Degenerate Bernoulli polynomials

DefinitionModified degenerate Bernoulli polynomials#✓fully proved

For a complex number \(\lambda\), a natural number \(n\) and a complex number \(x\), the modified degenerate Bernoulli polynomial is \(\mathfrak{B}_{n,\lambda}(x) = c^{n-1} B_n(x)\), where \(c = \log(1+\lambda)/\lambda\) with the principal branch of the logarithm, \(B_n\) is the \(n\)-th Bernoulli polynomial (with \(B_1(x) = x - 1/2\)), and the exponent \(n - 1\) is an integer, so that \(\mathfrak{B}_{0,\lambda}(x) = c^{-1}\). At \(\lambda = 0\) and \(\lambda = -1\), where this quotient is undefined, \(c\) is taken to be \(0\), with \(0^{-1} = 0\) and \(0^0 = 1\).

noncomputable def modifiedDegenerateBernoulli (lam : ℂ) (n : ℕ) (x : ℂ) : ℂ :=
  (Complex.log (1 + lam) / lam) ^ ((n : ℤ) - 1) * Polynomial.aeval x (Polynomial.bernoulli n)
Sources
  • Qi-Peng Su and Hao Pan, Symmetric Identities on Modified Degenerate Bernoulli Polynomials, Journal of Integer Sequences 27 (2024), Article 24.5.4, Section 1 (generating function of Dolgy, Kim, Kwon and Seo).
Used by 7
Used byGenerating function of the modified degenerate Bernoulli polynomials · A convolution lemma for modified degenerate Bernoulli polynomials · Matiyasevich-type identity for modified degenerate Bernoulli numbers · Miki-type identity for modified degenerate Bernoulli numbers · The first modified degenerate Bernoulli polynomial · Symmetric identity for modified degenerate Bernoulli polynomials · Three-term identity for modified degenerate Bernoulli polynomials

TheoremGenerating function of the modified degenerate Bernoulli polynomials#✓fully proved

Let \(\lambda\) and \(x\) be complex numbers such that \(c = \log(1+\lambda)/\lambda\) (principal branch) is nonzero, and write \(\mathfrak{B}_{n,\lambda}(x)\) for the modified degenerate Bernoulli polynomials. Then, as formal power series in \(t\) over \(\mathbb{C}\),

\[ \Bigl(\sum_{n \ge 0} \mathfrak{B}_{n,\lambda}(x) \frac{t^n}{n!}\Bigr) \bigl(e^{ct} - 1\bigr) = t\, e^{cxt}. \]

With \((1+\lambda)^w = e^{w \log(1+\lambda)}\), this says that the series times \((1+\lambda)^{t/\lambda} - 1\) equals \(t\,(1+\lambda)^{xt/\lambda}\).

theorem modifiedDegenerateBernoulli_generating_function (lam x : ℂ)
    (h : Complex.log (1 + lam) / lam ≠ 0) :
    PowerSeries.mk (fun n => modifiedDegenerateBernoulli lam n x / n.factorial) *
        (PowerSeries.rescale (Complex.log (1 + lam) / lam) (PowerSeries.exp ℂ) - 1) =
      PowerSeries.X *
        PowerSeries.rescale (Complex.log (1 + lam) / lam * x) (PowerSeries.exp ℂ)
Sources
  • Qi-Peng Su and Hao Pan, Symmetric Identities on Modified Degenerate Bernoulli Polynomials, Journal of Integer Sequences 27 (2024), Article 24.5.4, Section 1.
Uses 1
Statement usesModified degenerate Bernoulli polynomials

TheoremThe first modified degenerate Bernoulli polynomial#✓fully proved

For all complex numbers \(\lambda\) and \(x\), the modified degenerate Bernoulli polynomial \(\mathfrak{B}_{1,\lambda}\) of index \(1\) satisfies \(\mathfrak{B}_{1,\lambda}(x) = x - 1/2\).

theorem modifiedDegenerateBernoulli_one (lam x : ℂ) :
    modifiedDegenerateBernoulli lam 1 x = x - 1 / 2
Sources
  • Qi-Peng Su and Hao Pan, Symmetric Identities on Modified Degenerate Bernoulli Polynomials, Journal of Integer Sequences 27 (2024), Article 24.5.4, Section 1.
Uses 1
Statement usesModified degenerate Bernoulli polynomials

TheoremSymmetric identity for modified degenerate Bernoulli polynomials#●ready to prove

Let \(m \ge 2\) and \(n \ge 1\) be integers, let \(\lambda\) be a complex number with \(\lambda \ne 0\) and \(\lambda \ne -1\), let \(r_1, \ldots, r_m\) and \(x_1, \ldots, x_m\) be complex numbers, write \(\mathfrak{B}_{k,\lambda}(x)\) for the modified degenerate Bernoulli polynomials, and put \(r_0 = n - r_1 - \cdots - r_m\). Then

\[ r_0 \sum_{k_1 + \cdots + k_m = n} \prod_{j=1}^{m} \binom{r_j}{k_j} \mathfrak{B}_{k_j,\lambda}(x_j) = -\sum_{i=1}^{m} r_i \sum_{k_1 + \cdots + k_m = n} \binom{r_0}{k_i} \mathfrak{B}_{k_i,\lambda}(1 - x_i) \prod_{j \ne i} \binom{r_j}{k_j} \mathfrak{B}_{k_j,\lambda}(x_j - x_i + [j > i]), \]

where \(k_1, \ldots, k_m\) range over nonnegative integers, \(\binom{w}{k} = w(w-1) \cdots (w-k+1)/k!\) is the binomial coefficient with a complex upper argument, and \([j > i]\) is \(1\) if \(j > i\) and \(0\) otherwise.

public theorem_wanted modifiedDegenerateBernoulli_symmetric_identity (m n : ℕ) (hm : 2 ≤ m)
    (hn : 1 ≤ n) (r x : Fin m → ℂ) (lam : ℂ) (hlam : lam ≠ 0) (hlam1 : lam ≠ -1) :
    ((n : ℂ) - ∑ j, r j) * ∑ k ∈ Finset.Nat.antidiagonalTuple m n,
        ∏ j, Ring.choose (r j) (k j) * modifiedDegenerateBernoulli lam (k j) (x j) =
      -∑ i, r i * ∑ k ∈ Finset.Nat.antidiagonalTuple m n,
        Ring.choose ((n : ℂ) - ∑ j, r j) (k i) * modifiedDegenerateBernoulli lam (k i) (1 - x i) *
          ∏ j ∈ univ.erase i, Ring.choose (r j) (k j) *
            modifiedDegenerateBernoulli lam (k j) (x j - x i + if i < j then 1 else 0)
Sources
  • Qi-Peng Su and Hao Pan, Symmetric Identities on Modified Degenerate Bernoulli Polynomials, Journal of Integer Sequences 27 (2024), Article 24.5.4, Theorem 1 (jis-2024-24.5.4-thm-1).
Uses 1
Statement usesModified degenerate Bernoulli polynomials

TheoremThree-term identity for modified degenerate Bernoulli polynomials#●ready to prove

Let \(n \ge 1\), let \(\lambda\) be a complex number with \(\lambda \ne 0\) and \(\lambda \ne -1\), write \(\mathfrak{B}_{k,\lambda}(x)\) for the modified degenerate Bernoulli polynomials, and for complex numbers \(s, t, x, y\) put \(\mathfrak{D}_{s,t,\lambda}(x, y) = \sum_{k=0}^{n} (-1)^k \binom{s}{k} \binom{t}{n-k} \mathfrak{B}_{n-k,\lambda}(x) \mathfrak{B}_{k,\lambda}(y)\), where \(\binom{w}{k} = w(w-1) \cdots (w-k+1)/k!\) is the binomial coefficient with a complex upper argument. If \(r, s, t, x, y, z\) are complex numbers with \(r + s + t = n\) and \(x + y + z = 1\), then \(r\, \mathfrak{D}_{s,t,\lambda}(x, y) + s\, \mathfrak{D}_{t,r,\lambda}(y, z) + t\, \mathfrak{D}_{r,s,\lambda}(z, x) = 0\).

public theorem_wanted modifiedDegenerateBernoulli_three_term (n : ℕ) (hn : 1 ≤ n) (lam : ℂ)
    (hlam : lam ≠ 0) (hlam1 : lam ≠ -1) (r s t x y z : ℂ) (hrst : r + s + t = n)
    (hxyz : x + y + z = 1) :
    let D : ℂ → ℂ → ℂ → ℂ → ℂ := fun s t x y =>
      ∑ k ∈ range (n + 1), (-1) ^ k * Ring.choose s k * Ring.choose t (n - k) *
        modifiedDegenerateBernoulli lam (n - k) x * modifiedDegenerateBernoulli lam k y
    r * D s t x y + s * D t r y z + t * D r s z x = 0
Sources
  • Qi-Peng Su and Hao Pan, Symmetric Identities on Modified Degenerate Bernoulli Polynomials, Journal of Integer Sequences 27 (2024), Article 24.5.4, Corollary 2, equation (sunpanD) (jis-2024-24.5.4-cor-2).
Uses 1
Statement usesModified degenerate Bernoulli polynomials

TheoremMiki-type identity for modified degenerate Bernoulli numbers#●ready to prove

Let \(n \ge 4\), let \(\lambda\) be a complex number with \(\lambda \ne 0\) and \(\lambda \ne -1\), write \(\mathfrak{B}_{k,\lambda} = \mathfrak{B}_{k,\lambda}(0)\) for the value at \(0\) of the modified degenerate Bernoulli polynomial \(\mathfrak{B}_{k,\lambda}(x)\), and let \(H_n = \sum_{k=1}^{n} 1/k\) be the \(n\)-th harmonic number. Then

\[ \sum_{k=1}^{n-1} \frac{\mathfrak{B}_{k,\lambda} \mathfrak{B}_{n-k,\lambda}}{k} - \sum_{k=1}^{n-1} \binom{n}{k} \frac{\mathfrak{B}_{k,\lambda} \mathfrak{B}_{n-k,\lambda}}{k} = \frac{\lambda}{\log(1+\lambda)}\, H_n\, \mathfrak{B}_{n,\lambda} + \frac{n}{2}\, \mathfrak{B}_{n-1,\lambda}, \]

where \(\log\) is the principal branch.

public theorem_wanted modifiedDegenerateBernoulli_miki (n : ℕ) (hn : 4 ≤ n) (lam : ℂ)
    (hlam : lam ≠ 0) (hlam1 : lam ≠ -1) :
    ∑ k ∈ Icc 1 (n - 1), modifiedDegenerateBernoulli lam k 0 / k *
        modifiedDegenerateBernoulli lam (n - k) 0 -
      ∑ k ∈ Icc 1 (n - 1), (n.choose k : ℂ) * modifiedDegenerateBernoulli lam k 0 / k *
        modifiedDegenerateBernoulli lam (n - k) 0 =
      lam / Complex.log (1 + lam) * (harmonic n : ℂ) * modifiedDegenerateBernoulli lam n 0 +
        n / 2 * modifiedDegenerateBernoulli lam (n - 1) 0
Sources
  • Qi-Peng Su and Hao Pan, Symmetric Identities on Modified Degenerate Bernoulli Polynomials, Journal of Integer Sequences 27 (2024), Article 24.5.4, Corollary 2, equation (mikiD) (jis-2024-24.5.4-cor-2).
Uses 1
Statement usesModified degenerate Bernoulli polynomials

TheoremMatiyasevich-type identity for modified degenerate Bernoulli numbers#●ready to prove

Let \(n \ge 4\), let \(\lambda\) be a complex number with \(\lambda \ne 0\) and \(\lambda \ne -1\), and write \(\mathfrak{B}_{k,\lambda} = \mathfrak{B}_{k,\lambda}(0)\) for the value at \(0\) of the modified degenerate Bernoulli polynomial \(\mathfrak{B}_{k,\lambda}(x)\). Then

\[ (n+2) \sum_{k=2}^{n-2} \mathfrak{B}_{k,\lambda} \mathfrak{B}_{n-k,\lambda} = 2 \sum_{k=2}^{n-2} \binom{n+2}{k} \mathfrak{B}_{k,\lambda} \mathfrak{B}_{n-k,\lambda} + \frac{\lambda}{\log(1+\lambda)}\, n(n+1)\, \mathfrak{B}_{n,\lambda}, \]

where \(\log\) is the principal branch.

public theorem_wanted modifiedDegenerateBernoulli_matiyasevich (n : ℕ) (hn : 4 ≤ n) (lam : ℂ)
    (hlam : lam ≠ 0) (hlam1 : lam ≠ -1) :
    ((n : ℂ) + 2) * ∑ k ∈ Icc 2 (n - 2),
        modifiedDegenerateBernoulli lam k 0 * modifiedDegenerateBernoulli lam (n - k) 0 =
      2 * ∑ k ∈ Icc 2 (n - 2), ((n + 2).choose k : ℂ) *
          modifiedDegenerateBernoulli lam k 0 * modifiedDegenerateBernoulli lam (n - k) 0 +
        lam / Complex.log (1 + lam) * (n * (n + 1)) * modifiedDegenerateBernoulli lam n 0
Sources
  • Qi-Peng Su and Hao Pan, Symmetric Identities on Modified Degenerate Bernoulli Polynomials, Journal of Integer Sequences 27 (2024), Article 24.5.4, Corollary 2, equation (matiyasevichD) (jis-2024-24.5.4-cor-2).
Uses 1
Statement usesModified degenerate Bernoulli polynomials

TheoremA convolution lemma for modified degenerate Bernoulli polynomials#●ready to prove

Let \(m \ge 2\), let \(\lambda\) be a complex number with \(\lambda \ne 0\) and \(\lambda \ne -1\), let \(s_1, \ldots, s_m\) be nonnegative integers with \(S = s_1 + \cdots + s_m\), and let \(x_1, \ldots, x_m\) be complex numbers. Writing \(\mathfrak{B}_{n,\lambda}(x)\) for the modified degenerate Bernoulli polynomials,

\[ \sum_{i=1}^{m} s_i\, \mathfrak{B}_{s_i-1,\lambda}(x_i) \prod_{j \ne i} \mathfrak{B}_{s_j,\lambda}(x_j) = \sum_{i=1}^{m} s_i \sum_{k_1 + \cdots + k_m = S} \mathfrak{B}_{k_i-1,\lambda}(x_i) \prod_{j \ne i} \binom{s_j}{k_j} \mathfrak{B}_{k_j,\lambda}(x_j - x_i + [j > i]), \]

where \(k_1, \ldots, k_m\) range over nonnegative integers, \([j > i]\) is \(1\) if \(j > i\) and \(0\) otherwise, and \(\mathfrak{B}_{-1,\lambda}\) is read as \(0\) (on the left this term has the factor \(s_i = 0\)).

public theorem_wanted modifiedDegenerateBernoulli_lemma_three (m : ℕ) (hm : 2 ≤ m)
    (s : Fin m → ℕ) (x : Fin m → ℂ) (lam : ℂ) (hlam : lam ≠ 0) (hlam1 : lam ≠ -1) :
    ∑ i, (s i : ℂ) * modifiedDegenerateBernoulli lam (s i - 1) (x i) *
        ∏ j ∈ univ.erase i, modifiedDegenerateBernoulli lam (s j) (x j) =
      ∑ i, (s i : ℂ) * ∑ k ∈ Finset.Nat.antidiagonalTuple m (∑ j, s j),
        (if k i = 0 then 0 else modifiedDegenerateBernoulli lam (k i - 1) (x i)) *
          ∏ j ∈ univ.erase i, ((s j).choose (k j) : ℂ) *
            modifiedDegenerateBernoulli lam (k j) (x j - x i + if i < j then 1 else 0)
Sources
  • Qi-Peng Su and Hao Pan, Symmetric Identities on Modified Degenerate Bernoulli Polynomials, Journal of Integer Sequences 27 (2024), Article 24.5.4, Lemma 3 (jis-2024-24.5.4-lem-3).
Uses 1
Statement usesModified degenerate Bernoulli polynomials
3 fully proved5 ready to prove

The dependency explorer needs JavaScript and its graph data.

Download the complete graph data.