A115977 Expansion of elliptic modular function lambda in powers of the nome q.

16, -128, 704, -3072, 11488, -38400, 117632, -335872, 904784, -2320128, 5702208, -13504512, 30952544, -68901888, 149403264, -316342272, 655445792, -1331327616, 2655115712, -5206288384, 10049485312, -19115905536, 35867019904, -66437873664

Offset

1,1

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

References

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 591.

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 121.

A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, p. 23, eq. (37).

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from G. C. Greubel)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

A. Dieckmann, Collection of Infinite Products and Series

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Eric Weisstein's World of Mathematics, Elliptic Lambda Function

Wolfram Research Basic Algebraic Identities Relations involving squares, 1st formula

Formula

Expansion of Jacobi elliptic parameter m = k^2 = (theta_2(q) / theta_3(q))^4 in powers of the nome q.

Expansion of 16 * q * (psi(q^2) / phi(q))^4 = 16 * q * (psi(q^2) / psi(q))^8 = 16 * q * (psi(q) / phi(q))^8 = 16 * q * (psi(-q) / phi(-q^2))^8 = 16 * q / (chi(q) * chi(-q^2))^8 = 16 * q * (f(-q^4) / f(q))^8 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.

Expansion of 16 * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^8 in powers of q.

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 * (1 - v)^2 - 16 * v * (1 - u).

lambda( -1 / tau ) = 1 - lambda( tau ) (see A128692).

G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128692.

G.f.: 16 * q * (Product_{k>0} (1 + q^(2*k)) / (1 + q^(2*k - 1)))^8.

a(n) = 16 * A005798(n). a(n) = -(-1)^n * A014972(n) unless n=0.

a(n) = -(-1)^n * A132136(n). - Michael Somos, Jun 03 2015

Empirical: Sum_{n>=1}(exp(-2*Pi)^n*a(n)) = 17 - 12*sqrt(2). - Simon Plouffe, Feb 20 2011

a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n)) / (32 * n^(3/4)). - Vaclav Kotesovec, Apr 06 2018

The g.f. A(q) = 16*q - 128*q^2 + 704*q^3 - 3072*q^4 + ... satisfies A(q) + A(-q) = A(q)*A(-q). - Peter Bala, Sep 26 2023

Example

G.f. = 16*q - 128*q^2 + 704*q^3 - 3072*q^4 + 11488*q^5 - 38400*q^6 + 117632*q^7 - ...

Mathematica

a[ n_] := SeriesCoefficient[ InverseEllipticNomeQ @ x, {x, 0, n}];
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ ModularLambda[ Log[q] / (Pi I)], {q, 0, n}]];
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q] / EllipticTheta[ 3, 0, q])^4, {q, 0, n}];
a[ n_] := SeriesCoefficient[ 1/16 (EllipticTheta[ 2, 0, q] / EllipticTheta[ 3, 0, q^2])^8, {q, 0, n}]; (* Michael Somos, May 26 2016 *)

Prog

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); 16 * polcoeff( (eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3)^8, n))};

Crossrefs

Cf. A005798, A014972, A128692, A132136.

Sequence in context: A045651 A035473 A014972 * A128692 A132136 A163399

Adjacent sequences: A115974 A115975 A115976 * A115978 A115979 A115980

Keyword

sign

Author

Michael Somos, Feb 09 2006

Status

approved