A261975 Expansion of elliptic_E / elliptic_K in powers of q.

1, -8, 48, -224, 864, -2928, 9024, -25792, 69312, -176936, 432288, -1016736, 2312832, -5107504, 10983552, -23060544, 47373696, -95401872, 188637936, -366744160, 701930304, -1324016896, 2463662016, -4526174784, 8216376576, -14747939768, 26191413024, -46048199360

Offset: 0

Author

Joerg Arndt, Sep 07 2015

Keywords

• sign

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A004018 (elliptic K(q)), A194094 (elliptic E(q)), A115977 (elliptic k(q)^2).

Cf. A261975 (E/K), A261977 ((E/K)^(1/2)), A261978 ((E/K)^(1/4)).

Cf. A261976 (K/E), A261979 ((K/E)^(1/2)), A261980 ((K/E)^(1/4)).

Programs

Mathematica

nmax = 30; dtheta = D[Normal[Series[EllipticTheta[3, 0, x], {x, 0, nmax}]], x]; CoefficientList[Series[(EllipticTheta[4, 0, x]^4*EllipticTheta[3, 0, x] + 4*x*dtheta)/EllipticTheta[3, 0, x]^5, {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 10 2018 *)

Formula

G.f.: (T4^4 * T3 + 4*q * d/dq T3) / T3^5 where T3 = theta_3(q) and T4 = theta_4(q).