A274653 - OEIS

%I #12 Jul 10 2016 04:09:45

%S 0,1,21,185,18655,307503,12548151,305496477,138343008375,

%T 4464248592375,323592065474535,13015087974100485,2301190559547593805,

%U 110887163426713235625,11570760017278599886875,649837647729572203369125,1250848387902442801195686375,80233244659365977333374518375

%N Numerators of coefficients of z^n/n! for the expansion of Fricke's hypergeometric function F_1(1/2,1/2;z).

%C For the denominators see A274654.

%C The coefficients of z^n for the expansion of F_1(1/2,1/2;z) are A274655(n)/A274656(n).

%C Fricke's hypergeometric function F_1(a,b;z) = Sum_{n > = 0} f(a,b;n)*z^n/n!, satisfies the recurrence

%C   f(a,b,n) = ((a+n-1)*(b+n-1)/n)*f(a,b;n-1) + c(a,b;n)*(1/(a+n-1) + 1/ (b+n-1) - 2/n), with c(a,b;n) = [z^n/n!]hypergeometric([a,b],[1],z) = risefac(a,n) * risefac(b,n)/n!, where risefac is the rising factorial (Pochhammer's symbol) and the input is f(a,b;0)= 0. See the Fricke I reference, p. 114.

%C The hypergeometric function F_1(1/2,1/2;z) appears in the formula for (2/Pi) K'(k) + (1/Pi)*log(k^2/16)*(2/Pi)*K(k) = F_1(1/2,1/2;k^2), where K and sqrt(-1)*K' are the real and imaginary quarter periods, and k is the modulus (k^2 is the parameter) of elliptic functions. See the Fricke I reference p. 465, eq. (11), and also Fricke III, p. 2, eq. (3).

%C (2/Pi)*K(k) = hypergeometric([1/2,1/2],[1],k^2). For the expansion coefficients see A038534/A056982 and also A274657/A123854.

%H R. Fricke, <a href="http://link.springer.com/chapter/10.1007%2F978-3-642-19557-0_9">Die elliptischen Funktionen und ihre Anwendungen, Erster Teil</a>, Springer-Verlag, 2012, p. 465, eq. (11) with p.114, eq. (15).

%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, p. 2, eq. (3).

%F a(n) = numerator(r(n)), with the rationals (in lowest terms) r(n) = [z^n/n!]F_1(1/2,1/2;z), with the hypergeometric function F_1 given by Fricke. The recurrence of the coefficients r(n) = f(1/2,1/2;n) is obtained from the general one given above.

%F r(n) = ((2*n-1)^2/(4*n))*r(n-1) + 2*c(n)/(n*(2*n-1)), n >= 1, r(0) = 0, with c(n) = c(1/2,1/2;n) = ((2*n)!)^2 / (n!^3*2^(4*n)) (see A274657/A123854).

%F E.g.f. for r(n) is Fricke's F_1(1/2,1/2;z).

%e The sequence of rationals {r(n)} begins:

%e 0, 1/2, 21/32, 185/128, 18655/4096, 307503/16384, 12548151/131072, 305496477/524288, 138343008375/33554432, 4464248592375/134217728, 323592065474535/1073741824, ....

%e The expansion of F_1(1/2,1/2;z) begins:

%e (1/2)*z + (21/32)*z^2/2! + (185/128)*z^3/3! + (18655/4096)*z^4/4! + (307503/16384)*z^5/5! + ..., or

%e (1/2)*z + (21/64)*z^2 + (185/768)*z^3 + (18655/98304)*z^4 + (102501/655360)*z^5 + ...

%Y Cf. A038534, A056982, A274654, A274655, A274656, A274657, A123854.

%K nonn,easy,frac

%O 0,3

%A _Wolfdieter Lang_, Jul 07 2016