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Reversion of series for j-function.
6

%I #43 Dec 17 2021 20:58:08

%S 1,744,750420,872769632,1102652742882,1470561136292880,

%T 2037518752496883080,2904264865530359889600,4231393254051181981976079,

%U 6273346050902229242859370584,9433668720359866477436486024652

%N Reversion of series for j-function.

%C From _Peter Bala_, Dec 17 2013: (Start)

%C Given a formal Laurent series L(z) = 1/z + a + b*z + c*z^2 + ..., there exists a formal series L^<-1>(z) = 1/z + A/z^2 + B/z^3 + ... such that L(L^<-1>(z)) = L^<-1>(L(z)) = z. The series L^<-1>(z) is called the reversion of the series L(z).

%C To find L^<-1>(z), first find the series reversion of the reciprocal series 1/L(z) = z - a*z^2 + z^3*(a^2 - b) - ... with respect to z, and then replace the variable z with the variable 1/z. This is the approach used in the Maple program below. (End)

%C Invert j = 1/q + 744 + 196884*q + 21493760*q^2 + ... to get q = 1/j + 744/j^2 + 750420/j^3 + ....

%D J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Springer, see p. 482.

%H Vaclav Kotesovec, <a href="/A091406/b091406.txt">Table of n, a(n) for n = 1..300</a>

%H Y Abdelaziz, JM Maillard, <a href="http://arxiv.org/abs/1611.08493">Modular forms, Schwarzian conditions, and symmetries of differential equations in physics</a>, arXiv preprint arXiv:1611.08493, 2016

%H Y.-H. He and V. Jejjala, <a href="http://arXiv.org/abs/hep-th/0307293">Modular Matrix Models</a>, arXiv:hep-th/0307293, 2003.

%F a(n) ~ c * 1728^n / n^(3/2), where c = 2 * Gamma(3/4)^4 / (exp(2*Pi) * sqrt(3) * Pi^(3/2)) = 0.000873226754634291459391356302898297243945046378336447143... - _Vaclav Kotesovec_, Jun 28 2017, updated Mar 07 2018

%e G.f. = x + 744*x^2 + 750420*x^3 + 872769632*x^4 + 1102652742882*x^5 + ... - _Michael Somos_, Dec 17 2021

%p #A091406

%p with(numtheory):

%p Order := 12:

%p g2 := 4/3*(1 + 240*add(sigma[3](n)*q^n, n = 1..Order)):

%p g3 := 8/27*(1 - 504*add(sigma[5](n)*q^n, n = 1..Order)):

%p delta := series(g2^3 - 27*g3^2, q, Order):

%p #define the reciprocal of Klein's j_invariant

%p j_reciprocal := series(delta/(1728*g2^3), q, Order):

%p #find series reversion of j_reciprocal

%p j_inv := solve(series(j_reciprocal, q) = y, q):

%p seq(coeff(j_inv,y,n), n = 1..11); # _Peter Bala_, Dec 17 2013

%t max = 9; s1 = 1728*Series[ KleinInvariantJ[t], {t, 0, 2*max} ] /. t -> -2*I*(Pi/Log[q]); s2 = Normal[InverseSeries[ Series[ s1, {q, 0, max} ], j]] /. j -> 1/x; Rest[ CoefficientList[ s2, x ] ] (* _Jean-François Alcover_, Feb 16 2012, fixed by _Vaclav Kotesovec_, Jun 28 2017 *)

%t a[ n_] := If[ n < 1, 0, SeriesCoefficient[ InverseSeries[ Series[ 1/(1728 KleinInvariantJ[ Log[q]/(2 Pi I)]), {q, 0, n}]], {q, 0, n}]]; Table[a[n], {n, 12}] (* _Michael Somos_, Dec 17 2021 *)

%o (PARI) {a(n) = local(A); if( n<1, 0, A = O(x^n); A = x * (eta(x^2 + A) / eta(x + A))^24; polcoeff( serreverse( A / (1 + 256*A)^3), n))} /* _Michael Somos_, Jul 13 2004 */

%Y Cf. A000521, A178451. See A066396 for another version.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_, Mar 03 2004