OFFSET
1,2
COMMENTS
k' is the square root of the complementary parameter of elliptic functions. In the Abramowitz-Stegun (A-St) reference, p. 569, k'^2 is called m_1. The relation between k'^2 and k^2, the parameter (called m in A-St), is k'^2 = 1 - k^2.
The expansion of q in odd powers of (1/4)*(1 - k') / (1 + k') appears in the Kneser reference, p. 218, where it is attributed to L. Lindelöf. It is obtained from the expansion of sqrt(q) in odd powers of k/4, namely q^{1/2} = Sum_{n >= 0} a(n)*(k/4)^(2*n+1), which results from the expansion -Pi*K'/K = log(q) = log(k^2/16) + log(1 + Sum_{n>=1} A005797(n+1)*(k^2/16)^n) = log(k^2/16) + 8*(k^2/16) + 52*(k^2/16)^2 + ... (see A-St, p. 591, 17. 3.21, Kneser, p. 216, Fricke, eq. (4), p. 2, and A227505, A274345/A274346). The fact that a replacement of q by q^2 means a replacement of k by (1 - k')/(1 + k') is used (Landen transformation).
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..800
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], pp. 569, 591.
R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Dritter Teil, Springer-Verlag, 2012., p. 2, eq. (4).
A. Kneser, Neue Untersuchung einer Reihe aus der Theorie der elliptischen Funktionen, J. reine u. angew. Math. 157, 1927, 209 - 218.
FORMULA
q^{1/2} = Sum_{n >= 0} a(n)*(k/4)^(2*n+1).
q = Sum_{n >= 0} a(n)*((1/4)*(1 - k')/(1 + k'))^(2*n+1).
MATHEMATICA
CoefficientList[Series[Sqrt[EllipticNomeQ[16*x]/x], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 07 2019 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Jun 30 2016
STATUS
approved