Integral representations of k-Jacobsthal numbers

DefinitionLucas parameters of the k-Jacobsthal numbers#✓fully proved

For a natural number \(k\), the Lucas-sequence parameters \((P, Q) = (k, -2)\). The Lucas sequences \(J_{k,n} = U_n(k, -2)\) and \(j_{k,n} = V_n(k, -2)\) with these parameters are the \(k\)-Jacobsthal numbers (\(J_{k,0} = 0\), \(J_{k,1} = 1\)) and the \(k\)-Jacobsthal–Lucas numbers (\(j_{k,0} = 2\), \(j_{k,1} = k\)); both satisfy \(x_{n+2} = k x_{n+1} + 2 x_n\).

def kJacobsthalParams (k : ℕ) : LucasSequenceParams where
  p := k
  q := -2
  q_ne_zero := by decide

Sources

• Weerayuth Nilsrakoo and Achariya Nilsrakoo, An Integral Representation of k-Jacobsthal and k-Jacobsthal-Lucas Numbers, INTEGERS 25 (2025), Article A74, Section 1, recurrences (1) and (2).

Uses 1 · Used by 18

Statement usesParameters of a Lucas sequence

Used byCatalan's identity for the companion \(k\)-Jacobsthal numbers · Integral representation of the companion \(k\)-Jacobsthal numbers · The \(k\)-Jacobsthal numbers as companion numbers · Integral formulas for even- and odd-indexed Jacobsthal numbers · Integral representation of the Jacobsthal-Lucas numbers along arithmetic progressions · Integral representation of the Jacobsthal-Lucas numbers at multiples of an index · Integral representation of Jacobsthal numbers with an offset · Integral representation of Jacobsthal numbers at multiples · Addition formulas for k-Jacobsthal numbers · Binet-type identities for k-Jacobsthal numbers · Integral representation of even- and odd-indexed k-Jacobsthal numbers · Integral representation of k-Jacobsthal numbers at multiples · The second k-Jacobsthal number · Integral representation of even- and odd-indexed k-Jacobsthal–Lucas numbers · Integral representation of k-Jacobsthal–Lucas numbers at multiples · The \(k\)-Jacobsthal-Lucas number \(j_{k,2}\) · Ratio of consecutive \(k\)-Jacobsthal numbers · Companion \(k\)-Jacobsthal numbers via Lucas sequences

TheoremThe second k-Jacobsthal number#✓fully proved

For every natural number \(k\), the \(k\)-Jacobsthal number \(J_{k,2} = U_2(k, -2)\) equals \(k\).

theorem lucasU_kJacobsthalParams_two (k : ℕ) : lucasU (kJacobsthalParams k) 2 = k

Sources

• Weerayuth Nilsrakoo and Achariya Nilsrakoo, An Integral Representation of k-Jacobsthal and k-Jacobsthal-Lucas Numbers, INTEGERS 25 (2025), Article A74, proof of Theorem 2.

Uses 3

Statement usesLucas parameters of the k-Jacobsthal numbers · Lucas sequence \(U(P, Q)\)

Proof usesParameters of a Lucas sequence

TheoremThe \(k\)-Jacobsthal-Lucas number \(j_{k,2}\)#✓fully proved

For every natural number \(k\), the \(k\)-Jacobsthal-Lucas numbers \(j_{k,n} = V_n(k, -2)\), the Lucas sequence \(V\) with parameters \((P, Q) = (k, -2)\), satisfy \(j_{k,2} = k^2 + 4\).

theorem lucasV_kJacobsthalParams_two (k : ℕ) :
    lucasV (kJacobsthalParams k) 2 = (k : ℤ) ^ 2 + 4

Sources

• Weerayuth Nilsrakoo and Achariya Nilsrakoo, An Integral Representation of k-Jacobsthal and k-Jacobsthal-Lucas Numbers, INTEGERS 25 (2025), Article A74, proof of Theorem 2.

Uses 3

Statement usesLucas parameters of the k-Jacobsthal numbers · Companion Lucas sequence \(V(P, Q)\)

Proof usesParameters of a Lucas sequence

DefinitionCompanion \(k\)-Jacobsthal numbers#✓fully proved

For a natural number \(k\) and integers \(a, b\), the companion \(k\)-Jacobsthal numbers \(\mathcal{J}_{k,n}(a, b)\), \(n \in \mathbb{N}\), are the integers defined by \(\mathcal{J}_{k,0}(a, b) = a\), \(\mathcal{J}_{k,1}(a, b) = b\) and \(\mathcal{J}_{k,n+2}(a, b) = k\,\mathcal{J}_{k,n+1}(a, b) + 2\,\mathcal{J}_{k,n}(a, b)\).

def companionKJacobsthal (k : ℕ) (a b : ℤ) : ℕ → ℤ
  | 0 => a
  | 1 => b
  | n + 2 => k * companionKJacobsthal k a b (n + 1) + 2 * companionKJacobsthal k a b n

Sources

• Weerayuth Nilsrakoo and Achariya Nilsrakoo, An Integral Representation of k-Jacobsthal and k-Jacobsthal-Lucas Numbers, INTEGERS 25 (2025), Article A74, Section 4.

Used by 7

Used byRecurrence of the companion \(k\)-Jacobsthal numbers · Catalan's identity for the companion \(k\)-Jacobsthal numbers · Generating function of the companion \(k\)-Jacobsthal numbers · Integral representation of the companion \(k\)-Jacobsthal numbers · The second companion \(k\)-Jacobsthal number · The \(k\)-Jacobsthal numbers as companion numbers · Companion \(k\)-Jacobsthal numbers via Lucas sequences

TheoremRecurrence of the companion \(k\)-Jacobsthal numbers#✓fully proved

For every natural number \(k\), all integers \(a, b\) and every natural number \(n\), the companion \(k\)-Jacobsthal numbers satisfy \(\mathcal{J}_{k,n+2}(a, b) = k\,\mathcal{J}_{k,n+1}(a, b) + 2\,\mathcal{J}_{k,n}(a, b)\).

theorem companionKJacobsthal_add_two (k : ℕ) (a b : ℤ) (n : ℕ) :
    companionKJacobsthal k a b (n + 2) =
      k * companionKJacobsthal k a b (n + 1) + 2 * companionKJacobsthal k a b n

Sources

• Weerayuth Nilsrakoo and Achariya Nilsrakoo, An Integral Representation of k-Jacobsthal and k-Jacobsthal-Lucas Numbers, INTEGERS 25 (2025), Article A74, Section 4.

Uses 1 · Used by 1

Statement usesCompanion \(k\)-Jacobsthal numbers

Used byThe \(k\)-Jacobsthal numbers as companion numbers

TheoremThe second companion \(k\)-Jacobsthal number#✓fully proved

For every natural number \(k\) and all integers \(a, b\), the companion \(k\)-Jacobsthal numbers satisfy \(\mathcal{J}_{k,2}(a, b) = bk + 2a\).

theorem companionKJacobsthal_two (k : ℕ) (a b : ℤ) :
    companionKJacobsthal k a b 2 = b * k + 2 * a

Sources

• Weerayuth Nilsrakoo and Achariya Nilsrakoo, An Integral Representation of k-Jacobsthal and k-Jacobsthal-Lucas Numbers, INTEGERS 25 (2025), Article A74, Section 4, list of the first companion numbers.

Uses 1

Statement usesCompanion \(k\)-Jacobsthal numbers

TheoremThe \(k\)-Jacobsthal numbers as companion numbers#✓fully proved

For a natural number \(k\), let \(\mathcal{J}_{k,n}(a, b)\) denote the companion \(k\)-Jacobsthal numbers and \(J_{k,n} = U_n(k, -2)\) the \(k\)-Jacobsthal numbers, the Lucas sequence with parameters \((k, -2)\). Then for every natural number \(n\), \(\mathcal{J}_{k,n}(0, 1) = J_{k,n}\): the companion numbers with initial values \(0\) and \(1\) are the \(k\)-Jacobsthal numbers.

theorem companionKJacobsthal_zero_one (k n : ℕ) :
    companionKJacobsthal k 0 1 n = lucasU (kJacobsthalParams k) n

Sources

• Weerayuth Nilsrakoo and Achariya Nilsrakoo, An Integral Representation of k-Jacobsthal and k-Jacobsthal-Lucas Numbers, INTEGERS 25 (2025), Article A74, Section 4, special cases of the companion numbers.

Uses 6

Statement usesCompanion \(k\)-Jacobsthal numbers · Lucas parameters of the k-Jacobsthal numbers · Lucas sequence \(U(P, Q)\)

Proof usesParameters of a Lucas sequence · Recurrence of the companion \(k\)-Jacobsthal numbers · Recurrence for the Lucas sequence \(U(P, Q)\)

TheoremBinet-type identities for k-Jacobsthal numbers#●ready to prove

Let \(k\) and \(n\) be natural numbers, let \(J_{k,n} = U_n(k, -2)\) and \(j_{k,n} = V_n(k, -2)\) be the \(k\)-Jacobsthal and \(k\)-Jacobsthal–Lucas numbers, and let \(\Delta_k = \sqrt{k^2 + 8}\) and \(\sigma_k = (k + \Delta_k)/2\). Then, in \(\mathbb{R}\), \(j_{k,n} + \Delta_k J_{k,n} = 2\sigma_k^n\) and \(j_{k,n} - \Delta_k J_{k,n} = 2(-2)^n / \sigma_k^n\), and, in \(\mathbb{Z}\), \(j_{k,n}^2 - (k^2 + 8) J_{k,n}^2 = (-2)^{n+2}\).

public theorem_wanted kJacobsthal_binet_identities (k n : ℕ) :
    jL[k, n] + Δ[k] * J[k, n] = 2 * ((k + Δ[k]) / 2) ^ n ∧
      jL[k, n] - Δ[k] * J[k, n] = 2 * (-2) ^ n / ((k + Δ[k]) / 2) ^ n ∧
      lucasV (kJacobsthalParams k) n ^ 2 - ((k : ℤ) ^ 2 + 8) * lucasU (kJacobsthalParams k) n ^ 2 =
        (-2) ^ (n + 2)

Sources

• Weerayuth Nilsrakoo and Achariya Nilsrakoo, An Integral Representation of k-Jacobsthal and k-Jacobsthal-Lucas Numbers, INTEGERS 25 (2025), Article A74, Lemma 1.

Uses 3

Statement usesLucas parameters of the k-Jacobsthal numbers · Lucas sequence \(U(P, Q)\) · Companion Lucas sequence \(V(P, Q)\)

TheoremAddition formulas for k-Jacobsthal numbers#●ready to prove

Let \(k\), \(m\) and \(n\) be natural numbers and let \(J_{k,n} = U_n(k, -2)\) and \(j_{k,n} = V_n(k, -2)\) be the \(k\)-Jacobsthal and \(k\)-Jacobsthal–Lucas numbers. Then \(2J_{k,m+n} = J_{k,m} j_{k,n} + J_{k,n} j_{k,m}\) and \(2j_{k,m+n} = j_{k,m} j_{k,n} + (k^2 + 8) J_{k,m} J_{k,n}\).

public theorem_wanted kJacobsthal_add_formulas (k m n : ℕ) :
    2 * lucasU (kJacobsthalParams k) (m + n) =
        lucasU (kJacobsthalParams k) m * lucasV (kJacobsthalParams k) n +
          lucasU (kJacobsthalParams k) n * lucasV (kJacobsthalParams k) m ∧
      2 * lucasV (kJacobsthalParams k) (m + n) =
        lucasV (kJacobsthalParams k) m * lucasV (kJacobsthalParams k) n +
          ((k : ℤ) ^ 2 + 8) * lucasU (kJacobsthalParams k) m * lucasU (kJacobsthalParams k) n

Sources

• Weerayuth Nilsrakoo and Achariya Nilsrakoo, An Integral Representation of k-Jacobsthal and k-Jacobsthal-Lucas Numbers, INTEGERS 25 (2025), Article A74, Lemma 2.

Uses 3

Statement usesLucas parameters of the k-Jacobsthal numbers · Lucas sequence \(U(P, Q)\) · Companion Lucas sequence \(V(P, Q)\)

TheoremIntegral representation of k-Jacobsthal numbers at multiples#●ready to prove

Let \(k\), \(\ell\) and \(n\) be natural numbers, let \(J_{k,n} = U_n(k, -2)\) and \(j_{k,n} = V_n(k, -2)\) be the \(k\)-Jacobsthal and \(k\)-Jacobsthal–Lucas numbers, and let \(\Delta_k = \sqrt{k^2 + 8}\). Then

\[J_{k,\ell n} = \frac{n J_{k,\ell}}{2^n} \int_{-1}^{1} (j_{k,\ell} + \Delta_k J_{k,\ell} x)^{n-1} \, dx.\]

Here \(n - 1\) is an integer exponent, so for \(n = 0\) the power is a reciprocal.

public theorem_wanted lucasU_kJacobsthal_mul_eq_integral (k ℓ n : ℕ) :
    (lucasU (kJacobsthalParams k) (ℓ * n) : ℝ) =
      n * J[k, ℓ] / 2 ^ n *
        ∫ x in (-1 : ℝ)..1, (jL[k, ℓ] + Δ[k] * J[k, ℓ] * x) ^ ((n : ℤ) - 1)

Sources

• Weerayuth Nilsrakoo and Achariya Nilsrakoo, An Integral Representation of k-Jacobsthal and k-Jacobsthal-Lucas Numbers, INTEGERS 25 (2025), Article A74, Theorem 1.

Uses 3

Statement usesLucas parameters of the k-Jacobsthal numbers · Lucas sequence \(U(P, Q)\) · Companion Lucas sequence \(V(P, Q)\)

TheoremIntegral representation of even- and odd-indexed k-Jacobsthal numbers#●ready to prove

Let \(k\) and \(n\) be natural numbers, let \(J_{k,n} = U_n(k, -2)\) be the \(k\)-Jacobsthal numbers and let \(\Delta_k = \sqrt{k^2 + 8}\). Then

\[J_{k,2n} = \frac{kn}{2^n} \int_{-1}^{1} (k^2 + 4 + k \Delta_k x)^{n-1} \, dx\]

and

\[J_{k,2n+1} = \frac{1}{2^{n+1}} \int_{-1}^{1} (nk^2 + k^2 + 4 + k(n+1) \Delta_k x) (k^2 + 4 + k \Delta_k x)^{n-1} \, dx.\]

Here \(n - 1\) is an integer exponent, so for \(n = 0\) the power is a reciprocal.

public theorem_wanted lucasU_kJacobsthal_even_odd_eq_integral (k n : ℕ) :
    (lucasU (kJacobsthalParams k) (2 * n) : ℝ) =
        k * n / 2 ^ n * ∫ x in (-1 : ℝ)..1, ((k : ℝ) ^ 2 + 4 + k * Δ[k] * x) ^ ((n : ℤ) - 1) ∧
      (lucasU (kJacobsthalParams k) (2 * n + 1) : ℝ) =
        1 / 2 ^ (n + 1) * ∫ x in (-1 : ℝ)..1,
          (n * (k : ℝ) ^ 2 + k ^ 2 + 4 + k * (n + 1) * Δ[k] * x) *
            ((k : ℝ) ^ 2 + 4 + k * Δ[k] * x) ^ ((n : ℤ) - 1)

Sources

• Weerayuth Nilsrakoo and Achariya Nilsrakoo, An Integral Representation of k-Jacobsthal and k-Jacobsthal-Lucas Numbers, INTEGERS 25 (2025), Article A74, Theorem 2.

Uses 2

Statement usesLucas parameters of the k-Jacobsthal numbers · Lucas sequence \(U(P, Q)\)

TheoremIntegral representation of k-Jacobsthal–Lucas numbers at multiples#●ready to prove

Let \(k\), \(\ell\) and \(n\) be natural numbers, let \(J_{k,n} = U_n(k, -2)\) and \(j_{k,n} = V_n(k, -2)\) be the \(k\)-Jacobsthal and \(k\)-Jacobsthal–Lucas numbers, and let \(\Delta_k = \sqrt{k^2 + 8}\). Then

\[j_{k,\ell n} = \frac{1}{2^n} \int_{-1}^{1} (j_{k,\ell} + (n+1) \Delta_k J_{k,\ell} x) (j_{k,\ell} + \Delta_k J_{k,\ell} x)^{n-1} \, dx.\]

Here \(n - 1\) is an integer exponent, so for \(n = 0\) the power is a reciprocal.

public theorem_wanted lucasV_kJacobsthal_mul_eq_integral (k ℓ n : ℕ) :
    (lucasV (kJacobsthalParams k) (ℓ * n) : ℝ) =
      1 / 2 ^ n * ∫ x in (-1 : ℝ)..1,
        (jL[k, ℓ] + (n + 1) * Δ[k] * J[k, ℓ] * x) * (jL[k, ℓ] + Δ[k] * J[k, ℓ] * x) ^ ((n : ℤ) - 1)

Sources

• Weerayuth Nilsrakoo and Achariya Nilsrakoo, An Integral Representation of k-Jacobsthal and k-Jacobsthal-Lucas Numbers, INTEGERS 25 (2025), Article A74, Theorem 3.

Uses 3

Statement usesLucas parameters of the k-Jacobsthal numbers · Lucas sequence \(U(P, Q)\) · Companion Lucas sequence \(V(P, Q)\)

TheoremIntegral representation of even- and odd-indexed k-Jacobsthal–Lucas numbers#●ready to prove

Let \(k\) and \(n\) be natural numbers, let \(j_{k,n} = V_n(k, -2)\) be the \(k\)-Jacobsthal–Lucas numbers and let \(\Delta_k = \sqrt{k^2 + 8}\). Then

\[j_{k,2n} = \frac{1}{2^n} \int_{-1}^{1} (k^2 + 4 + k(n+1) \Delta_k x) (k^2 + 4 + k \Delta_k x)^{n-1} \, dx\]

and

\[j_{k,2n+1} = \frac{k}{2^{n+1}} \int_{-1}^{1} (n(k^2 + 8) + k^2 + 4 + k(n+1) \Delta_k x) (k^2 + 4 + k \Delta_k x)^{n-1} \, dx.\]

Here \(n - 1\) is an integer exponent, so for \(n = 0\) the power is a reciprocal.

public theorem_wanted lucasV_kJacobsthal_even_odd_eq_integral (k n : ℕ) :
    (lucasV (kJacobsthalParams k) (2 * n) : ℝ) =
        1 / 2 ^ n * ∫ x in (-1 : ℝ)..1,
          ((k : ℝ) ^ 2 + 4 + k * (n + 1) * Δ[k] * x) *
            ((k : ℝ) ^ 2 + 4 + k * Δ[k] * x) ^ ((n : ℤ) - 1) ∧
      (lucasV (kJacobsthalParams k) (2 * n + 1) : ℝ) =
        k / 2 ^ (n + 1) * ∫ x in (-1 : ℝ)..1,
          (n * ((k : ℝ) ^ 2 + 8) + k ^ 2 + 4 + k * (n + 1) * Δ[k] * x) *
            ((k : ℝ) ^ 2 + 4 + k * Δ[k] * x) ^ ((n : ℤ) - 1)

Sources

• Weerayuth Nilsrakoo and Achariya Nilsrakoo, An Integral Representation of k-Jacobsthal and k-Jacobsthal-Lucas Numbers, INTEGERS 25 (2025), Article A74, Theorem 6.

Uses 2

Statement usesLucas parameters of the k-Jacobsthal numbers · Companion Lucas sequence \(V(P, Q)\)

TheoremIntegral representation of Jacobsthal numbers at multiples#●ready to prove

Let \(\ell\) and \(n\) be natural numbers and let \(J_n = U_n(1, -2)\) and \(j_n = V_n(1, -2)\) be the Jacobsthal and Jacobsthal–Lucas numbers (the case \(k = 1\)). Then

\[J_{\ell n} = \frac{n J_\ell}{2^n} \int_{-1}^{1} (j_\ell + 3 J_\ell x)^{n-1} \, dx.\]

Here \(n - 1\) is an integer exponent, so for \(n = 0\) the power is a reciprocal.

public theorem_wanted jacobsthal_mul_eq_integral (ℓ n : ℕ) :
    (lucasU (kJacobsthalParams 1) (ℓ * n) : ℝ) =
      n * J[1, ℓ] / 2 ^ n * ∫ x in (-1 : ℝ)..1, (jL[1, ℓ] + 3 * J[1, ℓ] * x) ^ ((n : ℤ) - 1)

Sources

• Weerayuth Nilsrakoo and Achariya Nilsrakoo, An Integral Representation of k-Jacobsthal and k-Jacobsthal-Lucas Numbers, INTEGERS 25 (2025), Article A74, Corollary 1.

Uses 3

Statement usesLucas parameters of the k-Jacobsthal numbers · Lucas sequence \(U(P, Q)\) · Companion Lucas sequence \(V(P, Q)\)

TheoremIntegral formulas for even- and odd-indexed Jacobsthal numbers#●ready to prove

Let \(J_n = U_n(1, -2)\) be the Jacobsthal numbers (\(J_0 = 0\), \(J_1 = 1\), \(J_{n+2} = J_{n+1} + 2J_n\)). For every natural number \(n\), in \(\mathbb{R}\),

\[ J_{2n} = \frac{n}{2^n} \int_{-1}^{1} (5 + 3x)^{n-1}\,dx \quad\text{and}\quad J_{2n+1} = \frac{1}{2^{n+1}} \int_{-1}^{1} \bigl(n + 5 + 3(n+1)x\bigr)(5 + 3x)^{n-1}\,dx, \]

where \((5 + 3x)^{n-1}\) is an integer power (for \(n = 0\) it is \((5 + 3x)^{-1}\)).

public theorem_wanted jacobsthal_even_odd_eq_integral (n : ℕ) :
    (lucasU (kJacobsthalParams 1) (2 * n) : ℝ) =
        n / 2 ^ n * ∫ x in (-1 : ℝ)..1, (5 + 3 * x) ^ ((n : ℤ) - 1) ∧
      (lucasU (kJacobsthalParams 1) (2 * n + 1) : ℝ) =
        1 / 2 ^ (n + 1) * ∫ x in (-1 : ℝ)..1,
          (n + 5 + 3 * (n + 1) * x) * (5 + 3 * x) ^ ((n : ℤ) - 1)

Sources

• Weerayuth Nilsrakoo and Achariya Nilsrakoo, An Integral Representation of k-Jacobsthal and k-Jacobsthal-Lucas Numbers, INTEGERS 25 (2025), Article A74, Corollary 2.

Uses 2

Statement usesLucas parameters of the k-Jacobsthal numbers · Lucas sequence \(U(P, Q)\)

TheoremIntegral representation of the Jacobsthal-Lucas numbers at multiples of an index#●ready to prove

Let \(J_m = U_m(1, -2)\) be the Jacobsthal numbers and \(j_m = V_m(1, -2)\) the Jacobsthal-Lucas numbers (\(j_0 = 2\), \(j_1 = 1\), \(j_{m+2} = j_{m+1} + 2j_m\)). For all natural numbers \(\ell\) and \(n\), in \(\mathbb{R}\),

\[ j_{\ell n} = \frac{1}{2^n} \int_{-1}^{1} \bigl(j_\ell + 3(n+1) J_\ell\,x\bigr)(j_\ell + 3J_\ell x)^{n-1}\,dx, \]

where \((j_\ell + 3J_\ell x)^{n-1}\) is an integer power (for \(n = 0\) it is the reciprocal).

public theorem_wanted jacobsthalLucas_mul_eq_integral (ℓ n : ℕ) :
    (lucasV (kJacobsthalParams 1) (ℓ * n) : ℝ) =
      1 / 2 ^ n * ∫ x in (-1 : ℝ)..1,
        (jL[1, ℓ] + 3 * (n + 1) * J[1, ℓ] * x) * (jL[1, ℓ] + 3 * J[1, ℓ] * x) ^ ((n : ℤ) - 1)

Sources

• Weerayuth Nilsrakoo and Achariya Nilsrakoo, An Integral Representation of k-Jacobsthal and k-Jacobsthal-Lucas Numbers, INTEGERS 25 (2025), Article A74, Corollary 3.

Uses 3

Statement usesLucas parameters of the k-Jacobsthal numbers · Lucas sequence \(U(P, Q)\) · Companion Lucas sequence \(V(P, Q)\)

TheoremIntegral representation of Jacobsthal numbers with an offset#●ready to prove

Let \(\ell\), \(n\) and \(r\) be natural numbers and let \(J_n = U_n(1, -2)\) and \(j_n = V_n(1, -2)\) be the Jacobsthal and Jacobsthal–Lucas numbers (the case \(k = 1\)). Then

\[J_{\ell n + r} = \frac{1}{2^{n+1}} \int_{-1}^{1} (n J_\ell j_r + J_r j_\ell + 3(n+1) J_\ell J_r x) (j_\ell + 3 J_\ell x)^{n-1} \, dx.\]

Here \(n - 1\) is an integer exponent, so for \(n = 0\) the power is a reciprocal.

public theorem_wanted jacobsthal_mul_add_eq_integral (ℓ n r : ℕ) :
    (lucasU (kJacobsthalParams 1) (ℓ * n + r) : ℝ) =
      1 / 2 ^ (n + 1) * ∫ x in (-1 : ℝ)..1,
        (n * J[1, ℓ] * jL[1, r] + J[1, r] * jL[1, ℓ] +
            3 * (n + 1) * J[1, ℓ] * J[1, r] * x) * (jL[1, ℓ] + 3 * J[1, ℓ] * x) ^ ((n : ℤ) - 1)

Sources

• Weerayuth Nilsrakoo and Achariya Nilsrakoo, An Integral Representation of k-Jacobsthal and k-Jacobsthal-Lucas Numbers, INTEGERS 25 (2025), Article A74, Corollary 4.

Uses 3

Statement usesLucas parameters of the k-Jacobsthal numbers · Lucas sequence \(U(P, Q)\) · Companion Lucas sequence \(V(P, Q)\)

TheoremIntegral representation of the Jacobsthal-Lucas numbers along arithmetic progressions#●ready to prove

Let \(J_m = U_m(1, -2)\) be the Jacobsthal numbers and \(j_m = V_m(1, -2)\) the Jacobsthal-Lucas numbers (\(j_0 = 2\), \(j_1 = 1\), \(j_{m+2} = j_{m+1} + 2j_m\)). For all natural numbers \(\ell\), \(n\) and \(r\), in \(\mathbb{R}\),

\[ j_{\ell n + r} = \frac{1}{2^{n+1}} \int_{-1}^{1} \bigl(9n J_\ell J_r + j_\ell j_r + 3(n+1) J_\ell j_r\,x\bigr)(j_\ell + 3J_\ell x)^{n-1}\,dx, \]

where \((j_\ell + 3J_\ell x)^{n-1}\) is an integer power (for \(n = 0\) it is the reciprocal).

public theorem_wanted jacobsthalLucas_mul_add_eq_integral (ℓ n r : ℕ) :
    (lucasV (kJacobsthalParams 1) (ℓ * n + r) : ℝ) =
      1 / 2 ^ (n + 1) * ∫ x in (-1 : ℝ)..1,
        (9 * n * J[1, ℓ] * J[1, r] + jL[1, ℓ] * jL[1, r] +
            3 * (n + 1) * J[1, ℓ] * jL[1, r] * x) * (jL[1, ℓ] + 3 * J[1, ℓ] * x) ^ ((n : ℤ) - 1)

Sources

• Weerayuth Nilsrakoo and Achariya Nilsrakoo, An Integral Representation of k-Jacobsthal and k-Jacobsthal-Lucas Numbers, INTEGERS 25 (2025), Article A74, Corollary 5.

Uses 3

Statement usesLucas parameters of the k-Jacobsthal numbers · Lucas sequence \(U(P, Q)\) · Companion Lucas sequence \(V(P, Q)\)

TheoremRatio of consecutive \(k\)-Jacobsthal numbers#●ready to prove

For every positive integer \(k\), the \(k\)-Jacobsthal numbers \(J_{k,n} = U_n(k, -2)\), the Lucas sequence \(U\) with parameters \((P, Q) = (k, -2)\), satisfy \(J_{k,n+1}/J_{k,n} \to (k + \sqrt{k^2 + 8})/2\) as \(n \to \infty\), the quotients taken in \(\mathbb{R}\).

public theorem_wanted tendsto_lucasU_kJacobsthal_ratio (k : ℕ) (hk : 0 < k) :
    Filter.Tendsto
      (fun n : ℕ => (lucasU (kJacobsthalParams k) (n + 1) : ℝ) / lucasU (kJacobsthalParams k) n)
      Filter.atTop (nhds ((k + Δ[k]) / 2))

Sources

• Weerayuth Nilsrakoo and Achariya Nilsrakoo, An Integral Representation of k-Jacobsthal and k-Jacobsthal-Lucas Numbers, INTEGERS 25 (2025), Article A74, Theorem 8.

Uses 2

Statement usesLucas parameters of the k-Jacobsthal numbers · Lucas sequence \(U(P, Q)\)

TheoremCompanion \(k\)-Jacobsthal numbers via Lucas sequences#●ready to prove

For a natural number \(k\) and integers \(a, b\), let \(\mathcal{J}_{k,n}(a, b)\) be the companion \(k\)-Jacobsthal numbers, and let \(J_{k,n} = U_n(k, -2)\) and \(j_{k,n} = V_n(k, -2)\) be the \(k\)-Jacobsthal and \(k\)-Jacobsthal–Lucas numbers, the Lucas sequences \(U\) and \(V\) with parameters \((P, Q) = (k, -2)\). Then for every natural number \(n\), \(2\,\mathcal{J}_{k,n}(a, b) = a\,j_{k,n} + (2b - ak)\,J_{k,n}\).

The source states this as \(\mathcal{J}_{k,n} = \tfrac{a}{2} j_{k,n} + \tfrac{2b - ak}{2} J_{k,n}\).

public theorem_wanted two_mul_companionKJacobsthal (k : ℕ) (a b : ℤ) (n : ℕ) :
    2 * companionKJacobsthal k a b n =
      a * lucasV (kJacobsthalParams k) n + (2 * b - a * k) * lucasU (kJacobsthalParams k) n

Sources

• Weerayuth Nilsrakoo and Achariya Nilsrakoo, An Integral Representation of k-Jacobsthal and k-Jacobsthal-Lucas Numbers, INTEGERS 25 (2025), Article A74, Theorem 9.

Uses 4

Statement usesCompanion \(k\)-Jacobsthal numbers · Lucas parameters of the k-Jacobsthal numbers · Lucas sequence \(U(P, Q)\) · Companion Lucas sequence \(V(P, Q)\)

TheoremGenerating function of the companion \(k\)-Jacobsthal numbers#●ready to prove

For every natural number \(k\) and all integers \(a, b\), the companion \(k\)-Jacobsthal numbers \(\mathcal{J}_{k,n} = \mathcal{J}_{k,n}(a, b)\) satisfy, in the ring \(\mathbb{Z}[[x]]\) of formal power series,

\[\Bigl(\sum_{n \ge 0} \mathcal{J}_{k,n} x^n\Bigr)(1 - kx - 2x^2) = a + (b - ak)x.\]

This is the identity \(\sum_{n \ge 0} \mathcal{J}_{k,n} x^n = \frac{a + (b - ak)x}{1 - kx - 2x^2}\) with the denominator cleared.

public theorem_wanted companionKJacobsthal_generatingFunction (k : ℕ) (a b : ℤ) :
    PowerSeries.mk (companionKJacobsthal k a b) *
        (1 - PowerSeries.C (k : ℤ) * PowerSeries.X - PowerSeries.C 2 * PowerSeries.X ^ 2) =
      PowerSeries.C a + PowerSeries.C (b - a * k) * PowerSeries.X

Sources

• Weerayuth Nilsrakoo and Achariya Nilsrakoo, An Integral Representation of k-Jacobsthal and k-Jacobsthal-Lucas Numbers, INTEGERS 25 (2025), Article A74, Theorem 10.

Uses 1

Statement usesCompanion \(k\)-Jacobsthal numbers

TheoremCatalan's identity for the companion \(k\)-Jacobsthal numbers#●ready to prove

Let \(k\) be a natural number, \(a, b\) integers, \(\mathcal{J}_{k,m} = \mathcal{J}_{k,m}(a, b)\) the companion \(k\)-Jacobsthal numbers and \(J_{k,m} = U_m(k, -2)\) the \(k\)-Jacobsthal numbers. For all natural numbers \(r \le n\),

\[ \mathcal{J}_{k,n-r}\,\mathcal{J}_{k,n+r} - \mathcal{J}_{k,n}^2 = \bigl(a^2 (k^2 + 8) - (2b - ak)^2\bigr)\,(-2)^{n-r-2}\,J_{k,r}^2 \]

in \(\mathbb{Q}\), where \((-2)^{n-r-2}\) is an integer power.

With \(\Delta_k = \sqrt{k^2 + 8}\), the first factor on the right equals \((2b - ak + a\Delta_k)(ak - 2b + a\Delta_k)\), the form in which the source states it.

public theorem_wanted companionKJacobsthal_catalan (k : ℕ) (a b : ℤ) (n r : ℕ) (hrn : r ≤ n) :
    ((companionKJacobsthal k a b (n - r) * companionKJacobsthal k a b (n + r) -
        companionKJacobsthal k a b n ^ 2 : ℤ) : ℚ) =
      ((a ^ 2 * ((k : ℤ) ^ 2 + 8) - (2 * b - a * k) ^ 2 : ℤ) : ℚ) *
        (-2 : ℚ) ^ ((n : ℤ) - r - 2) * (lucasU (kJacobsthalParams k) r : ℚ) ^ 2

Sources

• Weerayuth Nilsrakoo and Achariya Nilsrakoo, An Integral Representation of k-Jacobsthal and k-Jacobsthal-Lucas Numbers, INTEGERS 25 (2025), Article A74, Theorem 11.

Uses 3

Statement usesCompanion \(k\)-Jacobsthal numbers · Lucas parameters of the k-Jacobsthal numbers · Lucas sequence \(U(P, Q)\)

TheoremIntegral representation of the companion \(k\)-Jacobsthal numbers#●ready to prove

Let \(k\), \(\ell\), \(n\) be natural numbers and \(a, b\) integers. Write \(\mathcal{J}_{k,m} = \mathcal{J}_{k,m}(a, b)\) for the companion \(k\)-Jacobsthal numbers, \(J_{k,m} = U_m(k, -2)\) and \(j_{k,m} = V_m(k, -2)\) for the \(k\)-Jacobsthal and \(k\)-Jacobsthal-Lucas numbers, and \(\Delta_k = \sqrt{k^2 + 8}\). Then, in \(\mathbb{R}\),

\[ \mathcal{J}_{k,\ell n} = \frac{1}{2^{n+1}} \int_{-1}^{1} \bigl(a\,j_{k,\ell} + (2b - ak)\,n\,J_{k,\ell} + a(n+1)\Delta_k J_{k,\ell}\,x\bigr) \bigl(j_{k,\ell} + \Delta_k J_{k,\ell}\,x\bigr)^{n-1}\,dx, \]

where the power \(n - 1\) is an integer power (for \(n = 0\) it is the reciprocal).

public theorem_wanted companionKJacobsthal_mul_eq_integral (k ℓ n : ℕ) (a b : ℤ) :
    (companionKJacobsthal k a b (ℓ * n) : ℝ) =
      1 / 2 ^ (n + 1) * ∫ x in (-1 : ℝ)..1,
        (a * jL[k, ℓ] + (2 * b - a * k) * n * J[k, ℓ] +
            a * (n + 1) * Δ[k] * J[k, ℓ] * x) * (jL[k, ℓ] + Δ[k] * J[k, ℓ] * x) ^ ((n : ℤ) - 1)

Sources

• Weerayuth Nilsrakoo and Achariya Nilsrakoo, An Integral Representation of k-Jacobsthal and k-Jacobsthal-Lucas Numbers, INTEGERS 25 (2025), Article A74, Theorem 12.

Uses 4

Statement usesCompanion \(k\)-Jacobsthal numbers · Lucas parameters of the k-Jacobsthal numbers · Lucas sequence \(U(P, Q)\) · Companion Lucas sequence \(V(P, Q)\)

7 fully proved16 ready to prove

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