It is a special case of Hypergeometric2F1 and JacobiP and satisfies a second-order ordinary differential equation in z .

Trinomial coefficients The coefficient of x n in the expansion of (1 + x + x 2 ) t is Sum[Binomial[n + t - 1 - 3k, n - 3k] Binomial[t, k] (-1) k , {k, 0, t}] which can be evaluated as Binomial[2t, n] Hypergeometric2F1[-n, n - 2t, 1/2 - t, 1/4] or finally GegenbauerC[n, -t, -1/2] .

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.

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.)

For the Thue–Morse sequence the result is 1/2 (-1) n + ((-3) n √ π Hypergeometric2F1[3/2, -n, 3/2 - n, -(1/3)])/(4 n!

.); 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 ).

Such a circle has area 2 π a 2 (1 - Cos[r/a]) = π r 2 (1 - r 2 /(12 a 2 ) + r 4 /(360a 4 ) - …) In the d -dimensional space corresponding to the surface of a (d + 1) -dimensional sphere of radius a , the volume of a d -dimensional sphere of radius r is similarly given by d s[d] a d Integrate[Sin[ θ ] d - 1 , { θ ,0, r/a}] = s[d] r d (1 - d (d - 1) r 2 /((6 (d + 2))a 2 + (d (5d 2 - 12d + 7))r 4 /((360 (d + 4))a 4 ) …) where Integrate[Sin[x] d - 1 , x] = -Cos[x] Hypergeometric2F1[1/2, (2 - d)/2, 3/2, Cos[x] 2 ] In an arbitrary d -dimensional space the volume of a sphere can depend on position, but in general it is given by s[d] r d (1 - RicciScalar r 2 /(6(d + 2)) + …) where the Ricci scalar curvature is evaluated at the position of the sphere.