Total positivity of Riordan arrays

TheoremBell-type Riordan arrays with a Pólya frequency A-sequence are totally positive#●ready to prove

Let \(g\) be a formal power series over \(\mathbb{R}\) with \(g(0) \neq 0\), write \([x^n]F\) for the coefficient of \(x^n\) in a power series \(F\), and let \(D = (d_{n,k})_{n,k \ge 0}\) be the Bell-type Riordan array \((g(x), x g(x))\), with entries \(d_{n,k} = [x^n]\, g(x) (x g(x))^k\). Let \((a_j)_{j \ge 0}\) be a real sequence, the \(A\)-sequence of \(D\), such that for all \(n, k \ge 0\)

\[d_{n+1,k+1} = \sum_{j=0}^{n} a_j\, d_{n,k+j}.\]

If \((a_j)\) is a Pólya frequency sequence, that is, every minor of its lower Toeplitz matrix (whose entry in row \(n\) and column \(k\) is \(a_{n-k}\) for \(k \le n\) and \(0\) otherwise) is nonnegative, then \(D\) is totally positive: every minor of \(D\) is nonnegative.

The sum stops at \(j = n\) because \(d_{n,m} = 0\) for \(m > n\).

public theorem_wanted bellRiordan_hasNonnegativeMinors_of_aSequence (g : PowerSeries ℝ)
    (hg : PowerSeries.constantCoeff g ≠ 0) (a : ℕ → ℝ)
    (ha : ∀ n k : ℕ, PowerSeries.coeff (n + 1) (g * (PowerSeries.X * g) ^ (k + 1)) =
      ∑ j ∈ Finset.range (n + 1),
        a j * PowerSeries.coeff n (g * (PowerSeries.X * g) ^ (k + j)))
    (hpf : Matrix.HasNonnegativeMinors (Matrix.lowerToeplitz a)) :
    Matrix.HasNonnegativeMinors
      (Matrix.of fun n k : ℕ => PowerSeries.coeff n (g * (PowerSeries.X * g) ^ k))

Sources

• Lili Mu, Yuanyuan Xing and Sai-Nan Zheng, A New Criterion for the Total Positivity of Riordan Arrays, Journal of Integer Sequences 28 (2025), Article 25.7.5, Proposition 5.

Uses 2

Statement usesLower-triangular Toeplitz matrix · Matrices with nonnegative minors

DefinitionRiordan array with given A- and Z-sequences#✓fully proved

For real sequences \((a_n)_{n \ge 0}\) and \((z_n)_{n \ge 0}\), an infinite real matrix \(R = (r_{n,k})_{n,k \ge 0}\) is the Riordan array with \(A\)-sequence \((a_n)\) and \(Z\)-sequence \((z_n)\) when \(r_{0,0} = 1\), \(r_{0,k} = 0\) for \(k \ge 1\), and for all \(n, k \ge 0\)

\[r_{n+1,0} = \sum_{j=0}^{n} z_j\, r_{n,j}, \qquad r_{n+1,k+1} = \sum_{j=0}^{n} a_j\, r_{n,k+j}.\]

The sums stop at \(j = n\) because the recurrence forces \(r_{n,m} = 0\) for \(m > n\).

def IsRiordanArrayAZ (a z : ℕ → ℝ) (R : ℕ → ℕ → ℝ) : Prop :=
  R 0 0 = 1 ∧ (∀ k, R 0 (k + 1) = 0) ∧
    (∀ n, R (n + 1) 0 = ∑ j ∈ Finset.range (n + 1), z j * R n j) ∧
    ∀ n k, R (n + 1) (k + 1) = ∑ j ∈ Finset.range (n + 1), a j * R n (k + j)

Sources

• Lili Mu, Yuanyuan Xing and Sai-Nan Zheng, A New Criterion for the Total Positivity of Riordan Arrays, Journal of Integer Sequences 28 (2025), Article 25.7.5, Section 1, equation (1).

Used by 4

Used byTotal positivity of Bell-type Riordan arrays from a Pólya frequency series · Entries in row 1 of a Riordan array · Total positivity of a Riordan array from its coefficient matrix · A total positivity criterion for Riordan arrays

TheoremEntries in row 1 of a Riordan array#✓fully proved

If \(R = (r_{n,k})_{n,k \ge 0}\) is the Riordan array with \(A\)-sequence \((a_n)\) and \(Z\)-sequence \((z_n)\), for real sequences \((a_n)\) and \((z_n)\), then \(r_{1,0} = z_0\) and \(r_{1,1} = a_0\).

theorem IsRiordanArrayAZ.one_zero_and_one_one {a z : ℕ → ℝ} {R : ℕ → ℕ → ℝ}
    (h : IsRiordanArrayAZ a z R) : R 1 0 = z 0 ∧ R 1 1 = a 0

Sources

• Lili Mu, Yuanyuan Xing and Sai-Nan Zheng, A New Criterion for the Total Positivity of Riordan Arrays, Journal of Integer Sequences 28 (2025), Article 25.7.5, Section 1, equation (1).

Uses 1

Statement usesRiordan array with given A- and Z-sequences

DefinitionCoefficient matrix of a Riordan array#✓fully proved

For real sequences \((z_n)_{n \ge 0}\) and \((a_n)_{n \ge 0}\) with generating functions \(Z(x)\) and \(A(x)\), the coefficient matrix \(J(Z(x), A(x))\) is the infinite real matrix, with rows and columns indexed by \(n, k \ge 0\), whose entry in row \(n\) and column \(k\) is \(z_n\) if \(k = 0\), is \(a_{n+1-k}\) if \(1 \le k \le n + 1\), and is \(0\) if \(k > n + 1\). Thus row \(n\) reads \(z_n, a_n, a_{n-1}, \ldots, a_0, 0, 0, \ldots\)

def riordanCoeffMatrix (z a : ℕ → ℝ) : Matrix ℕ ℕ ℝ :=
  Matrix.of fun n k => if k = 0 then z n else if k ≤ n + 1 then a (n + 1 - k) else 0

Sources

• Lili Mu, Yuanyuan Xing and Sai-Nan Zheng, A New Criterion for the Total Positivity of Riordan Arrays, Journal of Integer Sequences 28 (2025), Article 25.7.5, Section 1.

Used by 3

Used byTotal positivity of a Riordan array from its coefficient matrix · A total positivity criterion for Riordan arrays · First two columns of the coefficient matrix of a Riordan array

TheoremFirst two columns of the coefficient matrix of a Riordan array#✓fully proved

For real sequences \((z_n)\) and \((a_n)\) with generating functions \(Z(x)\) and \(A(x)\), and for every \(n \ge 0\), the coefficient matrix \(J(Z(x), A(x))\) has entry \(z_n\) in row \(n\), column \(0\), and entry \(a_n\) in row \(n\), column \(1\).

theorem riordanCoeffMatrix_zero_one (z a : ℕ → ℝ) (n : ℕ) :
    riordanCoeffMatrix z a n 0 = z n ∧ riordanCoeffMatrix z a n 1 = a n

Sources

• Lili Mu, Yuanyuan Xing and Sai-Nan Zheng, A New Criterion for the Total Positivity of Riordan Arrays, Journal of Integer Sequences 28 (2025), Article 25.7.5, Section 1.

Uses 1

Statement usesCoefficient matrix of a Riordan array

TheoremTotal positivity of a Riordan array from its coefficient matrix#●ready to prove

Let \((a_n)\) and \((z_n)\) be real sequences with generating functions \(A(x)\) and \(Z(x)\), and let \(R\) be the Riordan array with \(A\)-sequence \((a_n)\) and \(Z\)-sequence \((z_n)\). If the coefficient matrix \(J(Z(x), A(x))\) is totally positive, that is, every minor of it is nonnegative, then every minor of \(R\) is nonnegative.

public theorem_wanted riordanArray_hasNonnegativeMinors_of_coeffMatrix {a z : ℕ → ℝ}
    {R : ℕ → ℕ → ℝ} (hR : IsRiordanArrayAZ a z R)
    (hJ : Matrix.HasNonnegativeMinors (riordanCoeffMatrix z a)) :
    Matrix.HasNonnegativeMinors (Matrix.of R)

Sources

• Lili Mu, Yuanyuan Xing and Sai-Nan Zheng, A New Criterion for the Total Positivity of Riordan Arrays, Journal of Integer Sequences 28 (2025), Article 25.7.5, Proposition 1 (jis-2025-25.7.5-prop-1).

Uses 3

Statement usesRiordan array with given A- and Z-sequences · Matrices with nonnegative minors · Coefficient matrix of a Riordan array

TheoremA total positivity criterion for Riordan arrays#●ready to prove

Let \(P, Q, A, Z\) be formal power series over \(\mathbb{R}\) with \(P(0) \ne 0\), write \([x^n]F\) for the coefficient of \(x^n\) in a power series \(F\), and let \(a, b, r, s, t \ge 0\) be real numbers. Suppose that

\[A(x)\,(P(x) - rx) = P(x)^2 + sxP(x) + tx^2\]

and

\[Z(x)\,(P(x)^2 - rxP(x)) = (P(x)^2 + sxP(x) + tx^2)\,Q(x) + (aP(x) + bx)(P(x) - xQ(x)),\]

and let \(R\) be the Riordan array whose \(A\)-sequence is \(([x^n]A)_{n \ge 0}\) and whose \(Z\)-sequence is \(([x^n]Z)_{n \ge 0}\). Let \(J(Q(x), P(x))\) be the coefficient matrix whose row \(n \ge 0\) reads \([x^n]Q, [x^n]P, [x^{n-1}]P, \dots, [x^0]P, 0, 0, \dots\). Call a matrix totally positive when every minor of it is nonnegative. If \(J(Q(x), P(x))\) is totally positive, \(s^2 \ge 4t\) and \(as + a\sqrt{s^2 - 4t} \ge 2b\), then \(R\) is totally positive.

Since \(P(0) \ne 0\), the series \(P(x) - rx\) and \(P(x)^2 - rxP(x)\) are invertible, so the two hypotheses say \(A = (P^2 + sxP + tx^2)/(P - rx)\) and \(Z = ((P^2 + sxP + tx^2)Q + (aP + bx)(P - xQ))/(P^2 - rxP)\).

public theorem_wanted riordanArray_hasNonnegativeMinors_of_params (P Q A Z : PowerSeries ℝ)
    (hP : PowerSeries.constantCoeff P ≠ 0) {a b r s t : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
    (hr : 0 ≤ r) (hs : 0 ≤ s) (ht : 0 ≤ t)
    (hA : A * (P - PowerSeries.C r * PowerSeries.X) =
      P ^ 2 + PowerSeries.C s * PowerSeries.X * P + PowerSeries.C t * PowerSeries.X ^ 2)
    (hZ : Z * (P ^ 2 - PowerSeries.C r * PowerSeries.X * P) =
      (P ^ 2 + PowerSeries.C s * PowerSeries.X * P + PowerSeries.C t * PowerSeries.X ^ 2) * Q +
        (PowerSeries.C a * P + PowerSeries.C b * PowerSeries.X) * (P - PowerSeries.X * Q))
    {R : ℕ → ℕ → ℝ}
    (hR : IsRiordanArrayAZ (fun n => PowerSeries.coeff n A) (fun n => PowerSeries.coeff n Z) R)
    (hJ : Matrix.HasNonnegativeMinors
      (riordanCoeffMatrix (fun n => PowerSeries.coeff n Q) (fun n => PowerSeries.coeff n P)))
    (hst : 4 * t ≤ s ^ 2) (hab : 2 * b ≤ a * s + a * Real.sqrt (s ^ 2 - 4 * t)) :
    Matrix.HasNonnegativeMinors (Matrix.of R)

Sources

• Lili Mu, Yuanyuan Xing and Sai-Nan Zheng, A New Criterion for the Total Positivity of Riordan Arrays, Journal of Integer Sequences 28 (2025), Article 25.7.5, Theorem 2 (jis-2025-25.7.5-thm-2).

Uses 3

Statement usesRiordan array with given A- and Z-sequences · Matrices with nonnegative minors · Coefficient matrix of a Riordan array

TheoremTotal positivity of Bell-type Riordan arrays from a Pólya frequency series#●ready to prove

Let \(P\) and \(A\) be formal power series over \(\mathbb{R}\) with \(P(0) \ne 0\), write \([x^n]F\) for the coefficient of \(x^n\) in a power series \(F\), and let \(r, s, t \ge 0\) be real numbers with \(s^2 \ge 4t\). Suppose that \(P\) is a Pólya frequency series, that is, every minor of the lower Toeplitz matrix of its coefficients (whose entry in row \(n\) and column \(k\) is \([x^{n-k}]P\) for \(k \le n\) and \(0\) otherwise) is nonnegative, and that

\[A(x)\,(P(x) - rx) = P(x)^2 + sxP(x) + tx^2.\]

Let \(R\) be the Riordan array with \(A\)-sequence \((a_n)_{n \ge 0}\) and \(Z\)-sequence \((z_n)_{n \ge 0}\), where \(a_n = [x^n]A\) and \(z_n = a_{n+1}\) (a Bell-type Riordan array). Then \(R\) is totally positive: every minor of \(R\) is nonnegative.

Since \(P(0) \ne 0\), the series \(P(x) - rx\) is invertible, so the hypothesis on \(A\) says \(A = (P^2 + sxP + tx^2)/(P - rx)\).

public theorem_wanted bellRiordanArray_hasNonnegativeMinors_of_pf (P A : PowerSeries ℝ)
    (hP : PowerSeries.constantCoeff P ≠ 0) {r s t : ℝ} (hr : 0 ≤ r) (hs : 0 ≤ s) (ht : 0 ≤ t)
    (hpf : Matrix.HasNonnegativeMinors (Matrix.lowerToeplitz fun n => PowerSeries.coeff n P))
    (hA : A * (P - PowerSeries.C r * PowerSeries.X) =
      P ^ 2 + PowerSeries.C s * PowerSeries.X * P + PowerSeries.C t * PowerSeries.X ^ 2)
    {R : ℕ → ℕ → ℝ}
    (hR : IsRiordanArrayAZ (fun n => PowerSeries.coeff n A)
      (fun n => PowerSeries.coeff (n + 1) A) R)
    (hst : 4 * t ≤ s ^ 2) :
    Matrix.HasNonnegativeMinors (Matrix.of R)

Sources

• Lili Mu, Yuanyuan Xing and Sai-Nan Zheng, A New Criterion for the Total Positivity of Riordan Arrays, Journal of Integer Sequences 28 (2025), Article 25.7.5, Corollary 6 (jis-2025-25.7.5-cor-6).

Uses 3

Statement usesLower-triangular Toeplitz matrix · Riordan array with given A- and Z-sequences · Matrices with nonnegative minors

4 fully proved4 ready to prove

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