Catalan-type polynomials

DefinitionNegative real power of a trinomial as a power series#✓fully proved

For real numbers \(\lambda\), \(c\), \(\alpha\), \(\beta\) and a natural number \(m\), this is the formal power series in \(t\) with real coefficients

\[c^{-\lambda} \sum_{n \ge 0} \binom{-\lambda}{n} u^n, \qquad u = \frac{-\alpha t + \beta t^m}{c},\]

that is, the real power \(c^{-\lambda}\) times the binomial series \((1 + t)^{-\lambda} = \sum_{n \ge 0} \binom{-\lambda}{n} t^n\) with \(u\) substituted for \(t\) (for \(c = 0\) the division is read as \(0\), so \(u = 0\)). The substitution is meaningful when \(u\) has zero constant term, which holds for \(m \ge 1\); for \(c > 0\) and \(m \ge 1\) the series is the expansion of \((c - \alpha t + \beta t^m)^{-\lambda}\) in powers of \(t\).

For \(c > 0\) and \(m \ge 1\) this is the denominator factor of every generating function in Pathan and Usman's paper; Section 2 takes \(c = 1\), \(\alpha = m\) and \(\beta = x\).

noncomputable def trinomialNegPowSeries (lam c α β : ℝ) (m : ℕ) : PowerSeries ℝ :=
  C (c ^ (-lam)) *
    (binomialSeries ℝ (-lam)).subst (C (-α / c) * X + C (β / c) * X ^ m : PowerSeries ℝ)

Sources

• M. A. Pathan and T. Usman, Unified Catalan-Type Polynomials with Matrix Formulations and Applications, INTEGERS 26 (2026), Article A105, Equations (6), (12) and (15).

Used by 7

Used bySplitting the numerator of a unified Catalan-type generating function · Hermite-shifted Catalan-type polynomials as convolutions · Nieto–Hermite Catalan-type polynomials as convolutions · Constant term of the trinomial series · Hankel determinants of recurrent Catalan-type sequences vanish · Determinant of the lower-triangular Catalan-type matrix · The trinomial series at exponent one

TheoremConstant term of the trinomial series#✓fully proved

For all real numbers \(\lambda, c, \alpha, \beta\) and every natural number \(m \ne 0\), the negative real power of the trinomial \(c - \alpha t + \beta t^m\) with exponent \(\lambda\) as a power series in \(t\), that is, the real power \(c^{-\lambda}\) times the binomial series \((1 + u)^{-\lambda}\) with \(u = (-\alpha t + \beta t^m)/c\) substituted (where \(u = 0\) when \(c = 0\)), has constant term \(c^{-\lambda}\).

theorem constantCoeff_trinomialNegPowSeries (lam c α β : ℝ) {m : ℕ} (hm : m ≠ 0) :
    constantCoeff (trinomialNegPowSeries lam c α β m) = c ^ (-lam)

Sources

• M. A. Pathan and T. Usman, Unified Catalan-Type Polynomials with Matrix Formulations and Applications, INTEGERS 26 (2026), Article A105, Equation (6).

Uses 1

Statement usesNegative real power of a trinomial as a power series

TheoremThe trinomial series at exponent one#✓fully proved

For all real numbers \(\alpha, \beta\) and every natural number \(m \ne 0\), the formal expansion of \((1 - \alpha t + \beta t^m)^{-\lambda}\) in powers of \(t\) at \(\lambda = -1\) is the polynomial \(1 - \alpha t + \beta t^m\) itself.

theorem trinomialNegPowSeries_neg_one (α β : ℝ) {m : ℕ} (hm : m ≠ 0) :
    trinomialNegPowSeries (-1) 1 α β m = 1 - C α * X + C β * X ^ m

Sources

• M. A. Pathan and T. Usman, Unified Catalan-Type Polynomials with Matrix Formulations and Applications, INTEGERS 26 (2026), Article A105, Equation (6).

Uses 1

Statement usesNegative real power of a trinomial as a power series

DefinitionKampé de Fériet polynomials#✓fully proved

For natural numbers \(m, n\) and elements \(x, y\) of a commutative \(\mathbb{R}\)-algebra, the Kampé de Fériet (two-variable Hermite) polynomial is

\[H^m_n(x, y) = \sum_{r=0}^{q} \frac{n!}{r!\,(n - m r)!}\, y^r x^{n - m r},\]

where \(q = \lfloor n / m \rfloor\), read as \(q = 0\) when \(m = 0\). The paper writes \(H_n(x, y)\) for \(H^2_n(x, y)\); taking values in an algebra lets \(H_k(x + 2yt, y)\) be formed as a formal power series in \(t\).

noncomputable def kampeDeFerietHermite {A : Type*} [CommRing A] [Algebra ℝ A] (m n : ℕ)
    (x y : A) : A :=
  ∑ r ∈ Finset.range (n / m + 1),
    ((n.factorial : ℝ) / ((r.factorial : ℝ) * ((n - m * r).factorial : ℝ))) •
      (y ^ r * x ^ (n - m * r))

Sources

• M. A. Pathan and T. Usman, Unified Catalan-Type Polynomials with Matrix Formulations and Applications, INTEGERS 26 (2026), Article A105, Equation (2).

Used by 3

Used byHermite-shifted Catalan-type polynomials as convolutions · The second Kampé de Fériet polynomial · The zeroth Kampé de Fériet polynomial

TheoremThe zeroth Kampé de Fériet polynomial#✓fully proved

For every natural number \(m\) and all elements \(x, y\) of a commutative \(\mathbb{R}\)-algebra, \(H^m_0(x, y) = 1\).

theorem kampeDeFerietHermite_zero {A : Type*} [CommRing A] [Algebra ℝ A] (m : ℕ) (x y : A) :
    kampeDeFerietHermite m 0 x y = 1

Sources

• M. A. Pathan and T. Usman, Unified Catalan-Type Polynomials with Matrix Formulations and Applications, INTEGERS 26 (2026), Article A105, Equation (2).

Uses 1

Statement usesKampé de Fériet polynomials

TheoremThe second Kampé de Fériet polynomial#✓fully proved

For all real numbers \(x\) and \(y\), \(H^2_2(x, y) = x^2 + 2y\).

theorem kampeDeFerietHermite_two_two (x y : ℝ) :
    kampeDeFerietHermite 2 2 x y = x ^ 2 + 2 * y

Sources

• M. A. Pathan and T. Usman, Unified Catalan-Type Polynomials with Matrix Formulations and Applications, INTEGERS 26 (2026), Article A105, Equation (2).

Uses 1

Statement usesKampé de Fériet polynomials

TheoremHermite-shifted Catalan-type polynomials as convolutions#●ready to prove

Write \(H_n(x, y) = H^2_n(x, y)\) for the Kampé de Fériet polynomials. For a natural number \(m\) and real numbers \(\lambda, z\), write \((1 - m t + z t^m)^{-\lambda}\) and \((1 - m z t + t^m)^{-\lambda}\) for the negative real powers of the trinomials \(1 - m t + z t^m\) and \(1 - m z t + t^m\) with exponent \(\lambda\) as power series in \(t\) (for \(m \ge 1\) these are their expansions in powers of \(t\)), and let \(P^\lambda_{r,m}(z)\) and \(h^\lambda_{r,m}(z)\) be their coefficients of \(t^r\). For all natural numbers \(k, m, n\) and real numbers \(\lambda, x, y, z\), as formal power series in \(t\) (with \(e^{xt + yt^2}\) the exponential series and \(H_k(x + 2yt, y)\) evaluated in power series), the coefficient of \(t^n\) in \(e^{xt + yt^2} H_k(x + 2yt, y) (1 - m t + z t^m)^{-\lambda}\) is \(\sum_{r=0}^{n} H_{n-r+k}(x, y) P^\lambda_{r,m}(z) / (n - r)!\), and the coefficient of \(t^n\) in \(e^{xt + yt^2} H_k(x + 2yt, y) (1 - m z t + t^m)^{-\lambda}\) is \(\sum_{r=0}^{n} H_{n-r+k}(x, y) h^\lambda_{r,m}(z) / (n - r)!\).

These coefficients are the paper's \(H^{k;\lambda}_{n,m}(x, y, z)\) and \(h^{k;\lambda}_{n,m}(x, y, z)\) (Definition 3, Equations (8) and (9)).

public theorem_wanted coeff_hermiteShift_mul_trinomialNegPowSeries (k m n : ℕ)
    (lam x y z : ℝ) :
    let G : PowerSeries ℝ := ((exp ℝ).subst (C x * X + C y * X ^ 2 : PowerSeries ℝ) :
        PowerSeries ℝ) * kampeDeFerietHermite 2 k (C x + C (2 * y) * X) (C y)
    coeff n (G * trinomialNegPowSeries lam 1 m z m) =
        ∑ r ∈ Finset.range (n + 1), kampeDeFerietHermite 2 (n - r + k) x y /
          ((n - r).factorial : ℝ) * coeff r (trinomialNegPowSeries lam 1 m z m) ∧
      coeff n (G * trinomialNegPowSeries lam 1 (m * z) 1 m) =
        ∑ r ∈ Finset.range (n + 1), kampeDeFerietHermite 2 (n - r + k) x y /
          ((n - r).factorial : ℝ) * coeff r (trinomialNegPowSeries lam 1 (m * z) 1 m)

Sources

• M. A. Pathan and T. Usman, Unified Catalan-Type Polynomials with Matrix Formulations and Applications, INTEGERS 26 (2026), Article A105, Theorem 2.

Uses 2

Statement usesKampé de Fériet polynomials · Negative real power of a trinomial as a power series

TheoremNieto–Hermite Catalan-type polynomials as convolutions#●ready to prove

Let \(j, k, m, r\) be natural numbers with \(k \le r\) and let \(\lambda, x, y\) be real numbers. Let \(E(t) = e^{2yt - t^2}\) be the exponential series, so that its coefficient of \(t^n\) is \(H_n(y) / n!\) with \(H_n\) the Hermite polynomials, and let \(F(t)\) be the power series whose coefficient of \(t^n\) is \(H_n(y) / n!\) when \(n \ge k\) and \(j \mid n - k\), and \(0\) otherwise (for \(j \ge 1\) this is the series \(F(j, k, y, t) = \sum_n H_{jn+k}(y) t^{jn+k} / (jn + k)!\) of Equation (1)). Write \((1 - m t + x t^m)^{-\lambda}\) and \((1 - m x t + t^m)^{-\lambda}\) for the negative real powers of the trinomials \(1 - m t + x t^m\) and \(1 - m x t + t^m\) with exponent \(\lambda\) as power series in \(t\) (for \(m \ge 1\) these are their expansions in powers of \(t\)), let \(P^\lambda_{s,m}(x)\) and \(h^\lambda_{s,m}(x)\) be their coefficients of \(t^s\), and let \(q = \lfloor (r - k) / j \rfloor\), read as \(q = 0\) when \(j = 0\). Then the coefficient of \(t^r\) in \(F(t) (1 - m t + x t^m)^{-\lambda}\) is \(\sum_{n=0}^{q} H_{jn+k}(y) P^\lambda_{r-jn-k,m}(x) / (jn + k)!\), and the coefficient of \(t^r\) in \(F(t) (1 - m x t + t^m)^{-\lambda}\) is the same sum with \(h^\lambda_{r-jn-k,m}(x)\) in place of \(P^\lambda_{r-jn-k,m}(x)\).

These are the paper's \(F^{j.k;\lambda}_{r,m}(x, y)\) and \(f^{j.k;\lambda}_{r,m}(x, y)\) (Equations (10) and (11)). The paper takes \(j, m \ge 1\); for \(r < k\) its sum is empty.

public theorem_wanted coeff_nietoHermiteSeries_mul_trinomialNegPowSeries (j k m r : ℕ)
    (lam x y : ℝ) (hkr : k ≤ r) :
    let E : PowerSeries ℝ :=
      ((exp ℝ).subst (C (2 * y) * X - X ^ 2 : PowerSeries ℝ) : PowerSeries ℝ)
    let F : PowerSeries ℝ := PowerSeries.mk fun n ↦ if k ≤ n ∧ j ∣ n - k then coeff n E else 0
    coeff r (F * trinomialNegPowSeries lam 1 m x m) =
        ∑ n ∈ Finset.range ((r - k) / j + 1),
          coeff (j * n + k) E * coeff (r - j * n - k) (trinomialNegPowSeries lam 1 m x m) ∧
      coeff r (F * trinomialNegPowSeries lam 1 (m * x) 1 m) =
        ∑ n ∈ Finset.range ((r - k) / j + 1),
          coeff (j * n + k) E * coeff (r - j * n - k) (trinomialNegPowSeries lam 1 (m * x) 1 m)

Sources

• M. A. Pathan and T. Usman, Unified Catalan-Type Polynomials with Matrix Formulations and Applications, INTEGERS 26 (2026), Article A105, Theorem 3.

Uses 1

Statement usesNegative real power of a trinomial as a power series

TheoremSplitting the numerator of a unified Catalan-type generating function#●ready to prove

Let \(j, k, m, d, n\) be natural numbers, let \(a, b, c, \lambda, x, y\) be real numbers and let \(A\) be a polynomial with integer coefficients. Let \(K(t)\) be the negative real power of the trinomial \(c - a x t + b (2x - 1)^d t^m\) with exponent \(\lambda\) as a power series in \(t\) (for \(c > 0\) and \(m \ge 1\) this is the expansion of \((c - a x t + b (2x - 1)^d t^m)^{-\lambda}\) in powers of \(t\)), and let \(S(t) = \sum_i y^i t^i / i!\), the sum running over the natural numbers \(i \ge k\) with \(j \mid i - k\) (for \(j \ge 1\) this is the pseudo-Bessel function \(S(j, k, yt)\) of Equation (5)). Then the coefficient of \(t^n\) in \((S(t) + A(x)\, t)\, K(t)\) equals the coefficient of \(t^n\) in \(S(t) K(t)\), plus the coefficient of \(t^n\) in \((1 + A(x)\, t)\, K(t)\), minus the coefficient of \(t^n\) in \(K(t)\).

In the paper's notation this is \(P^{j.k;\lambda,A}_{n,m}(x, y) = P^{j.k;\lambda,0}_{n,m}(x, y) + P^{1.0;\lambda,A}_{n,m}(x) - P^{1.0;\lambda,0}_{n,m}(x)\); the kernel is the paper's for \(c > 0\).

public theorem_wanted coeff_catalanTypeKernel_decomposition (j k m d n : ℕ)
    (a b c lam x y : ℝ) (A : Polynomial ℤ) :
    let K : PowerSeries ℝ := trinomialNegPowSeries lam c (a * x) (b * (2 * x - 1) ^ d) m
    let S : PowerSeries ℝ :=
      PowerSeries.mk fun i ↦ if k ≤ i ∧ j ∣ i - k then y ^ i / (i.factorial : ℝ) else 0
    let At : PowerSeries ℝ := C (Polynomial.aeval x A) * X
    coeff n ((S + At) * K) = coeff n (S * K) + coeff n ((1 + At) * K) - coeff n K

Sources

• M. A. Pathan and T. Usman, Unified Catalan-Type Polynomials with Matrix Formulations and Applications, INTEGERS 26 (2026), Article A105, Theorem 6.

Uses 1

Statement usesNegative real power of a trinomial as a power series

TheoremDeterminant of the lower-triangular Catalan-type matrix#●ready to prove

Let \(m, N\) be natural numbers, let \(\lambda, x\) be real numbers, let \(A\) be a polynomial with integer coefficients, and let \(P_n\) be the coefficient of \(t^n\) in \((1 + A(x)\, t)(1 - m t + x t^m)^{-\lambda}\), where \((1 - m t + x t^m)^{-\lambda}\) denotes the negative real power of the trinomial \(1 - m t + x t^m\) with exponent \(\lambda\) as a power series in \(t\) (for \(m \ge 1\), its expansion in powers of \(t\)). Then the \((N + 1) \times (N + 1)\) matrix whose \((i, j)\) entry, for \(0 \le i, j \le N\), is \(P_{i-j}\) when \(j \le i\) and \(0\) otherwise has determinant \(P_0^{N+1}\).

In the paper's notation \(P_n = P^{\lambda,A}_{n,m}(x)\) (Definition 6 at \(y = 0\)) and the matrix is \(P_N^{(m,\lambda,A)}(x)\) of Equation (18). The paper's proviso \(P_0 \ne 0\) is dropped.

public theorem_wanted det_catalanTypeLowerTriangular (m N : ℕ) (lam x : ℝ)
    (A : Polynomial ℤ) :
    let P : ℕ → ℝ := fun n ↦
      coeff n ((1 + C (Polynomial.aeval x A) * X) * trinomialNegPowSeries lam 1 m x m)
    (Matrix.of fun i j : Fin (N + 1) ↦ if j ≤ i then P (i - j) else 0).det = P 0 ^ (N + 1)

Sources

• M. A. Pathan and T. Usman, Unified Catalan-Type Polynomials with Matrix Formulations and Applications, INTEGERS 26 (2026), Article A105, Theorem 7.

Uses 1

Statement usesNegative real power of a trinomial as a power series

TheoremHankel determinants of recurrent Catalan-type sequences vanish#●ready to prove

Let \(m, N, r\) be natural numbers with \(r \le N\), let \(\lambda, x\) be real numbers, let \(A\) be a polynomial with integer coefficients, and let \(P_n\) be the coefficient of \(t^n\) in \((1 + A(x)\, t)(1 - m t + x t^m)^{-\lambda}\), where \((1 - m t + x t^m)^{-\lambda}\) denotes the negative real power of the trinomial \(1 - m t + x t^m\) with exponent \(\lambda\) as a power series in \(t\) (for \(m \ge 1\), its expansion in powers of \(t\)). Suppose real numbers \(c_0, \dots, c_r\), not all zero, satisfy \(\sum_{k=0}^{r} c_k P_{n-k} = 0\) for every \(n \ge r\). Then the \((N + 1) \times (N + 1)\) Hankel matrix with \((i, j)\) entry \(P_{i+j}\) for \(0 \le i, j \le N\) has determinant \(0\).

In the paper's notation \(P_n = P^{\lambda,A}_{n,m}(x)\), the recurrence is Equation (22) and the Hankel matrix is \(H_N^{(m,\lambda,A)}(x)\) of Equation (20). The paper's assumption \(1 \le r\) is dropped.

public theorem_wanted det_catalanTypeHankel_eq_zero (m N r : ℕ) (lam x : ℝ)
    (A : Polynomial ℤ) (cs : ℕ → ℝ) :
    let P : ℕ → ℝ := fun n ↦
      coeff n ((1 + C (Polynomial.aeval x A) * X) * trinomialNegPowSeries lam 1 m x m)
    r ≤ N → (∃ k ≤ r, cs k ≠ 0) →
      (∀ n, r ≤ n → ∑ k ∈ Finset.range (r + 1), cs k * P (n - k) = 0) →
      (Matrix.of fun i j : Fin (N + 1) ↦ P (i + j)).det = 0

Sources

• M. A. Pathan and T. Usman, Unified Catalan-Type Polynomials with Matrix Formulations and Applications, INTEGERS 26 (2026), Article A105, Theorem 8.

Uses 1

Statement usesNegative real power of a trinomial as a power series

6 fully proved5 ready to prove

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