, Journal of Integer Sequences 29 (2026), Article 26.2.5, Proposition 7, quoting Ait-Amrane et al. (jis-2026-26.2.5-prop-7).

Uses 1

Statement usesGeneralized hyper-Fibonacci numbers

TheoremEventual positivity of hyper-Fibonacci Toeplitz determinants#●ready to prove

Let \(p, a, b\) be integers with \(p \ge 1\), \(a \ge 0\) and \(b > 0\), and let \(W^{(r)}_{p,n}\) denote the generalized hyper-Fibonacci numbers. For every natural number \(m \ge 1\) there is a natural number \(N\) such that for every \(n \ge N\) the \(m \times m\) Toeplitz matrix with entry \(W^{(m-1)}_{p,n+i-j}\) in row \(i\) and column \(j\) (\(0 \le i, j < m\)) has positive determinant.

public theorem_wanted generalizedHyperFibonacci_toeplitz_det_eventually_pos (p a b : ℤ)
    (hp : 1 ≤ p) (ha : 0 ≤ a) (hb : 0 < b) (m : ℕ) (hm : 1 ≤ m) :
    ∃ N : ℕ, ∀ n ≥ N, 0 < (Matrix.of fun i j : Fin m =>
      generalizedHyperFibonacci p a b (m - 1) (n + i - j)).det

Sources

• Abdelhak Taane, Ihab-Eddine Djellas and Mohammed Mekkaoui, Total Positivity of Toeplitz Matrices Involving Generalized Hyper-Fibonacci Numbers, Journal of Integer Sequences 29 (2026), Article 26.2.5, Theorem 8 (jis-2026-26.2.5-thm-8).

Uses 1

Statement usesGeneralized hyper-Fibonacci numbers

TheoremEventual positivity of lower-order hyper-Fibonacci Toeplitz determinants#●ready to prove

Let \(p, a, b\) be integers with \(p \ge 1\), \(a \ge 0\) and \(b > 0\), and let \(W^{(r)}_{p,n}\) denote the generalized hyper-Fibonacci numbers. For natural numbers \(r\), \(m\) and \(n\) write \(T^{(r)}_{m,n}\) for the \(m \times m\) Toeplitz matrix with entry \(W^{(r)}_{p,n+i-j}\) in row \(i\) and column \(j\) (\(0 \le i, j < m\)). If \(r \ge 1\) and \(1 \le m \le r + 1\), then there is a natural number \(N\) such that \(\det T^{(r)}_{m,n} > 0\) for every \(n \ge N\).

public theorem_wanted generalizedHyperFibonacci_toeplitz_det_eventually_pos_of_le (p a b : ℤ)
    (hp : 1 ≤ p) (ha : 0 ≤ a) (hb : 0 < b) (r m : ℕ) (hr : 1 ≤ r) (hm : 1 ≤ m)
    (hmr : m ≤ r + 1) :
    ∃ N : ℕ, ∀ n ≥ N, 0 < (Matrix.of fun i j : Fin m =>
      generalizedHyperFibonacci p a b r (n + i - j)).det

Sources

• Abdelhak Taane, Ihab-Eddine Djellas and Mohammed Mekkaoui, Total Positivity of Toeplitz Matrices Involving Generalized Hyper-Fibonacci Numbers, Journal of Integer Sequences 29 (2026), Article 26.2.5, Corollary 9 (jis-2026-26.2.5-cor-9).

Uses 1

Statement usesGeneralized hyper-Fibonacci numbers

TheoremEventual log-concavity of generalized hyper-Fibonacci numbers#●ready to prove

Let \(p, a, b\) be integers with \(p \ge 1\), \(a \ge 0\) and \(b > 0\), let \(W^{(r)}_{p,n}\) denote the generalized hyper-Fibonacci numbers, and let \(r \ge 1\) be a natural number. Then there is a natural number \(N\) such that for every \(n \ge N\) we have \(W^{(r)}_{p,n} > 0\) and \(W^{(r)}_{p,n} W^{(r)}_{p,n+2} \le (W^{(r)}_{p,n+1})^2\).

public theorem_wanted generalizedHyperFibonacci_eventually_logConcave (p a b : ℤ)
    (hp : 1 ≤ p) (ha : 0 ≤ a) (hb : 0 < b) (r : ℕ) (hr : 1 ≤ r) :
    ∃ N : ℕ, ∀ n ≥ N, 0 < generalizedHyperFibonacci p a b r n ∧
      generalizedHyperFibonacci p a b r n * generalizedHyperFibonacci p a b r (n + 2) ≤
        generalizedHyperFibonacci p a b r (n + 1) ^ 2

Sources

• Abdelhak Taane, Ihab-Eddine Djellas and Mohammed Mekkaoui, Total Positivity of Toeplitz Matrices Involving Generalized Hyper-Fibonacci Numbers, Journal of Integer Sequences 29 (2026), Article 26.2.5, Corollary 10 (jis-2026-26.2.5-cor-10).

Uses 1

Statement usesGeneralized hyper-Fibonacci numbers

TheoremEventual log-concavity of hyper \(p\)-Fibonacci and hyper \(p\)-Lucas numbers#●ready to prove

Let \(p \ge 1\) be an integer and \(r \ge 1\) a natural number, and let \(F_{p,n}^{(r)}\) and \(L_{p,n}^{(r)}\) be the generalized hyper-Fibonacci numbers \(W_{p,n}^{(r)}\) with initial values \((a, b) = (0, 1)\) and \((a, b) = (2, p)\) respectively (the hyper \(p\)-Fibonacci and hyper \(p\)-Lucas numbers). Both sequences are eventually log-concave: there is \(N\) such that for all \(n \ge N\), \(F_{p,n}^{(r)} > 0\) and \(F_{p,n}^{(r)} F_{p,n+2}^{(r)} \le (F_{p,n+1}^{(r)})^2\), and there is \(N'\) such that for all \(n \ge N'\), \(L_{p,n}^{(r)} > 0\) and \(L_{p,n}^{(r)} L_{p,n+2}^{(r)} \le (L_{p,n+1}^{(r)})^2\).

public theorem_wanted hyperPFibonacci_hyperPLucas_eventually_logConcave (p : ℤ) (hp : 1 ≤ p)
    (r : ℕ) (hr : 1 ≤ r) :
    (∃ N : ℕ, ∀ n ≥ N, 0 < generalizedHyperFibonacci p 0 1 r n ∧
      generalizedHyperFibonacci p 0 1 r n * generalizedHyperFibonacci p 0 1 r (n + 2) ≤
        generalizedHyperFibonacci p 0 1 r (n + 1) ^ 2) ∧
    (∃ N : ℕ, ∀ n ≥ N, 0 < generalizedHyperFibonacci p 2 p r n ∧
      generalizedHyperFibonacci p 2 p r n * generalizedHyperFibonacci p 2 p r (n + 2) ≤
        generalizedHyperFibonacci p 2 p r (n + 1) ^ 2)

Sources

• Abdelhak Taane, Ihab-Eddine Djellas and Mohammed Mekkaoui, Total Positivity of Toeplitz Matrices Involving Generalized Hyper-Fibonacci Numbers, Journal of Integer Sequences 29 (2026), Article 26.2.5, Corollary 11 (jis-2026-26.2.5-cor-11).

Uses 1

Statement usesGeneralized hyper-Fibonacci numbers

TheoremEventual total positivity of hyper-Fibonacci Toeplitz matrices#●ready to prove

Let \(p, a, b\) be integers with \(p \ge 1\), \(a \ge 0\) and \(b > 0\), put \(\Delta = a^2 + pab - b^2\), let \(W^{(r)}_{p,n}\) denote the generalized hyper-Fibonacci numbers, and let \(r \ge 1\) be a natural number. For a natural number \(s\) write \(T_s\) for the real \((r+2) \times (r+2)\) Toeplitz matrix with entry \(W^{(r)}_{p,s+i-j}\) in row \(i\) and column \(j\) (\(0 \le i, j \le r+1\)). If \(\Delta > 0\), there is a natural number \(N\) such that \(T_{2n}\) has positive minors (is totally positive) for every \(n \ge N\); if \(\Delta < 0\), there is a natural number \(N\) such that \(T_{2n+1}\) has positive minors for every \(n \ge N\).

public theorem_wanted generalizedHyperFibonacci_toeplitz_hasPositiveMinors (p a b : ℤ)
    (hp : 1 ≤ p) (ha : 0 ≤ a) (hb : 0 < b) (r : ℕ) (hr : 1 ≤ r) :
    (0 < a ^ 2 + p * a * b - b ^ 2 → ∃ N : ℕ, ∀ n ≥ N,
      Matrix.HasPositiveMinors (Matrix.of fun i j : Fin (r + 2) =>
        (generalizedHyperFibonacci p a b r (2 * n + i - j) : ℝ))) ∧
    (a ^ 2 + p * a * b - b ^ 2 < 0 → ∃ N : ℕ, ∀ n ≥ N,
      Matrix.HasPositiveMinors (Matrix.of fun i j : Fin (r + 2) =>
        (generalizedHyperFibonacci p a b r (2 * n + 1 + i - j) : ℝ)))

Sources

• Abdelhak Taane, Ihab-Eddine Djellas and Mohammed Mekkaoui, Total Positivity of Toeplitz Matrices Involving Generalized Hyper-Fibonacci Numbers, Journal of Integer Sequences 29 (2026), Article 26.2.5, Theorem 12 (jis-2026-26.2.5-thm-12).

Uses 2

Statement usesMatrices with positive minors · Generalized hyper-Fibonacci numbers

TheoremEventually positive bounded-order minors of hyper-Fibonacci Toeplitz matrices#●ready to prove

Let \(p, a, b\) be integers with \(p \ge 1\), \(a \ge 0\) and \(b > 0\), let \(W^{(r)}_{p,n}\) denote the generalized hyper-Fibonacci numbers, and let \(r \ge 1\) be a natural number. For a natural number \(n\) let \(T_n\) be the real \((r+2) \times (r+2)\) Toeplitz matrix with entry \(W^{(r)}_{p,n+i-j}\) in row \(i\) and column \(j\) (\(0 \le i, j \le r+1\)). Then there is a natural number \(N\) such that for every \(n \ge N\), every natural number \(k \le r + 1\) and all strictly increasing maps \(\rho, \gamma : \{0, \dots, k-1\} \to \{0, \dots, r+1\}\), the \(k \times k\) submatrix \(((T_n)_{\rho(u), \gamma(v)})_{0 \le u, v < k}\) has positive determinant; that is, every minor of \(T_n\) of order at most \(r + 1\) is positive.

public theorem_wanted generalizedHyperFibonacci_toeplitz_positiveMinors_le (p a b : ℤ)
    (hp : 1 ≤ p) (ha : 0 ≤ a) (hb : 0 < b) (r : ℕ) (hr : 1 ≤ r) :
    ∃ N : ℕ, ∀ n ≥ N, ∀ k ≤ r + 1, ∀ (rows cols : Fin k → Fin (r + 2)),
      StrictMono rows → StrictMono cols →
        0 < ((Matrix.of fun i j : Fin (r + 2) =>
          (generalizedHyperFibonacci p a b r (n + i - j) : ℝ)).submatrix rows cols).det

Sources

• Abdelhak Taane, Ihab-Eddine Djellas and Mohammed Mekkaoui, Total Positivity of Toeplitz Matrices Involving Generalized Hyper-Fibonacci Numbers, Journal of Integer Sequences 29 (2026), Article 26.2.5, Corollary 13 (jis-2026-26.2.5-cor-13).

Uses 1

Statement usesGeneralized hyper-Fibonacci numbers

TheoremTotal positivity of hyper \(p\)-Fibonacci Toeplitz matrices#●ready to prove

Let \(p \ge 1\) be an integer and \(r \ge 1\) a natural number, and let \(F_{p,n}^{(r)}\) be the hyper \(p\)-Fibonacci numbers, the generalized hyper-Fibonacci numbers \(W_{p,n}^{(r)}\) with initial values \(a = 0\) and \(b = 1\). Consider real \((r + 2) \times (r + 2)\) matrices with rows and columns indexed by \(0, \dots, r + 1\). For all sufficiently large \(n\), the Toeplitz matrix with entry \(F_{p,2n+1+i-j}^{(r)}\) in row \(i\) and column \(j\) is totally positive. Moreover, for all sufficiently large \(n\), the Toeplitz matrix with entry \(F_{p,n+i-j}^{(r)}\) in row \(i\) and column \(j\) is \(\mathrm{TP}_{r+1}\): each of its minors of order at most \(r + 1\) is positive.

public theorem_wanted hyperPFibonacci_toeplitz_hasPositiveMinors (p : ℤ) (hp : 1 ≤ p) (r : ℕ)
    (hr : 1 ≤ r) :
    (∃ N : ℕ, ∀ n ≥ N, Matrix.HasPositiveMinors (Matrix.of fun i j : Fin (r + 2) =>
      (generalizedHyperFibonacci p 0 1 r (2 * n + 1 + i - j) : ℝ))) ∧
    (∃ N : ℕ, ∀ n ≥ N, ∀ k ≤ r + 1, ∀ (rows cols : Fin k → Fin (r + 2)),
      StrictMono rows → StrictMono cols →
        0 < ((Matrix.of fun i j : Fin (r + 2) =>
          (generalizedHyperFibonacci p 0 1 r (n + i - j) : ℝ)).submatrix rows cols).det)

Sources

• Abdelhak Taane, Ihab-Eddine Djellas and Mohammed Mekkaoui, Total Positivity of Toeplitz Matrices Involving Generalized Hyper-Fibonacci Numbers, Journal of Integer Sequences 29 (2026), Article 26.2.5, Corollary 14 (jis-2026-26.2.5-cor-14).

Uses 2

Statement usesMatrices with positive minors · Generalized hyper-Fibonacci numbers

TheoremTotal positivity of hyper \(p\)-Lucas Toeplitz matrices#●ready to prove

Let \(p \ge 1\) be an integer and \(r \ge 1\) a natural number, and let \(L_{p,n}^{(r)}\) be the hyper \(p\)-Lucas numbers, the generalized hyper-Fibonacci numbers \(W_{p,n}^{(r)}\) with initial values \(a = 2\) and \(b = p\). Consider real \((r + 2) \times (r + 2)\) matrices with rows and columns indexed by \(0, \dots, r + 1\). For all sufficiently large \(n\), the Toeplitz matrix with entry \(L_{p,2n+i-j}^{(r)}\) in row \(i\) and column \(j\) is totally positive. Moreover, for all sufficiently large \(n\), the Toeplitz matrix with entry \(L_{p,n+i-j}^{(r)}\) in row \(i\) and column \(j\) is \(\mathrm{TP}_{r+1}\): each of its minors of order at most \(r + 1\) is positive.

public theorem_wanted hyperPLucas_toeplitz_hasPositiveMinors (p : ℤ) (hp : 1 ≤ p) (r : ℕ)
    (hr : 1 ≤ r) :
    (∃ N : ℕ, ∀ n ≥ N, Matrix.HasPositiveMinors (Matrix.of fun i j : Fin (r + 2) =>
      (generalizedHyperFibonacci p 2 p r (2 * n + i - j) : ℝ))) ∧
    (∃ N : ℕ, ∀ n ≥ N, ∀ k ≤ r + 1, ∀ (rows cols : Fin k → Fin (r + 2)),
      StrictMono rows → StrictMono cols →
        0 < ((Matrix.of fun i j : Fin (r + 2) =>
          (generalizedHyperFibonacci p 2 p r (n + i - j) : ℝ)).submatrix rows cols).det)

Sources

• Abdelhak Taane, Ihab-Eddine Djellas and Mohammed Mekkaoui, Total Positivity of Toeplitz Matrices Involving Generalized Hyper-Fibonacci Numbers, Journal of Integer Sequences 29 (2026), Article 26.2.5, Corollary 15 (jis-2026-26.2.5-cor-15).

Uses 2

Statement usesMatrices with positive minors · Generalized hyper-Fibonacci numbers

8 fully proved12 ready to prove

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