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Expansion of Auxiliary function L(1-m) / 4 in powers of m / 16.
2

%I #32 May 08 2018 15:11:55

%S 0,1,21,740,37310,2460024,200770416,19551774528,2213488134000,

%T 285711909912000,41419784380740480,6663725042739448320,

%U 1178209566488368028160,227096910697908706560000

%N Expansion of Auxiliary function L(1-m) / 4 in powers of m / 16.

%C Nome q(m) = x exp(8 * (Sum_{n>0} a(n) * x^n / n!) / (Sum_{n>=0} binomial(2n, n)^2 * x^n)) where x = m / 16.

%C The Fricke reference on page 2 has equation "(3) Pi i omega = -Pi K'/K = log k^2 - 4 log 2 + F_1(1/2, 1/2; k^2) / F(1/2, 1/2, 1; k^2), wo F_1 und F ..." where F_1 = 8 * Sum_{n>0} a(n) * x^n / n! with x = m / 16 = (k / 4)^2. - _Michael Somos_, Jul 14 2013

%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 591.

%D J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 9.

%H G. C. Greubel, <a href="/A078791/b078791.txt">Table of n, a(n) for n = 0..300</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H R. Fricke, <a href="http://dx.doi.org/10.1007/978-3-642-20954-3_1">Die elliptischen Funktionen und ihre Anwendungen</a>, Dritter Teil, Springer-Verlag, 2012.

%F E.g.f.: L(1-m) = log(16 / m) (K(m) / Pi) - K(1-m) = 4 Sum_{n>0} a(n) (m/16)^n / n!.

%F 2 * a(n) = A098118(n) * A000984(n). - _Michael Somos_, Apr 14 2015

%F a(n) ~ log(2) * 2^(4*n - 1/2) * n^n / (sqrt(Pi*n) * exp(n)). - _Vaclav Kotesovec_, Jul 10 2016

%e G.f. = x + 21*x^2 + 740*x^3 + 37310*x^4 + 2460024*x^5 + 200770416*x^6 + 19551774528*x^7 + ...

%t a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Log[ EllipticNomeQ[ 16 x] / x] Hypergeometric2F1[ 1/2, 1/2, 1, 16 x] / 8, {x, 0, n}]]; (* _Michael Somos_, Jul 14 2013 *)

%t a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Log[ EllipticNomeQ[ 16 x] / x] EllipticK[ 16 x] / (4 Pi), {x, 0, n}]]; (* _Michael Somos_, Jul 14 2013 *)

%t a[ n_] := If[ n < 0, 0, n! Binomial[ 2 n, n]^2 Sum[ 1/k, {k, n + 1, 2 n}] / 2]; (* _Michael Somos_, Jul 14 2013 *)

%t a[ n_] := If[ n < 0, 0, n! Binomial[ 2 n, n]^2 (HarmonicNumber[2 n] - HarmonicNumber[n]) / 2]; (* _Michael Somos_, Apr 14 2015 *)

%o (PARI) {a(n) = if( n<0, 0, sum( k=1, n, 1 / (2*k - 1) / k) / 4 * (2*n)!^2 / n!^3)};

%Y Cf. A000984, A005797, A098118.

%K nonn,easy

%O 0,3

%A _Michael Somos_, Dec 05 2002