Representation numbers of quaternary quadratic forms

TheoremRepresentation numbers of the quaternary forms \((1,1,10,10)\) and \((2,2,5,5)\)#●ready to prove

For a complex number \(q\) write \(f_k = \prod_{m \ge 1} (1 - q^{km})\). For integers \(a, b, c, d\) and \(n\), let \(N(a,b,c,d;n)\) be the number of \((x_1, x_2, x_3, x_4) \in \mathbb{Z}^4\) with \(a x_1^2 + b x_2^2 + c x_3^2 + d x_4^2 = n\), and let \(\sigma(n/k)\) be the sum of the positive divisors of \(n/k\) if \(k \mid n\) and \(0\) otherwise. Let \(b_1, b_2, b_3\) be sequences of complex numbers indexed by \(n \ge 0\) such that for every complex \(q\) with \(|q| < 1\) the series \(\sum_{n \ge 0} b_i(n) q^n\) converge, with sums \(q f_2^2 f_{10}^2\), \(q^2 f_4^2 f_{20}^2\) and \(q^3 f_4^5 f_{10} f_{40}^2 / (f_2 f_8^2 f_{20})\) for \(i = 1, 2, 3\) respectively. For \(n \ge 1\) put

\[ T(n) = \tfrac{2}{3}\sigma(n) - \tfrac{2}{3}\sigma(n/2) + \tfrac{4}{3}\sigma(n/4) + \tfrac{10}{3}\sigma(n/5) - \tfrac{16}{3}\sigma(n/8) - \tfrac{10}{3}\sigma(n/10) + \tfrac{20}{3}\sigma(n/20) - \tfrac{80}{3}\sigma(n/40). \]

Then for every \(n \ge 1\),

\[ N(1,1,10,10;n) = T(n) + \tfrac{10}{3} b_1(n) + \tfrac{8}{3} b_2(n) + 4b_3(n) \]

and

\[ N(2,2,5,5;n) = T(n) - \tfrac{2}{3} b_1(n) + \tfrac{8}{3} b_2(n) - 4b_3(n). \]

With \(\eta_k = q^{k/24} f_k\), the three generating functions are the eta quotients \(\eta_2^2\eta_{10}^2\), \(\eta_4^2\eta_{20}^2\) and \(\eta_4^5\eta_{10}\eta_{40}^2/(\eta_2\eta_8^2\eta_{20})\). These formulas are due to Alaca and Altiary; Vasuki and Nagendra reprove them elementarily.

MetaMathlibExt.quaternaryRep_one_one_ten_ten_and_two_two_five_five · 19 lines 
public theorem_wanted quaternaryRep_one_one_ten_ten_and_two_two_five_five
    (b₁ b₂ b₃ : ℕ → ℂ)
    (hb₁ : ∀ q : ℂ, ‖q‖ < 1 →
      HasSum (fun n => b₁ n * q ^ n) (q * (𝔣 (q ^ 2)) ^ 2 * (𝔣 (q ^ 10)) ^ 2))
    (hb₂ : ∀ q : ℂ, ‖q‖ < 1 →
      HasSum (fun n => b₂ n * q ^ n) (q ^ 2 * (𝔣 (q ^ 4)) ^ 2 * (𝔣 (q ^ 20)) ^ 2))
    (hb₃ : ∀ q : ℂ, ‖q‖ < 1 →
      HasSum (fun n => b₃ n * q ^ n)
        (q ^ 3 * (𝔣 (q ^ 4)) ^ 5 * 𝔣 (q ^ 10) * (𝔣 (q ^ 40)) ^ 2 /
          (𝔣 (q ^ 2) * (𝔣 (q ^ 8)) ^ 2 * 𝔣 (q ^ 20))))
    (n : ℕ) (hn : 0 < n) :
    𝐍 1 1 10 10 n =
        2 / 3 * σdiv 1 n - 2 / 3 * σdiv 2 n + 4 / 3 * σdiv 4 n + 10 / 3 * σdiv 5 n
          - 16 / 3 * σdiv 8 n - 10 / 3 * σdiv 10 n + 20 / 3 * σdiv 20 n - 80 / 3 * σdiv 40 n
          + 10 / 3 * b₁ n + 8 / 3 * b₂ n + 4 * b₃ n ∧
      𝐍 2 2 5 5 n =
        2 / 3 * σdiv 1 n - 2 / 3 * σdiv 2 n + 4 / 3 * σdiv 4 n + 10 / 3 * σdiv 5 n
          - 16 / 3 * σdiv 8 n - 10 / 3 * σdiv 10 n + 20 / 3 * σdiv 20 n - 80 / 3 * σdiv 40 n
          - 2 / 3 * b₁ n + 8 / 3 * b₂ n - 4 * b₃ n
Sources
  • K. R. Vasuki and P. Nagendra, On Certain Quaternary Quadratic Forms, INTEGERS 25 (2025), Article A24, Theorem 1.1 (Alaca and Altiary).
Uses 1
Statement usesRaw infinite q-Pochhammer product

TheoremRepresentation numbers of the quaternary forms \((1,1,15,15)\), \((3,3,5,5)\) and \((1,3,5,15)\)#●ready to prove

Representation numbers of the quaternary forms \((1,1,15,15)\), \((3,3,5,5)\) and \((1,3,5,15)\). For a complex number \(q\) write \(f_k = \prod_{m \ge 1} (1 - q^{km})\). For integers \(a, b, c, d\) and \(n\), let \(N(a,b,c,d;n)\) be the number of \((x_1, x_2, x_3, x_4) \in \mathbb{Z}^4\) with \(a x_1^2 + b x_2^2 + c x_3^2 + d x_4^2 = n\), and let \(\sigma(n/k)\) be the sum of the positive divisors of \(n/k\) if \(k \mid n\) and \(0\) otherwise. Let \(a_1, \dots, a_5\) be sequences of complex numbers indexed by \(n \ge 0\) such that for every complex \(q\) with \(|q| < 1\) the series \(\sum_{n \ge 0} a_i(n) q^n\) converge, with sums \(q f_1 f_3 f_5 f_{15}\), \(q^2 f_2 f_6 f_{10} f_{30}\), \(q^4 f_4 f_{12} f_{20} f_{60}\), \(q f_3 f_5 f_6 f_{10}\) and \(q^2 f_6 f_{10} f_{12} f_{20}\) for \(i = 1, \dots, 5\) respectively. For \(n \ge 1\) put

\[ S(n) = \tfrac{2}{3}\sigma(n) - \tfrac{4}{3}\sigma(n/2) + \tfrac{8}{3}\sigma(n/4) - 2\sigma(n/3) + 4\sigma(n/6) - 8\sigma(n/12) + \tfrac{10}{3}\sigma(n/5) - \tfrac{20}{3}\sigma(n/10) + \tfrac{40}{3}\sigma(n/20) - 10\sigma(n/15) + 20\sigma(n/30) - 40\sigma(n/60). \]

Then for every \(n \ge 1\),

\[ N(1,1,15,15;n) = S(n) + \tfrac{2}{3}(a_1(n) - 2a_2(n) + 4a_3(n) + 4a_4(n) + 8a_5(n)), \]
\[ N(3,3,5,5;n) = S(n) + \tfrac{2}{3}(-5a_1(n) - 14a_2(n) - 20a_3(n) + 4a_4(n) + 8a_5(n)) \]

and

\[ N(1,3,5,15;n) = \tfrac{1}{2}(\sigma(n) - 2\sigma(n/2) + 4\sigma(n/4) + 3\sigma(n/3) - 6\sigma(n/6) + 12\sigma(n/12) - 5\sigma(n/5) + 10\sigma(n/10) - 20\sigma(n/20) - 15\sigma(n/15) + 30\sigma(n/30) - 60\sigma(n/60)) + \tfrac{3}{2} a_1(n) + a_2(n) + 6a_3(n). \]

With \(\eta_k = q^{k/24} f_k\), the five generating functions are the eta quotients \(\eta_1\eta_3\eta_5\eta_{15}\), \(\eta_2\eta_6\eta_{10}\eta_{30}\), \(\eta_4\eta_{12}\eta_{20}\eta_{60}\), \(\eta_3\eta_5\eta_6\eta_{10}\) and \(\eta_6\eta_{10}\eta_{12}\eta_{20}\). These formulas are due to Alaca; Vasuki and Nagendra reprove them elementarily.

MetaMathlibExt.quaternaryRep_one_one_fifteen_fifteen_three_three_five_five_one_three_five_fifteen · 31 lines 
public theorem_wanted quaternaryRep_one_one_fifteen_fifteen_three_three_five_five_one_three_five_fifteen
    (a₁ a₂ a₃ a₄ a₅ : ℕ → ℂ)
    (ha₁ : ∀ q : ℂ, ‖q‖ < 1 →
      HasSum (fun n => a₁ n * q ^ n) (q * 𝔣 q * 𝔣 (q ^ 3) * 𝔣 (q ^ 5) * 𝔣 (q ^ 15)))
    (ha₂ : ∀ q : ℂ, ‖q‖ < 1 →
      HasSum (fun n => a₂ n * q ^ n)
        (q ^ 2 * 𝔣 (q ^ 2) * 𝔣 (q ^ 6) * 𝔣 (q ^ 10) * 𝔣 (q ^ 30)))
    (ha₃ : ∀ q : ℂ, ‖q‖ < 1 →
      HasSum (fun n => a₃ n * q ^ n)
        (q ^ 4 * 𝔣 (q ^ 4) * 𝔣 (q ^ 12) * 𝔣 (q ^ 20) * 𝔣 (q ^ 60)))
    (ha₄ : ∀ q : ℂ, ‖q‖ < 1 →
      HasSum (fun n => a₄ n * q ^ n) (q * 𝔣 (q ^ 3) * 𝔣 (q ^ 5) * 𝔣 (q ^ 6) * 𝔣 (q ^ 10)))
    (ha₅ : ∀ q : ℂ, ‖q‖ < 1 →
      HasSum (fun n => a₅ n * q ^ n)
        (q ^ 2 * 𝔣 (q ^ 6) * 𝔣 (q ^ 10) * 𝔣 (q ^ 12) * 𝔣 (q ^ 20)))
    (n : ℕ) (hn : 0 < n) :
    𝐍 1 1 15 15 n =
        2 / 3 * σdiv 1 n - 4 / 3 * σdiv 2 n + 8 / 3 * σdiv 4 n - 2 * σdiv 3 n
          + 4 * σdiv 6 n - 8 * σdiv 12 n + 10 / 3 * σdiv 5 n - 20 / 3 * σdiv 10 n
          + 40 / 3 * σdiv 20 n - 10 * σdiv 15 n + 20 * σdiv 30 n - 40 * σdiv 60 n
          + 2 / 3 * (a₁ n - 2 * a₂ n + 4 * a₃ n + 4 * a₄ n + 8 * a₅ n) ∧
      𝐍 3 3 5 5 n =
        2 / 3 * σdiv 1 n - 4 / 3 * σdiv 2 n + 8 / 3 * σdiv 4 n - 2 * σdiv 3 n
          + 4 * σdiv 6 n - 8 * σdiv 12 n + 10 / 3 * σdiv 5 n - 20 / 3 * σdiv 10 n
          + 40 / 3 * σdiv 20 n - 10 * σdiv 15 n + 20 * σdiv 30 n - 40 * σdiv 60 n
          + 2 / 3 * (-5 * a₁ n - 14 * a₂ n - 20 * a₃ n + 4 * a₄ n + 8 * a₅ n) ∧
      𝐍 1 3 5 15 n =
        1 / 2 * (σdiv 1 n - 2 * σdiv 2 n + 4 * σdiv 4 n + 3 * σdiv 3 n - 6 * σdiv 6 n
            + 12 * σdiv 12 n - 5 * σdiv 5 n + 10 * σdiv 10 n - 20 * σdiv 20 n
            - 15 * σdiv 15 n + 30 * σdiv 30 n - 60 * σdiv 60 n)
          + 3 / 2 * a₁ n + a₂ n + 6 * a₃ n
Sources
  • K. R. Vasuki and P. Nagendra, On Certain Quaternary Quadratic Forms, INTEGERS 25 (2025), Article A24, Theorem 1.2 (Alaca).
Uses 1
Statement usesRaw infinite q-Pochhammer product

TheoremA theta identity for \(\varphi(q)^2\varphi(q^{10})^2 + 5\varphi(q^2)^2\varphi(q^5)^2\)#●ready to prove

Let \(q\) be a complex number with \(|q| < 1\). Write \(\varphi(x) = \sum_{n \in \mathbb{Z}} x^{n^2}\) and \(\psi(x) = \sum_{n \ge 0} x^{n(n+1)/2}\) for Ramanujan's theta functions, and \(f_k = \prod_{m \ge 1} (1 - q^{km})\). Then

\[ \varphi(q)^2\varphi(q^{10})^2 + 5\varphi(q^2)^2\varphi(q^5)^2 = \varphi(q^2)^4 + 5\varphi(q^{10})^4 + 4q\psi(q^2)^4 + 20q^5\psi(q^{10})^4 + 16q^2 f_4^2 f_{20}^2 - 16q^3 \frac{f_4^5 f_{10} f_{40}^2}{f_2 f_8^2 f_{20}}. \]

The power \(q^3\) of the last term is not legible in the available text of the paper; it is the one consistent with Theorem 1.1 and with Equation (3.13).

public theorem_wanted phi_sq_phi_sq_ten_add_five_phi_sq_two_phi_sq_five
    (q : ℂ) (hq : ‖q‖ < 1) :
    (φ q) ^ 2 * (φ (q ^ 10)) ^ 2 + 5 * (φ (q ^ 2)) ^ 2 * (φ (q ^ 5)) ^ 2 =
      (φ (q ^ 2)) ^ 4 + 5 * (φ (q ^ 10)) ^ 4 + 4 * q * (ψ (q ^ 2)) ^ 4
        + 20 * q ^ 5 * (ψ (q ^ 10)) ^ 4 + 16 * q ^ 2 * (𝔣 (q ^ 4)) ^ 2 * (𝔣 (q ^ 20)) ^ 2
        - 16 * q ^ 3 * ((𝔣 (q ^ 4)) ^ 5 * 𝔣 (q ^ 10) * (𝔣 (q ^ 40)) ^ 2
          / (𝔣 (q ^ 2) * (𝔣 (q ^ 8)) ^ 2 * 𝔣 (q ^ 20)))
Sources
  • K. R. Vasuki and P. Nagendra, On Certain Quaternary Quadratic Forms, INTEGERS 25 (2025), Article A24, Lemma 3.1.
Uses 2
Statement usesRaw infinite q-Pochhammer product · Raw Ramanujan theta sum

TheoremEisenstein-series formula for \(\varphi(q)^2\varphi(q^{10})^2\)#●ready to prove

Let \(q\) be a complex number with \(|q| < 1\). Write \(\varphi(x) = \sum_{n \in \mathbb{Z}} x^{n^2}\) for Ramanujan's theta function, \(f_k = \prod_{m \ge 1} (1 - q^{km})\), and \(P_k = P(q^k)\), where \(P(x) = 1 - 24 \sum_{m \ge 1} m x^m / (1 - x^m)\) is the weight-two Eisenstein series. Then

\[ \varphi(q)^2\varphi(q^{10})^2 = -\tfrac{1}{36}(P_1 - P_2 + 2P_4 + 5P_5 - 8P_8 - 5P_{10} + 10P_{20} - 40P_{40}) + \tfrac{10}{3} q f_2^2 f_{10}^2 + \tfrac{8}{3} q^2 f_4^2 f_{20}^2 + 4q^3 \frac{f_4^5 f_{10} f_{40}^2}{f_2 f_8^2 f_{20}}. \]
public theorem_wanted phi_sq_mul_phi_sq_ten_eisenstein (q : ℂ) (hq : ‖q‖ < 1) :
    (φ q) ^ 2 * (φ (q ^ 10)) ^ 2 =
      -(1 / 36) * (𝐏 q - 𝐏 (q ^ 2) + 2 * 𝐏 (q ^ 4) + 5 * 𝐏 (q ^ 5) - 8 * 𝐏 (q ^ 8)
          - 5 * 𝐏 (q ^ 10) + 10 * 𝐏 (q ^ 20) - 40 * 𝐏 (q ^ 40))
        + 10 / 3 * q * (𝔣 (q ^ 2)) ^ 2 * (𝔣 (q ^ 10)) ^ 2
        + 8 / 3 * q ^ 2 * (𝔣 (q ^ 4)) ^ 2 * (𝔣 (q ^ 20)) ^ 2
        + 4 * q ^ 3 * ((𝔣 (q ^ 4)) ^ 5 * 𝔣 (q ^ 10) * (𝔣 (q ^ 40)) ^ 2
          / (𝔣 (q ^ 2) * (𝔣 (q ^ 8)) ^ 2 * 𝔣 (q ^ 20)))
Sources
  • K. R. Vasuki and P. Nagendra, On Certain Quaternary Quadratic Forms, INTEGERS 25 (2025), Article A24, Lemma 3.3, Equation (3.13).
Uses 2
Statement usesRaw infinite q-Pochhammer product · Raw Ramanujan theta sum

TheoremEisenstein-series formula for \(\varphi(q^2)^2\varphi(q^5)^2\)#●ready to prove

Let \(q\) be a complex number with \(|q| < 1\). Write \(\varphi(x) = \sum_{n \in \mathbb{Z}} x^{n^2}\) for Ramanujan's theta function, \(f_k = \prod_{m \ge 1} (1 - q^{km})\), and \(P_k = P(q^k)\), where \(P(x) = 1 - 24 \sum_{m \ge 1} m x^m / (1 - x^m)\) is the weight-two Eisenstein series. Then

\[ \varphi(q^2)^2\varphi(q^5)^2 = -\tfrac{1}{36}(P_1 - P_2 + 2P_4 + 5P_5 - 8P_8 - 5P_{10} + 10P_{20} - 40P_{40}) - \tfrac{2}{3} q f_2^2 f_{10}^2 + \tfrac{8}{3} q^2 f_4^2 f_{20}^2 - 4q^3 \frac{f_4^5 f_{10} f_{40}^2}{f_2 f_8^2 f_{20}}. \]
public theorem_wanted phi_sq_two_mul_phi_sq_five_eisenstein (q : ℂ) (hq : ‖q‖ < 1) :
    (φ (q ^ 2)) ^ 2 * (φ (q ^ 5)) ^ 2 =
      -(1 / 36) * (𝐏 q - 𝐏 (q ^ 2) + 2 * 𝐏 (q ^ 4) + 5 * 𝐏 (q ^ 5) - 8 * 𝐏 (q ^ 8)
          - 5 * 𝐏 (q ^ 10) + 10 * 𝐏 (q ^ 20) - 40 * 𝐏 (q ^ 40))
        - 2 / 3 * q * (𝔣 (q ^ 2)) ^ 2 * (𝔣 (q ^ 10)) ^ 2
        + 8 / 3 * q ^ 2 * (𝔣 (q ^ 4)) ^ 2 * (𝔣 (q ^ 20)) ^ 2
        - 4 * q ^ 3 * ((𝔣 (q ^ 4)) ^ 5 * 𝔣 (q ^ 10) * (𝔣 (q ^ 40)) ^ 2
          / (𝔣 (q ^ 2) * (𝔣 (q ^ 8)) ^ 2 * 𝔣 (q ^ 20)))
Sources
  • K. R. Vasuki and P. Nagendra, On Certain Quaternary Quadratic Forms, INTEGERS 25 (2025), Article A24, Lemma 3.3, Equation (3.14).
Uses 2
Statement usesRaw infinite q-Pochhammer product · Raw Ramanujan theta sum

TheoremA theta identity for \(12\varphi(q)^2\varphi(q^{10})^2 + 12\varphi(q^2)^2\varphi(q^5)^2\)#●ready to prove

Let \(q\) be a complex number with \(|q| < 1\). Write \(\varphi(x) = \sum_{n \in \mathbb{Z}} x^{n^2}\) for Ramanujan's theta function and \(f_k = \prod_{m \ge 1} (1 - q^{km})\). Then

\[ 12\varphi(q)^2\varphi(q^{10})^2 + 12\varphi(q^2)^2\varphi(q^5)^2 = \varphi(q)^4 + 5\varphi(q^5)^4 + 4\varphi(q^2)^4 + 20\varphi(q^{10})^4 - \varphi(-q)^4 - 5\varphi(-q^5)^4 + 32q f_2^2 f_{10}^2 + 64q^2 f_4^2 f_{20}^2. \]
public theorem_wanted twelve_phi_sq_phi_sq_ten_add_twelve_phi_sq_two_phi_sq_five
    (q : ℂ) (hq : ‖q‖ < 1) :
    12 * (φ q) ^ 2 * (φ (q ^ 10)) ^ 2 + 12 * (φ (q ^ 2)) ^ 2 * (φ (q ^ 5)) ^ 2 =
      (φ q) ^ 4 + 5 * (φ (q ^ 5)) ^ 4 + 4 * (φ (q ^ 2)) ^ 4 + 20 * (φ (q ^ 10)) ^ 4
        - (φ (-q)) ^ 4 - 5 * (φ (-q ^ 5)) ^ 4
        + 32 * q * (𝔣 (q ^ 2)) ^ 2 * (𝔣 (q ^ 10)) ^ 2
        + 64 * q ^ 2 * (𝔣 (q ^ 4)) ^ 2 * (𝔣 (q ^ 20)) ^ 2
Sources
  • K. R. Vasuki and P. Nagendra, On Certain Quaternary Quadratic Forms, INTEGERS 25 (2025), Article A24, Lemma 3.2.
Uses 2
Statement usesRaw infinite q-Pochhammer product · Raw Ramanujan theta sum

TheoremEisenstein-series formula for \(\varphi(q)^2\varphi(q^{15})^2\)#●ready to prove

Let \(q\) be a complex number with \(|q| < 1\). Write \(\varphi(x) = \sum_{n \in \mathbb{Z}} x^{n^2}\) for Ramanujan's theta function, \(f_k = \prod_{m \ge 1} (1 - q^{km})\), and \(P_k = P(q^k)\), where \(P(x) = 1 - 24 \sum_{m \ge 1} m x^m / (1 - x^m)\) is the weight-two Eisenstein series. Then

\[ \varphi(q)^2\varphi(q^{15})^2 = -\tfrac{1}{36}(P_1 - 2P_2 - 3P_3 + 4P_4 + 5P_5 + 6P_6 - 10P_{10} - 12P_{12} - 15P_{15} + 20P_{20} + 30P_{30} - 60P_{60}) + \tfrac{2}{3}(q f_1 f_3 f_5 f_{15} - 2q^2 f_2 f_6 f_{10} f_{30} + 4q^4 f_4 f_{12} f_{20} f_{60} + 4q f_3 f_5 f_6 f_{10} + 8q^2 f_6 f_{10} f_{12} f_{20}). \]
public theorem_wanted phi_sq_mul_phi_sq_fifteen_eisenstein (q : ℂ) (hq : ‖q‖ < 1) :
    (φ q) ^ 2 * (φ (q ^ 15)) ^ 2 =
      -(1 / 36) * (𝐏 q - 2 * 𝐏 (q ^ 2) - 3 * 𝐏 (q ^ 3) + 4 * 𝐏 (q ^ 4) + 5 * 𝐏 (q ^ 5)
          + 6 * 𝐏 (q ^ 6) - 10 * 𝐏 (q ^ 10) - 12 * 𝐏 (q ^ 12) - 15 * 𝐏 (q ^ 15)
          + 20 * 𝐏 (q ^ 20) + 30 * 𝐏 (q ^ 30) - 60 * 𝐏 (q ^ 60))
        + 2 / 3 * (q * 𝔣 q * 𝔣 (q ^ 3) * 𝔣 (q ^ 5) * 𝔣 (q ^ 15)
          - 2 * q ^ 2 * 𝔣 (q ^ 2) * 𝔣 (q ^ 6) * 𝔣 (q ^ 10) * 𝔣 (q ^ 30)
          + 4 * q ^ 4 * 𝔣 (q ^ 4) * 𝔣 (q ^ 12) * 𝔣 (q ^ 20) * 𝔣 (q ^ 60)
          + 4 * q * 𝔣 (q ^ 3) * 𝔣 (q ^ 5) * 𝔣 (q ^ 6) * 𝔣 (q ^ 10)
          + 8 * q ^ 2 * 𝔣 (q ^ 6) * 𝔣 (q ^ 10) * 𝔣 (q ^ 12) * 𝔣 (q ^ 20))
Sources
  • K. R. Vasuki and P. Nagendra, On Certain Quaternary Quadratic Forms, INTEGERS 25 (2025), Article A24, Lemma 4.2.
Uses 2
Statement usesRaw infinite q-Pochhammer product · Raw Ramanujan theta sum

TheoremA product identity for Ramanujan's \(f(q)f(q^3)f(q^5)f(q^{15})\)#●ready to prove

Let \(q\) be a complex number with \(|q| < 1\). Write \(f_k = \prod_{m \ge 1} (1 - q^{km})\), and let \(f(x) = \prod_{m \ge 1} (1 - (-x)^m) = (-x; -x)_\infty\) be Ramanujan's function. Then

\[ q f(q) f(q^3) f(q^5) f(q^{15}) = q f_1 f_3 f_5 f_{15} + 2q^2 f_2 f_6 f_{10} f_{30} + 4q^4 f_4 f_{12} f_{20} f_{60}. \]
public theorem_wanted q_mul_ramanujanF_one_three_five_fifteen (q : ℂ) (hq : ‖q‖ < 1) :
    q * 𝔣 (-q) * 𝔣 (-q ^ 3) * 𝔣 (-q ^ 5) * 𝔣 (-q ^ 15) =
      q * 𝔣 q * 𝔣 (q ^ 3) * 𝔣 (q ^ 5) * 𝔣 (q ^ 15)
        + 2 * q ^ 2 * 𝔣 (q ^ 2) * 𝔣 (q ^ 6) * 𝔣 (q ^ 10) * 𝔣 (q ^ 30)
        + 4 * q ^ 4 * 𝔣 (q ^ 4) * 𝔣 (q ^ 12) * 𝔣 (q ^ 20) * 𝔣 (q ^ 60)
Sources
  • K. R. Vasuki and P. Nagendra, On Certain Quaternary Quadratic Forms, INTEGERS 25 (2025), Article A24, Lemma 4.3.
Uses 1
Statement usesRaw infinite q-Pochhammer product

TheoremEisenstein-series formula for \(\varphi(q^3)^2\varphi(q^5)^2\)#●ready to prove

Let \(q\) be a complex number with \(|q| < 1\). Write \(\varphi(x) = \sum_{n \in \mathbb{Z}} x^{n^2}\) for Ramanujan's theta function, \(f_k = \prod_{m \ge 1} (1 - q^{km})\), and \(P_k = P(q^k)\), where \(P(x) = 1 - 24 \sum_{m \ge 1} m x^m / (1 - x^m)\) is the weight-two Eisenstein series. Then

\[ \varphi(q^3)^2\varphi(q^5)^2 = -\tfrac{1}{36}(P_1 - 2P_2 - 3P_3 + 4P_4 + 5P_5 + 6P_6 - 10P_{10} - 12P_{12} - 15P_{15} + 20P_{20} + 30P_{30} - 60P_{60}) - \tfrac{2}{3}(5q f_1 f_3 f_5 f_{15} + 14q^2 f_2 f_6 f_{10} f_{30} + 20q^4 f_4 f_{12} f_{20} f_{60} - 4q f_3 f_5 f_6 f_{10} - 8q^2 f_6 f_{10} f_{12} f_{20}). \]
public theorem_wanted phi_sq_three_mul_phi_sq_five_eisenstein (q : ℂ) (hq : ‖q‖ < 1) :
    (φ (q ^ 3)) ^ 2 * (φ (q ^ 5)) ^ 2 =
      -(1 / 36) * (𝐏 q - 2 * 𝐏 (q ^ 2) - 3 * 𝐏 (q ^ 3) + 4 * 𝐏 (q ^ 4) + 5 * 𝐏 (q ^ 5)
          + 6 * 𝐏 (q ^ 6) - 10 * 𝐏 (q ^ 10) - 12 * 𝐏 (q ^ 12) - 15 * 𝐏 (q ^ 15)
          + 20 * 𝐏 (q ^ 20) + 30 * 𝐏 (q ^ 30) - 60 * 𝐏 (q ^ 60))
        - 2 / 3 * (5 * q * 𝔣 q * 𝔣 (q ^ 3) * 𝔣 (q ^ 5) * 𝔣 (q ^ 15)
          + 14 * q ^ 2 * 𝔣 (q ^ 2) * 𝔣 (q ^ 6) * 𝔣 (q ^ 10) * 𝔣 (q ^ 30)
          + 20 * q ^ 4 * 𝔣 (q ^ 4) * 𝔣 (q ^ 12) * 𝔣 (q ^ 20) * 𝔣 (q ^ 60)
          - 4 * q * 𝔣 (q ^ 3) * 𝔣 (q ^ 5) * 𝔣 (q ^ 6) * 𝔣 (q ^ 10)
          - 8 * q ^ 2 * 𝔣 (q ^ 6) * 𝔣 (q ^ 10) * 𝔣 (q ^ 12) * 𝔣 (q ^ 20))
Sources
  • K. R. Vasuki and P. Nagendra, On Certain Quaternary Quadratic Forms, INTEGERS 25 (2025), Article A24, Lemma 4.4.
Uses 2
Statement usesRaw infinite q-Pochhammer product · Raw Ramanujan theta sum

TheoremEisenstein-series formula for \(\varphi(q)\varphi(q^3)\varphi(q^5)\varphi(q^{15})\)#●ready to prove

Let \(q\) be a complex number with \(|q| < 1\). Write \(\varphi(x) = \sum_{n \in \mathbb{Z}} x^{n^2}\) for Ramanujan's theta function, \(f_k = \prod_{m \ge 1} (1 - q^{km})\), and \(P_k = P(q^k)\), where \(P(x) = 1 - 24 \sum_{m \ge 1} m x^m / (1 - x^m)\) is the weight-two Eisenstein series. Then

\[ \varphi(q)\varphi(q^3)\varphi(q^5)\varphi(q^{15}) = -\tfrac{1}{48}(P_1 - 2P_2 + 3P_3 + 4P_4 - 5P_5 - 6P_6 + 10P_{10} + 12P_{12} - 15P_{15} - 20P_{20} + 30P_{30} - 60P_{60}) + \tfrac{3}{2} q f_1 f_3 f_5 f_{15} + q^2 f_2 f_6 f_{10} f_{30} + 6q^4 f_4 f_{12} f_{20} f_{60}. \]
public theorem_wanted phi_mul_phi_three_mul_phi_five_mul_phi_fifteen_eisenstein
    (q : ℂ) (hq : ‖q‖ < 1) :
    φ q * φ (q ^ 3) * φ (q ^ 5) * φ (q ^ 15) =
      -(1 / 48) * (𝐏 q - 2 * 𝐏 (q ^ 2) + 3 * 𝐏 (q ^ 3) + 4 * 𝐏 (q ^ 4) - 5 * 𝐏 (q ^ 5)
          - 6 * 𝐏 (q ^ 6) + 10 * 𝐏 (q ^ 10) + 12 * 𝐏 (q ^ 12) - 15 * 𝐏 (q ^ 15)
          - 20 * 𝐏 (q ^ 20) + 30 * 𝐏 (q ^ 30) - 60 * 𝐏 (q ^ 60))
        + 3 / 2 * q * 𝔣 q * 𝔣 (q ^ 3) * 𝔣 (q ^ 5) * 𝔣 (q ^ 15)
        + q ^ 2 * 𝔣 (q ^ 2) * 𝔣 (q ^ 6) * 𝔣 (q ^ 10) * 𝔣 (q ^ 30)
        + 6 * q ^ 4 * 𝔣 (q ^ 4) * 𝔣 (q ^ 12) * 𝔣 (q ^ 20) * 𝔣 (q ^ 60)
Sources
  • K. R. Vasuki and P. Nagendra, On Certain Quaternary Quadratic Forms, INTEGERS 25 (2025), Article A24, Lemma 4.6.
Uses 2
Statement usesRaw infinite q-Pochhammer product · Raw Ramanujan theta sum
10 ready to prove

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