Degenerate harmonic numbers and Riordan arrays

DefinitionDegenerate harmonic terms#✓fully proved

For real numbers \(\lambda\) and \(\alpha\) and a natural number \(n\), the degenerate harmonic term is \(0\) when \(n = 0\) and \((-1)^{n-1} \binom{\lambda}{n} / (\lambda \alpha^n)\) when \(n \ge 1\), where \(\binom{\lambda}{n} = \lambda(\lambda - 1)\cdots(\lambda - n + 1)/n!\).

These are the coefficients of \(t^n\) in \(-\log_\lambda(1 - t/\alpha)\), where \(\log_\lambda(1 + t) = \sum_{n \ge 1} \binom{\lambda}{n} t^n / \lambda\); the series has no constant term. For \(n \ge 1\) the term is the paper's \(H^0_{n,\lambda}(\alpha)\); at \(n = 0\) the value \(0\) follows the paper's convention \(H^r_{0,\lambda}(\alpha) = 0\) and the generating function.

noncomputable def degenerateHarmonicTerm (lam α : ℝ) (n : ℕ) : ℝ :=
  if n = 0 then 0 else (-1) ^ (n - 1) * Ring.choose lam n / (lam * α ^ n)

Sources

• Ömer Duran, Neşe Ömür and Sibel Koparal, Generalized Degenerate Harmonic Numbers and Their Applications with Riordan Arrays, INTEGERS 24 (2024), Article A41, Section 1 (H⁰_{n,λ}(α) and Equation (2)).

Used by 7

Used byGeneralized degenerate harmonic numbers of a given rank · Degenerate harmonic numbers of rank vanish at index zero · Degenerate harmonic numbers of rank zero · Degenerate harmonic term of index one · Degenerate harmonic term of index zero · Generalized degenerate hyperharmonic numbers · Generalized degenerate hyperharmonic numbers of order one

TheoremDegenerate harmonic term of index zero#✓fully proved

For all real \(\lambda\) and \(\alpha\), the degenerate harmonic term of index \(0\), \(H^0_{0,\lambda}(\alpha)\), is \(0\).

theorem degenerateHarmonicTerm_zero (lam α : ℝ) : degenerateHarmonicTerm lam α 0 = 0

Sources

• Ömer Duran, Neşe Ömür and Sibel Koparal, Generalized Degenerate Harmonic Numbers and Their Applications with Riordan Arrays, INTEGERS 24 (2024), Article A41, Section 1.

Uses 1 · Used by 1

Statement usesDegenerate harmonic terms

Used byDegenerate harmonic numbers of rank vanish at index zero

TheoremDegenerate harmonic term of index one#✓fully proved

For every real \(\lambda \ne 0\) and every real \(\alpha\), the degenerate harmonic term of index \(1\), \(H^0_{1,\lambda}(\alpha) = \binom{\lambda}{1} / (\lambda \alpha)\), equals \(1/\alpha\).

At \(\alpha = 0\) both sides are \(0\), under the convention that a quotient by \(0\) is \(0\).

theorem degenerateHarmonicTerm_one {lam : ℝ} (hlam : lam ≠ 0) (α : ℝ) :
    degenerateHarmonicTerm lam α 1 = 1 / α

Sources

• Ömer Duran, Neşe Ömür and Sibel Koparal, Generalized Degenerate Harmonic Numbers and Their Applications with Riordan Arrays, INTEGERS 24 (2024), Article A41, Section 1.

Uses 1

Statement usesDegenerate harmonic terms

DefinitionGeneralized degenerate harmonic numbers#✓fully proved

For real numbers \(\lambda\) and \(\alpha\) and a natural number \(n\), Dağlı's generalized degenerate harmonic number is

\[ H_{n,\lambda}(\alpha) = \sum_{k=1}^{n} \frac{(-1)^{k-1} \binom{\lambda}{k}}{\lambda\, \alpha^k}, \]

where \(\binom{\lambda}{k} = \lambda(\lambda - 1)\cdots(\lambda - k + 1)/k!\); in particular \(H_{0,\lambda}(\alpha) = 0\). At \(\alpha = 1\) these are the degenerate harmonic numbers \(H_{n,\lambda}\).

noncomputable def generalizedDegenerateHarmonic (lam α : ℝ) (n : ℕ) : ℝ :=
  ∑ k ∈ Finset.Icc 1 n, (-1) ^ (k - 1) * Ring.choose lam k / (lam * α ^ k)

Sources

• Ömer Duran, Neşe Ömür and Sibel Koparal, Generalized Degenerate Harmonic Numbers and Their Applications with Riordan Arrays, INTEGERS 24 (2024), Article A41, Section 1.

Used by 9

Used byAlternating binomial sum of degenerate harmonic numbers · Degenerate Bernoulli and Euler sums of higher-rank degenerate harmonic numbers · Alternating degenerate exponential sum of degenerate harmonic numbers of rank · Partial sums of degenerate harmonic numbers of rank as a convolution · Degenerate harmonic numbers of rank zero · Alternating and binomial sums of degenerate hyperharmonic numbers · Recurrence for generalized degenerate harmonic numbers · The zeroth generalized degenerate harmonic number · Generalized degenerate hyperharmonic numbers of order one

TheoremThe zeroth generalized degenerate harmonic number#✓fully proved

For all real numbers \(\lambda\) and \(\alpha\), the generalized degenerate harmonic number \(H_{0,\lambda}(\alpha)\) is \(0\).

theorem generalizedDegenerateHarmonic_zero (lam α : ℝ) :
    generalizedDegenerateHarmonic lam α 0 = 0

Sources

• Ömer Duran, Neşe Ömür and Sibel Koparal, Generalized Degenerate Harmonic Numbers and Their Applications with Riordan Arrays, INTEGERS 24 (2024), Article A41, Section 1.

Uses 1 · Used by 1

Statement usesGeneralized degenerate harmonic numbers

Used byDegenerate harmonic numbers of rank zero

TheoremRecurrence for generalized degenerate harmonic numbers#✓fully proved

For all real numbers \(\lambda\) and \(\alpha\) and every natural number \(n\), the generalized degenerate harmonic numbers satisfy

\[ H_{n+1,\lambda}(\alpha) = H_{n,\lambda}(\alpha) + \frac{(-1)^n \binom{\lambda}{n+1}}{\lambda\, \alpha^{n+1}}, \]

where \(\binom{\lambda}{n+1} = \lambda(\lambda - 1)\cdots(\lambda - n)/(n+1)!\).

theorem generalizedDegenerateHarmonic_succ (lam α : ℝ) (n : ℕ) :
    generalizedDegenerateHarmonic lam α (n + 1) =
      generalizedDegenerateHarmonic lam α n +
        (-1) ^ n * Ring.choose lam (n + 1) / (lam * α ^ (n + 1))

Sources

• Ömer Duran, Neşe Ömür and Sibel Koparal, Generalized Degenerate Harmonic Numbers and Their Applications with Riordan Arrays, INTEGERS 24 (2024), Article A41, Section 1.

Uses 1 · Used by 1

Statement usesGeneralized degenerate harmonic numbers

Used byDegenerate harmonic numbers of rank zero

DefinitionGeneralized degenerate hyperharmonic numbers#✓fully proved

For real numbers \(\lambda\) and \(\alpha\), the generalized degenerate hyperharmonic numbers \(H^r_{n,\lambda}(\alpha)\), indexed by natural numbers \(r\) (the order) and \(n\), are defined by recursion on \(r\). At order \(0\) they are the degenerate harmonic terms: \(H^0_{0,\lambda}(\alpha) = 0\) and \(H^0_{n,\lambda}(\alpha) = (-1)^{n-1} \binom{\lambda}{n} / (\lambda \alpha^n)\) for \(n \ge 1\), where \(\binom{\lambda}{n}\) is the generalized binomial coefficient. For every order \(r\), \(H^{r+1}_{n,\lambda}(\alpha) = \sum_{k=1}^{n} H^r_{k,\lambda}(\alpha)\).

In particular \(H^r_{0,\lambda}(\alpha) = 0\) for \(r \ge 1\), and \(H^1_{n,\lambda}(\alpha)\) is the generalized degenerate harmonic number \(H_{n,\lambda}(\alpha) = \sum_{k=1}^{n} (-1)^{k-1} \binom{\lambda}{k} / (\lambda \alpha^k)\).

noncomputable def generalizedDegenerateHyperharmonic (lam α : ℝ) : ℕ → ℕ → ℝ
  | 0, n => degenerateHarmonicTerm lam α n
  | r + 1, n => ∑ k ∈ Finset.Icc 1 n, generalizedDegenerateHyperharmonic lam α r k

Sources

• Ömer Duran, Neşe Ömür and Sibel Koparal, Generalized Degenerate Harmonic Numbers and Their Applications with Riordan Arrays, INTEGERS 24 (2024), Article A41, Section 1.

Uses 1 · Used by 4

Statement usesDegenerate harmonic terms

Used byAlternating and binomial sums of degenerate hyperharmonic numbers · A binomial sum equal to a degenerate hyperharmonic number · Generalized degenerate hyperharmonic numbers of order one · Generalized degenerate hyperharmonic numbers of positive order vanish at zero

TheoremGeneralized degenerate hyperharmonic numbers of positive order vanish at zero#✓fully proved

For all real numbers \(\lambda\) and \(\alpha\) and every natural number \(r\), the generalized degenerate hyperharmonic number of order \(r + 1\) at index \(0\) is zero: \(H^{r+1}_{0,\lambda}(\alpha) = 0\).

theorem generalizedDegenerateHyperharmonic_succ_zero (lam α : ℝ) (r : ℕ) :
    generalizedDegenerateHyperharmonic lam α (r + 1) 0 = 0

Sources

• Ömer Duran, Neşe Ömür and Sibel Koparal, Generalized Degenerate Harmonic Numbers and Their Applications with Riordan Arrays, INTEGERS 24 (2024), Article A41, Section 1.

Uses 1

Statement usesGeneralized degenerate hyperharmonic numbers

TheoremGeneralized degenerate hyperharmonic numbers of order one#✓fully proved

For all real numbers \(\lambda\) and \(\alpha\) and every natural number \(n\), the generalized degenerate hyperharmonic number of order \(1\) equals the generalized degenerate harmonic number: \(H^1_{n,\lambda}(\alpha) = H_{n,\lambda}(\alpha)\).

theorem generalizedDegenerateHyperharmonic_one (lam α : ℝ) (n : ℕ) :
    generalizedDegenerateHyperharmonic lam α 1 n = generalizedDegenerateHarmonic lam α n

Sources

• Ömer Duran, Neşe Ömür and Sibel Koparal, Generalized Degenerate Harmonic Numbers and Their Applications with Riordan Arrays, INTEGERS 24 (2024), Article A41, Section 1.

Uses 3

Statement usesGeneralized degenerate harmonic numbers · Generalized degenerate hyperharmonic numbers

Proof usesDegenerate harmonic terms

DefinitionGeneralized degenerate harmonic numbers of a given rank#✓fully proved

For real \(\lambda\) and \(\alpha\), let \(\ell_0 = 0\) and \(\ell_m = (-1)^{m-1} \binom{\lambda}{m} / (\lambda \alpha^m)\) for \(m \ge 1\), where \(\binom{\lambda}{m}\) is the generalized binomial coefficient and a quotient by \(0\) is read as \(0\), and let \(L(t) = \sum_{m \ge 0} \ell_m t^m\) as a formal power series. For \(\lambda, \alpha > 0\) this is the expansion of \(-\log_\lambda(1 - t/\alpha)\), where \(\log_\lambda(1 + t) = ((1 + t)^\lambda - 1)/\lambda\) is the degenerate logarithm. For natural numbers \(n\) and \(r\), the generalized degenerate harmonic number of rank \(r\), written \(H_\lambda(n, r, \alpha)\), is the sum of the coefficients of \(t^0, t^1, \dots, t^n\) in \(L(t)^{r+1}\). Equivalently, \(\sum_{n \ge 0} H_\lambda(n, r, \alpha) t^n = L(t)^{r+1} / (1 - t)\).

Expanding the power, \(H_\lambda(n, r, \alpha)\) is the sum of \(\ell_{k_0} \cdots \ell_{k_r}\) over all \(k_0, \dots, k_r \ge 1\) with \(k_0 + \cdots + k_r \le n\), which is the explicit definition in the source.

noncomputable def degenerateHarmonicRank (lam α : ℝ) (n r : ℕ) : ℝ :=
  ∑ m ∈ Finset.range (n + 1),
    PowerSeries.coeff m (PowerSeries.mk (degenerateHarmonicTerm lam α) ^ (r + 1))

Sources

• Ömer Duran, Neşe Ömür and Sibel Koparal, Generalized Degenerate Harmonic Numbers and Their Applications with Riordan Arrays, INTEGERS 24 (2024), Article A41, Section 1, Equation (5).

Uses 1 · Used by 8

Statement usesDegenerate harmonic terms

Used byDegenerate Bernoulli and Euler sums of higher-rank degenerate harmonic numbers · Degenerate harmonic numbers of rank and degenerate Stirling numbers of the first kind · Alternating degenerate exponential sum of degenerate harmonic numbers of rank · Shifted alternating degenerate exponential sum of degenerate harmonic numbers of rank · Degenerate harmonic numbers of rank and degenerate Stirling numbers of the second kind · Partial sums of degenerate harmonic numbers of rank as a convolution · Degenerate harmonic numbers of rank vanish at index zero · Degenerate harmonic numbers of rank zero

TheoremDegenerate harmonic numbers of rank vanish at index zero#✓fully proved

For all real \(\lambda\) and \(\alpha\) and every natural number \(r\), the generalized degenerate harmonic number of rank \(r\) satisfies \(H_\lambda(0, r, \alpha) = 0\).

theorem degenerateHarmonicRank_zero_left (lam α : ℝ) (r : ℕ) :
    degenerateHarmonicRank lam α 0 r = 0

Sources

• Ömer Duran, Neşe Ömür and Sibel Koparal, Generalized Degenerate Harmonic Numbers and Their Applications with Riordan Arrays, INTEGERS 24 (2024), Article A41, Section 1.

Uses 3 · Used by 1

Statement usesGeneralized degenerate harmonic numbers of a given rank

Proof usesDegenerate harmonic terms · Degenerate harmonic term of index zero

Used byDegenerate harmonic numbers of rank zero

TheoremDegenerate harmonic numbers of rank zero#✓fully proved

For all real \(\lambda\) and \(\alpha\) and every natural number \(n\), the generalized degenerate harmonic number of rank \(0\) equals the generalized degenerate harmonic number: \(H_\lambda(n, 0, \alpha) = H_{n,\lambda}(\alpha)\).

theorem degenerateHarmonicRank_zero_right (lam α : ℝ) (n : ℕ) :
    degenerateHarmonicRank lam α n 0 = generalizedDegenerateHarmonic lam α n

Sources

• Ömer Duran, Neşe Ömür and Sibel Koparal, Generalized Degenerate Harmonic Numbers and Their Applications with Riordan Arrays, INTEGERS 24 (2024), Article A41, Section 1.

Uses 6

Statement usesGeneralized degenerate harmonic numbers of a given rank · Generalized degenerate harmonic numbers

Proof usesDegenerate harmonic numbers of rank vanish at index zero · Degenerate harmonic terms · Recurrence for generalized degenerate harmonic numbers · The zeroth generalized degenerate harmonic number

DefinitionDegenerate exponential series#✓fully proved

For a real number \(\lambda\), the degenerate exponential series is the formal power series \(e_\lambda(t) = \sum_{n \ge 0} (1)_{n,\lambda} \, t^n / n!\) over \(\mathbb{R}\), where \((1)_{0,\lambda} = 1\) and, for \(n \ge 1\), \((1)_{n,\lambda} = 1 \cdot (1 - \lambda) \cdots (1 - (n - 1)\lambda)\) is the generalized falling factorial of \(1\) with increment \(\lambda\).

noncomputable def degenerateExpSeries (lam : ℝ) : PowerSeries ℝ :=
  PowerSeries.mk fun n => generalizedFallingFactorial 1 lam n / n.factorial

Sources

• Ömer Duran, Neşe Ömür and Sibel Koparal, Generalized Degenerate Harmonic Numbers and Their Applications with Riordan Arrays, INTEGERS 24 (2024), Article A41, Equation (1) at x = 1.

Uses 1 · Used by 5

Statement usesGeneralized falling factorial

Used byConstant term of the degenerate exponential series · Degenerate Euler numbers · The zeroth degenerate Euler number · Degenerate Stirling numbers of the second kind · Degenerate Stirling numbers of the second kind in column zero

TheoremConstant term of the degenerate exponential series#✓fully proved

For every real number \(\lambda\), the constant term of the degenerate exponential series \(e_\lambda(t)\) is \(1\).

theorem constantCoeff_degenerateExpSeries (lam : ℝ) :
    PowerSeries.constantCoeff (degenerateExpSeries lam) = 1

Sources

• Ömer Duran, Neşe Ömür and Sibel Koparal, Generalized Degenerate Harmonic Numbers and Their Applications with Riordan Arrays, INTEGERS 24 (2024), Article A41, Equation (1).

Uses 1 · Used by 1

Statement usesDegenerate exponential series

Used byThe zeroth degenerate Euler number

DefinitionDegenerate Bernoulli numbers#✓fully proved

For a real number \(\lambda\), let \((1)_{0,\lambda} = 1\) and \((1)_{m,\lambda} = 1 \cdot (1 - \lambda) \cdots (1 - (m - 1)\lambda)\) for \(m \ge 1\) (the generalized falling factorial of \(1\) with increment \(\lambda\)). For \(n \in \mathbb{N}\), the degenerate Bernoulli number \(\beta_{n,\lambda}\) is \(n!\) times the coefficient of \(t^n\) in the inverse of the formal power series \(\sum_{m \ge 0} (1)_{m+1,\lambda} \, t^m / (m + 1)!\) over \(\mathbb{R}\), whose constant term is \(1\).

Since this power series is \((e_\lambda(t) - 1) / t\) for the degenerate exponential series \(e_\lambda(t)\), the numbers \(\beta_{n,\lambda}\) satisfy \(t / (e_\lambda(t) - 1) = \sum_{n \ge 0} \beta_{n,\lambda} \, t^n / n!\).

noncomputable def degenerateBernoulli (lam : ℝ) (n : ℕ) : ℝ :=
  n.factorial * PowerSeries.coeff n
    (PowerSeries.mk fun m => generalizedFallingFactorial 1 lam (m + 1) / (m + 1).factorial)⁻¹

Sources

• Ömer Duran, Neşe Ömür and Sibel Koparal, Generalized Degenerate Harmonic Numbers and Their Applications with Riordan Arrays, INTEGERS 24 (2024), Article A41, Equation (6).

Uses 1 · Used by 2

Statement usesGeneralized falling factorial

Used byThe zeroth degenerate Bernoulli number · Degenerate Bernoulli and Euler sums of higher-rank degenerate harmonic numbers

TheoremThe zeroth degenerate Bernoulli number#✓fully proved

For every real number \(\lambda\), the degenerate Bernoulli number \(\beta_{0,\lambda}\) equals \(1\).

theorem degenerateBernoulli_zero (lam : ℝ) : degenerateBernoulli lam 0 = 1

Sources

• Ömer Duran, Neşe Ömür and Sibel Koparal, Generalized Degenerate Harmonic Numbers and Their Applications with Riordan Arrays, INTEGERS 24 (2024), Article A41, Equation (6).

Uses 2

Statement usesDegenerate Bernoulli numbers

Proof usesGeneralized falling factorial

DefinitionDegenerate Euler numbers#✓fully proved

For a real number \(\lambda\) and \(n \in \mathbb{N}\), the degenerate Euler number \(\varepsilon_{n,\lambda}\) is \(n!\) times the coefficient of \(t^n\) in the formal power series \(2 / (e_\lambda(t) + 1)\) over \(\mathbb{R}\), where \(e_\lambda(t)\) is the degenerate exponential series; that is, \(2 / (e_\lambda(t) + 1) = \sum_{n \ge 0} \varepsilon_{n,\lambda} \, t^n / n!\).

noncomputable def degenerateEuler (lam : ℝ) (n : ℕ) : ℝ :=
  n.factorial * PowerSeries.coeff n (PowerSeries.C (2 : ℝ) * (degenerateExpSeries lam + 1)⁻¹)

Sources

• Ömer Duran, Neşe Ömür and Sibel Koparal, Generalized Degenerate Harmonic Numbers and Their Applications with Riordan Arrays, INTEGERS 24 (2024), Article A41, Equation (7).

Uses 1 · Used by 2

Statement usesDegenerate exponential series

Used byThe zeroth degenerate Euler number · Degenerate Bernoulli and Euler sums of higher-rank degenerate harmonic numbers

TheoremThe zeroth degenerate Euler number#✓fully proved

For every real number \(\lambda\), the degenerate Euler number \(\varepsilon_{0,\lambda}\) equals \(1\).

theorem degenerateEuler_zero (lam : ℝ) : degenerateEuler lam 0 = 1

Sources

• Ömer Duran, Neşe Ömür and Sibel Koparal, Generalized Degenerate Harmonic Numbers and Their Applications with Riordan Arrays, INTEGERS 24 (2024), Article A41, Equation (7).

Uses 3

Statement usesDegenerate Euler numbers

Proof usesConstant term of the degenerate exponential series · Degenerate exponential series

DefinitionDegenerate Stirling numbers of the first kind#✓fully proved

For a real number \(\lambda\), let \(\log_\lambda(1 + t)\) be the formal power series \(\sum_{m \ge 1} \binom{\lambda}{m} t^m / \lambda\), where \(\binom{\lambda}{m} = \lambda(\lambda - 1)\cdots(\lambda - m + 1)/m!\). For natural numbers \(n\) and \(k\), the degenerate Stirling number of the first kind \(S_{1,\lambda}(n, k)\) is \(n!/k!\) times the coefficient of \(t^n\) in \((\log_\lambda(1 + t))^k\). Equivalently, \((\log_\lambda(1 + t))^k / k! = \sum_{n \ge k} S_{1,\lambda}(n, k) t^n / n!\), and \(S_{1,\lambda}(n, k) = 0\) for \(n < k\).

noncomputable def degenerateStirlingFirst (lam : ℝ) (n k : ℕ) : ℝ :=
  n.factorial / k.factorial * PowerSeries.coeff n
    ((PowerSeries.mk fun m => if m = 0 then 0 else Ring.choose lam m / lam) ^ k)

Sources

• Ömer Duran, Neşe Ömür and Sibel Koparal, Generalized Degenerate Harmonic Numbers and Their Applications with Riordan Arrays, INTEGERS 24 (2024), Article A41, Equations (2) and (8).

Used by 2

Used byDegenerate harmonic numbers of rank and degenerate Stirling numbers of the first kind · Degenerate Stirling numbers of the first kind in column zero

TheoremDegenerate Stirling numbers of the first kind in column zero#✓fully proved

For every real number \(\lambda\) and every natural number \(n\), the degenerate Stirling number of the first kind \(S_{1,\lambda}(n, 0)\) equals \(1\) if \(n = 0\) and \(0\) otherwise.

theorem degenerateStirlingFirst_zero_right (lam : ℝ) (n : ℕ) :
    degenerateStirlingFirst lam n 0 = if n = 0 then 1 else 0

Sources

• Ömer Duran, Neşe Ömür and Sibel Koparal, Generalized Degenerate Harmonic Numbers and Their Applications with Riordan Arrays, INTEGERS 24 (2024), Article A41, Equation (8).

Uses 1

Statement usesDegenerate Stirling numbers of the first kind

DefinitionDegenerate Stirling numbers of the second kind#✓fully proved

For a real number \(\lambda\), let \(e_\lambda(t) = \sum_{n \ge 0} (1)_{n,\lambda} t^n / n!\) be the degenerate exponential series, where \((1)_{0,\lambda} = 1\) and \((1)_{n,\lambda} = 1 (1 - \lambda) \cdots (1 - (n - 1)\lambda)\). For natural numbers \(n\) and \(k\), the degenerate Stirling number of the second kind \(S_{2,\lambda}(n, k)\) is \(n!/k!\) times the coefficient of \(t^n\) in \((e_\lambda(t) - 1)^k\). Equivalently, \((e_\lambda(t) - 1)^k / k! = \sum_{n \ge k} S_{2,\lambda}(n, k) t^n / n!\), and \(S_{2,\lambda}(n, k) = 0\) for \(n < k\).

noncomputable def degenerateStirlingSecond (lam : ℝ) (n k : ℕ) : ℝ :=
  n.factorial / k.factorial * PowerSeries.coeff n ((degenerateExpSeries lam - 1) ^ k)

Sources

• Ömer Duran, Neşe Ömür and Sibel Koparal, Generalized Degenerate Harmonic Numbers and Their Applications with Riordan Arrays, INTEGERS 24 (2024), Article A41, Equation (8).

Uses 1 · Used by 2

Statement usesDegenerate exponential series

Used byDegenerate harmonic numbers of rank and degenerate Stirling numbers of the second kind · Degenerate Stirling numbers of the second kind in column zero

TheoremDegenerate Stirling numbers of the second kind in column zero#✓fully proved

For every real number \(\lambda\) and every natural number \(n\), the degenerate Stirling number of the second kind \(S_{2,\lambda}(n, 0)\) equals \(1\) if \(n = 0\) and \(0\) otherwise.

theorem degenerateStirlingSecond_zero_right (lam : ℝ) (n : ℕ) :
    degenerateStirlingSecond lam n 0 = if n = 0 then 1 else 0

Sources

• Ömer Duran, Neşe Ömür and Sibel Koparal, Generalized Degenerate Harmonic Numbers and Their Applications with Riordan Arrays, INTEGERS 24 (2024), Article A41, Equation (8).

Uses 2

Statement usesDegenerate Stirling numbers of the second kind

Proof usesDegenerate exponential series

DefinitionDegenerate Daehee numbers#✓fully proved

For a real number \(\lambda\) and \(n \in \mathbb{N}\), the degenerate Daehee number is \(D_{n,\lambda} = n! \binom{\lambda}{n+1} / \lambda\), where \(\binom{\lambda}{k} = \lambda (\lambda - 1) \cdots (\lambda - k + 1) / k!\) is the generalized binomial coefficient (for \(\lambda = 0\) the value is \(0\), by the convention \(x / 0 = 0\)).

For \(\lambda \ne 0\) the degenerate logarithm \(\log_\lambda(1 + t) = ((1 + t)^\lambda - 1) / \lambda\) satisfies \(\log_\lambda(1 + t) / t = \sum_{n \ge 0} \binom{\lambda}{n+1} t^n / \lambda\), so the numbers \(D_{n,\lambda}\) are given by \(\log_\lambda(1 + t) / t = \sum_{n \ge 0} D_{n,\lambda} \, t^n / n!\).

noncomputable def degenerateDaehee (lam : ℝ) (n : ℕ) : ℝ :=
  n.factorial * Ring.choose lam (n + 1) / lam

Sources

• Ömer Duran, Neşe Ömür and Sibel Koparal, Generalized Degenerate Harmonic Numbers and Their Applications with Riordan Arrays, INTEGERS 24 (2024), Article A41, Equations (2) and (9).

Used by 2

Used byThe zeroth degenerate Daehee number · Alternating and binomial sums of degenerate hyperharmonic numbers

TheoremThe zeroth degenerate Daehee number#✓fully proved

For every real number \(\lambda \ne 0\), the degenerate Daehee number \(D_{0,\lambda}\) equals \(1\).

theorem degenerateDaehee_zero {lam : ℝ} (hlam : lam ≠ 0) : degenerateDaehee lam 0 = 1

Sources

• Ömer Duran, Neşe Ömür and Sibel Koparal, Generalized Degenerate Harmonic Numbers and Their Applications with Riordan Arrays, INTEGERS 24 (2024), Article A41, Equation (9).

Uses 1

Statement usesDegenerate Daehee numbers

TheoremAlternating and binomial sums of degenerate hyperharmonic numbers#●ready to prove

Let \(\lambda > 0\) and \(\alpha > 0\) be real numbers and let \(n\) be a natural number. Write \(H_{m,\lambda}(\alpha)\) for the generalized degenerate harmonic numbers, \(H^s_{m,\lambda}(\alpha)\) for the generalized degenerate hyperharmonic numbers of order \(s\), and \(D_{n,\lambda}\) for the degenerate Daehee numbers. Then

\[ \sum_{k=0}^{n} (-1)^k H^{k+1}_{n-k+1,\lambda}(\alpha) = \frac{(-1)^n D_{n,\lambda}}{\alpha^{n+1}\, n!}, \]

and for every natural number \(r\),

\[ \sum_{k=0}^{n} \binom{k+r}{r} H_{n-k+1,\lambda}(\alpha) = H^{r+2}_{n+1,\lambda}(\alpha). \]

public theorem_wanted degenerateHyperharmonic_alternating_and_binomial_sums {lam α : ℝ} (hlam : 0 < lam)
    (hα : 0 < α) (n : ℕ) :
    ∑ k ∈ Finset.range (n + 1),
        (-1 : ℝ) ^ k * generalizedDegenerateHyperharmonic lam α (k + 1) (n - k + 1) =
      (-1) ^ n * degenerateDaehee lam n / (α ^ (n + 1) * n.factorial) ∧
    ∀ r : ℕ, ∑ k ∈ Finset.range (n + 1),
        ((k + r).choose r : ℝ) * generalizedDegenerateHarmonic lam α (n - k + 1) =
      generalizedDegenerateHyperharmonic lam α (r + 2) (n + 1)

Sources

• Ömer Duran, Neşe Ömür and Sibel Koparal, Generalized Degenerate Harmonic Numbers and Their Applications with Riordan Arrays, INTEGERS 24 (2024), Article A41, Theorem 1.

Uses 3

Statement usesDegenerate Daehee numbers · Generalized degenerate harmonic numbers · Generalized degenerate hyperharmonic numbers

TheoremAlternating binomial sum of degenerate harmonic numbers#●ready to prove

Let \(\lambda > 0\) be real and let \(H_{k,\lambda} = H_{k,\lambda}(1)\) be the degenerate harmonic numbers, that is, the generalized degenerate harmonic numbers at \(\alpha = 1\). Then for every integer \(n \ge 1\), \(\sum_{k=0}^{n} (-1)^{k+1} \binom{n}{k} H_{k,\lambda} = \binom{n + \lambda - 1}{n} / \lambda\), where \(\binom{n + \lambda - 1}{n}\) is the generalized binomial coefficient.

public theorem_wanted degenerateHarmonic_alternating_binomial_sum {lam : ℝ} (hlam : 0 < lam) {n : ℕ}
    (hn : 1 ≤ n) :
    ∑ k ∈ Finset.range (n + 1),
        (-1 : ℝ) ^ (k + 1) * (n.choose k : ℝ) * generalizedDegenerateHarmonic lam 1 k =
      Ring.choose ((n : ℝ) + lam - 1) n / lam

Sources

• Ömer Duran, Neşe Ömür and Sibel Koparal, Generalized Degenerate Harmonic Numbers and Their Applications with Riordan Arrays, INTEGERS 24 (2024), Article A41, Theorem 2.

Uses 1

Statement usesGeneralized degenerate harmonic numbers

TheoremAlternating degenerate exponential sum of degenerate harmonic numbers of rank#●ready to prove

Let \(\lambda > 0\) and \(\alpha > 0\) be real and let \(n\) be a natural number. Write \((1)_{k,\lambda} = 1 \cdot (1 - \lambda) \cdots (1 - (k-1)\lambda)\) for the generalized falling factorial (so \((1)_{0,\lambda} = 1\)), \(H_\lambda(m, r, \alpha)\) for the generalized degenerate harmonic numbers of rank \(r\), and \(H_{m,\lambda}(\alpha)\) for the generalized degenerate harmonic numbers. Then

\[\sum_{k=0}^{n} (-1)^k \frac{(1)_{k,\lambda}}{k!} H_\lambda(n+1, k, \alpha) = H_{n+1,\lambda}(\alpha) - \frac{H_{n,\lambda}(\alpha)}{\alpha}.\]

public theorem_wanted degenerateHarmonicRank_exp_weighted_sum {lam α : ℝ} (hlam : 0 < lam) (hα : 0 < α)
    (n : ℕ) :
    ∑ k ∈ Finset.range (n + 1),
        (-1 : ℝ) ^ k * generalizedFallingFactorial 1 lam k / k.factorial *
          degenerateHarmonicRank lam α (n + 1) k =
      generalizedDegenerateHarmonic lam α (n + 1) - generalizedDegenerateHarmonic lam α n / α

Sources

• Ömer Duran, Neşe Ömür and Sibel Koparal, Generalized Degenerate Harmonic Numbers and Their Applications with Riordan Arrays, INTEGERS 24 (2024), Article A41, Theorem 3.

Uses 3

Statement usesGeneralized degenerate harmonic numbers of a given rank · Generalized degenerate harmonic numbers · Generalized falling factorial

TheoremPartial sums of degenerate harmonic numbers of rank as a convolution#●ready to prove

Let \(\lambda > 0\) and \(\alpha > 0\) be real and let \(n\) and \(r\) be natural numbers. Write \(H_\lambda(k, r, \alpha)\) for the generalized degenerate harmonic numbers of rank \(r\) and \(H_{m,\lambda}(\alpha)\) for the generalized degenerate harmonic numbers. Then

\[\sum_{k=0}^{n} H_\lambda(k, r+1, \alpha) = \sum_{k=0}^{n} H_{n-k,\lambda}(\alpha) \, H_\lambda(k, r, \alpha).\]

public theorem_wanted degenerateHarmonicRank_succ_partial_sum {lam α : ℝ} (hlam : 0 < lam) (hα : 0 < α)
    (n r : ℕ) :
    ∑ k ∈ Finset.range (n + 1), degenerateHarmonicRank lam α k (r + 1) =
      ∑ k ∈ Finset.range (n + 1),
        generalizedDegenerateHarmonic lam α (n - k) * degenerateHarmonicRank lam α k r

Sources

• Ömer Duran, Neşe Ömür and Sibel Koparal, Generalized Degenerate Harmonic Numbers and Their Applications with Riordan Arrays, INTEGERS 24 (2024), Article A41, Theorem 4.

Uses 2

Statement usesGeneralized degenerate harmonic numbers of a given rank · Generalized degenerate harmonic numbers

TheoremShifted alternating degenerate exponential sum of degenerate harmonic numbers of rank#●ready to prove

Let \(\lambda > 0\) and \(\alpha > 0\) be real and let \(n\) be a natural number. Write \((1)_{k,\lambda} = 1 \cdot (1 - \lambda) \cdots (1 - (k-1)\lambda)\) for the generalized falling factorial and \(H_\lambda(m, r, \alpha)\) for the generalized degenerate harmonic numbers of rank \(r\). Then

\[\sum_{k=0}^{n} (-1)^k \frac{(1)_{k+1,\lambda}}{(k+1)!} H_\lambda(n+1, k, \alpha) = \frac{1}{\alpha}.\]

public theorem_wanted degenerateHarmonicRank_shifted_exp_weighted_sum {lam α : ℝ} (hlam : 0 < lam)
    (hα : 0 < α) (n : ℕ) :
    ∑ k ∈ Finset.range (n + 1),
        (-1 : ℝ) ^ k * generalizedFallingFactorial 1 lam (k + 1) / (k + 1).factorial *
          degenerateHarmonicRank lam α (n + 1) k =
      1 / α

Sources

• Ömer Duran, Neşe Ömür and Sibel Koparal, Generalized Degenerate Harmonic Numbers and Their Applications with Riordan Arrays, INTEGERS 24 (2024), Article A41, Theorem 5.

Uses 2

Statement usesGeneralized degenerate harmonic numbers of a given rank · Generalized falling factorial

TheoremA binomial sum equal to a degenerate hyperharmonic number#●ready to prove

Let \(\lambda > 0\) and \(\alpha > 0\) be real numbers and let \(r \le n\) be natural numbers. Write \(\binom{\lambda}{j} = \lambda(\lambda - 1)\cdots(\lambda - j + 1)/j!\) and \(H^s_{m,\lambda}(\alpha)\) for the generalized degenerate hyperharmonic numbers of order \(s\). Then

\[ \sum_{k=r}^{n} (-1)^{n-k} \frac{\binom{\lambda}{n-k+1}}{\lambda\, \alpha^{n-k+1}} \binom{k}{r} = H^{r+1}_{n-r+1,\lambda}(\alpha). \]

The hypothesis \(r \le n\) makes the summation range \(r \le k \le n\) nonempty, as in the paper.

public theorem_wanted degenerateHyperharmonic_eq_binomial_sum {lam α : ℝ} (hlam : 0 < lam) (hα : 0 < α)
    {n r : ℕ} (hrn : r ≤ n) :
    ∑ k ∈ Finset.Icc r n,
        (-1 : ℝ) ^ (n - k) * Ring.choose lam (n - k + 1) / (lam * α ^ (n - k + 1)) *
          (k.choose r : ℝ) =
      generalizedDegenerateHyperharmonic lam α (r + 1) (n - r + 1)

Sources

• Ömer Duran, Neşe Ömür and Sibel Koparal, Generalized Degenerate Harmonic Numbers and Their Applications with Riordan Arrays, INTEGERS 24 (2024), Article A41, Theorem 6.

Uses 1

Statement usesGeneralized degenerate hyperharmonic numbers

TheoremDegenerate harmonic numbers of rank and degenerate Stirling numbers of the first kind#●ready to prove

Let \(\lambda > 0\) and \(\alpha > 0\) be real and let \(n \ge 1\) be a natural number. Write \((1)_{k,\lambda} = 1 \cdot (1 - \lambda) \cdots (1 - (k-1)\lambda)\) for the generalized falling factorial, \(H_\lambda(m, r, \alpha)\) for the generalized degenerate harmonic numbers of rank \(r\), and \(S_{1,\lambda}(k, i)\) for the degenerate Stirling numbers of the first kind. Then

\[\sum_{k=1}^{n} \frac{(1)_{k,\lambda}}{k!} H_\lambda(n, k-1, \alpha) = \sum_{k=1}^{n} \sum_{i=1}^{k} \frac{(-1)^{k+i} (1)_{i,\lambda} S_{1,\lambda}(k, i)}{\alpha^k k!}.\]

public theorem_wanted degenerateHarmonicRank_exp_sum_eq_stirlingFirst {lam α : ℝ} (hlam : 0 < lam)
    (hα : 0 < α) {n : ℕ} (hn : 1 ≤ n) :
    ∑ k ∈ Finset.Icc 1 n,
        generalizedFallingFactorial 1 lam k / k.factorial * degenerateHarmonicRank lam α n (k - 1) =
      ∑ k ∈ Finset.Icc 1 n, ∑ i ∈ Finset.Icc 1 k,
        (-1 : ℝ) ^ (k + i) * generalizedFallingFactorial 1 lam i * degenerateStirlingFirst lam k i /
          (α ^ k * k.factorial)

Sources

• Ömer Duran, Neşe Ömür and Sibel Koparal, Generalized Degenerate Harmonic Numbers and Their Applications with Riordan Arrays, INTEGERS 24 (2024), Article A41, Theorem 7.

Uses 3

Statement usesGeneralized degenerate harmonic numbers of a given rank · Degenerate Stirling numbers of the first kind · Generalized falling factorial

TheoremDegenerate Bernoulli and Euler sums of higher-rank degenerate harmonic numbers#●ready to prove

Let \(\lambda > 0\) and \(\alpha > 0\) be real and let \(n \ge 1\) be an integer. Write \(\beta_{k,\lambda}\) and \(\varepsilon_{k,\lambda}\) for the degenerate Bernoulli and Euler numbers, \(H_{m,\lambda}(\alpha)\) for the generalized degenerate harmonic numbers, and \(H_\lambda(m, r, \alpha)\) for the generalized degenerate harmonic numbers of rank \(r\). Then \(\sum_{k=1}^{n} (-1)^k \beta_{k,\lambda} H_\lambda(n, k - 1, \alpha) / k! = \alpha H_{n+1,\lambda}(\alpha) - 1\), and if moreover \(\alpha \ne 1/2\), then \(\sum_{k=1}^{n} (-1)^{k+1} \varepsilon_{k,\lambda} H_\lambda(n, k - 1, \alpha) / k! = ((2\alpha)^{-n} - 1) / (2\alpha - 1)\).

public theorem_wanted degenerateHarmonicRank_bernoulli_euler_sums {lam α : ℝ} (hlam : 0 < lam)
    (hα : 0 < α) {n : ℕ} (hn : 1 ≤ n) :
    ∑ k ∈ Finset.Icc 1 n,
        (-1 : ℝ) ^ k * degenerateBernoulli lam k * degenerateHarmonicRank lam α n (k - 1) /
          k.factorial =
      α * generalizedDegenerateHarmonic lam α (n + 1) - 1 ∧
    (α ≠ 1 / 2 →
      ∑ k ∈ Finset.Icc 1 n,
          (-1 : ℝ) ^ (k + 1) * degenerateEuler lam k / k.factorial *
            degenerateHarmonicRank lam α n (k - 1) =
        (((2 * α) ^ n)⁻¹ - 1) / (2 * α - 1))

Sources

• Ömer Duran, Neşe Ömür and Sibel Koparal, Generalized Degenerate Harmonic Numbers and Their Applications with Riordan Arrays, INTEGERS 24 (2024), Article A41, Theorem 8.

Uses 4

Statement usesDegenerate Bernoulli numbers · Degenerate Euler numbers · Generalized degenerate harmonic numbers of a given rank · Generalized degenerate harmonic numbers

TheoremDegenerate harmonic numbers of rank and degenerate Stirling numbers of the second kind#●ready to prove

Let \(\lambda > 0\) and \(\alpha > 0\) be real and let \(m\) and \(n\) be natural numbers with \(1 \le m \le n\). Write \(S_{2,\lambda}(k, m)\) for the degenerate Stirling numbers of the second kind and \(H_\lambda(n, r, \alpha)\) for the generalized degenerate harmonic numbers of rank \(r\). Then

\[\sum_{k=m}^{n} \frac{(-1)^k S_{2,\lambda}(k, m) H_\lambda(n, k-1, \alpha)}{k!} = \frac{(-1)^m}{m! \, \alpha^m}.\]

public theorem_wanted degenerateHarmonicRank_stirlingSecond_sum {lam α : ℝ} (hlam : 0 < lam)
    (hα : 0 < α) {n m : ℕ} (hm : 1 ≤ m) (hmn : m ≤ n) :
    ∑ k ∈ Finset.Icc m n,
        (-1 : ℝ) ^ k * degenerateStirlingSecond lam k m * degenerateHarmonicRank lam α n (k - 1) /
          k.factorial =
      (-1) ^ m / (m.factorial * α ^ m)

Sources

• Ömer Duran, Neşe Ömür and Sibel Koparal, Generalized Degenerate Harmonic Numbers and Their Applications with Riordan Arrays, INTEGERS 24 (2024), Article A41, Theorem 9.

Uses 2

Statement usesGeneralized degenerate harmonic numbers of a given rank · Degenerate Stirling numbers of the second kind

24 fully proved9 ready to prove

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