Theta functions and q-series

Jacobi and Ramanujan theta functions, q-Pochhammer symbols and Euler products, Rogers–Ramanujan and related continued fractions, mock theta functions, Lambert series and q-series identities.

• Glaisher's divisor functions and infinite products
• Hyperbolicity of Jensen polynomials
• Lambert series
• q-Pochhammer symbol
• Coefficients of mock theta functions
• Jacobi triple product
• Ramanujan's theta function
• Ramanujan's continued fractions of order sixteen
• Ramanujan's function k
• Rogers–Ramanujan continued fraction
• Rogers–Ramanujan–Slater functions of order eighteen

TheoremJacobi's identity for the cube of Euler's product#✓fully proved

For every complex number \(q\) with \(|q| < 1\), the series \(\sum_{n=0}^{\infty} (-1)^n (2n+1) q^{n(n+1)/2}\) converges unconditionally to \((q;q)_\infty^3\), the cube of the infinite q-Pochhammer symbol \((q;q)_\infty = \prod_{n=0}^{\infty} (1 - q^{n+1})\).

theorem jacobi_cube_euler_product_hasSum (q : Complex) (hq : ‖q‖ < 1) :
    HasSum (fun n : Nat => (-1 : Complex) ^ n * ((2 * n + 1 : Nat) : Complex) *
      q ^ (n * (n + 1) / 2)) ((qPochhammerInf q q hq) ^ 3)

Sources

• Robson da Silva and James A. Sellers, "Parity Considerations for the Mex-Related Partition Functions of Andrews and Newman", https://cs.uwaterloo.ca/journals/JIS/VOL23/Sellers/sellers52.tex, lines 150-161 (equation Jacobi of the lemma; Eq. (1.3.24) of Berndt).

Uses 6

Statement usesInfinite q-Pochhammer symbol

Proof usesRaw infinite q-Pochhammer product · Vanishing coefficients of Jacobi's cube series · Coefficients of Jacobi's cube series at triangular numbers · Jacobi's cube series · Jacobi's cube identity

The dependency explorer needs JavaScript and its graph data.

Download the complete graph data.