Elliptic theta and elliptic modular functions are exceptions. … Like other elliptic modular functions, ModularLambda satisfies f[z]  f[(a + b z)/(c + d z)] with a , b , c , d integers such that a c - b d  1 . … Nested behavior is also found for example in EllipticTheta[3, 0, z] , which is given essentially by Sum[z n 2 , {n, ∞ }] .

The number with run lengths corresponding to successive integers (so that the n th digit is Mod[Floor[1/2 + Sqrt[2n]], 2] ) turns out to be (1 - 2 1/4 EllipticTheta[2, 0, 1/2] + EllipticTheta[3, 0, 1/2])/2 , and appears at least not to be algebraic.

[History of] exact solutions Some notable cases where closed-form analytical results have been found in terms of standard mathematical functions include: quadratic equations (~2000 BC) ( Sqrt ); cubic, quartic equations (1530s) ( x 1/n ); 2-body problem (1687) ( Cos ); catenary (1690) ( Cosh ); brachistochrone (1696) ( Sin ); spinning top (1849; 1888; 1888) ( JacobiSN ; WeierstrassP ; hyperelliptic functions); quintic equations (1858) ( EllipticTheta ); half-plane diffraction (1896) ( FresnelC ); Mie scattering (1908) ( BesselJ , BesselY , LegendreP ); Einstein equations (Schwarzschild (1916), Reissner–Nordström (1916), Kerr (1963) solutions) (rational and trigonometric functions); quantum hydrogen atom and harmonic oscillator (1927) ( LaguerreL , HermiteH ); 2D Ising model (1944) ( Sinh , EllipticK ); various Feynman diagrams (1960s-1980s) ( PolyLog ); KdV equation (1967) ( Sech etc.); Toda lattice (1967) ( Sech ); six-vertex spin model (1967) ( Sinh integrals); Calogero–Moser model (1971) ( Hypergeometric1F1 ); Yang–Mills instantons (1975) (rational functions); hard-hexagon spin model (1979) ( EllipticTheta ); additive cellular automata (1984) ( MultiplicativeOrder ); Seiberg–Witten supersymmetric theory (1994) ( Hypergeometric2F1 ).

For degrees 5 and 6 it was shown in the late 1800s that EllipticTheta or Hypergeometric2F1 are sufficient, although for degrees 5 and 6 respectively the necessary formulas have a LeafCount in the billions.

EllipticTheta[3, π , x]/2 gives a sequence with 1's at positions m 2 .

In the mid-1800s it became clear that despite their different origins most of these functions could be viewed as special cases of Hypergeometric2F1[a, b, c, z] , and that the functions covered the solutions to all linear differential equations of a certain type. ( Zeta and PolyLog are parametric derivatives of Hypergeometric2F1 ; elliptic modular functions are inverses.) … Thus, for example, generalized elliptic theta functions represent solutions to arbitrary polynomial equations, while multivariate hypergeometric functions represent arbitrary conformal mappings.