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It is a special case of Hypergeometric2F1 and JacobiP and satisfies a second-order ordinary differential equation in z . … The GegenbauerC[n, 1/2, z] obtained for d = 3 are LegendreP[n, z] .

For sequences involving only two distinct integers flat spectra are rare; with ± 1 those equivalent to {1, 1, 1, -1} seem to be the only examples. ( {r 2 , r s, s 2 , -r s} works for any r and s , as do all lists obtained working modulo x n - 1 from p[x]/p[1/x] where p[x] is any invertible polynomial.) … It is also flat for maximal length LFSR sequences (see page 1084 ) and for sequences JacobiSymbol[Range[0, p - 1], p] with prime p (see page 870 ). … If Mod[p, 4] 1 JacobiSymbol sequences also satisfy Fourier[list] list .

The so-called Paley family of Hadamard matrices for n = 4k = p + 1 with p prime are given by
PadLeft[Array[JacobiSymbol[#2 - #1, n - 1]&, {n, n} - 1] - IdentityMatrix[n - 1], {n, n}, 1]
Originally introduced by Jacques Hadamard in 1893 as the matrices with elements Abs[a] ≤ 1 which attain the maximal possible determinant ± n n/2 , Hadamard matrices appear in various combinatorial problems, particularly design of exhaustive combinations of experiments and Reed–Muller error-correcting codes.

Quadratic residue sequences
As an outgrowth of ideas related to RSA cryptography it was shown in 1982 by Lenore Blum , Manuel Blum and Michael Shub that the sequence
Mod[NestList[Mod[# 2 , m] &, x0, n], 2]
discussed on page 975 has the property that if m=p q with p and q primes (congruent to 3 modulo 4) then any systematic regularities detected in the sequence can eventually be used to discover factors of m . … If m is a prime p , then the simple tests JacobiSymbol[x, p] 1 (see page 1081 ) or Mod[x (p - 1)/2 , p] 1 determine whether x is a quadratic residue. But with m = p q , one has to factor m and find p and q in order to carry out similar tests.

[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 ).