A274654 -id:A274654 - cp's OEIS frontend

A274653 Numerators of coefficients of z^n/n! for the expansion of Fricke's hypergeometric function F_1(1/2,1/2;z).

0, 1, 21, 185, 18655, 307503, 12548151, 305496477, 138343008375, 4464248592375, 323592065474535, 13015087974100485, 2301190559547593805, 110887163426713235625, 11570760017278599886875, 649837647729572203369125, 1250848387902442801195686375, 80233244659365977333374518375

Offset: 0

Wolfdieter Lang, Jul 07 2016

For the denominators see A274654.
The coefficients of z^n for the expansion of F_1(1/2,1/2;z) are A274655(n)/A274656(n).
Fricke's hypergeometric function F_1(a,b;z) = Sum_{n > = 0} f(a,b;n)*z^n/n!, satisfies the recurrence
  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.
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).
(2/Pi)*K(k) = hypergeometric([1/2,1/2],[1],k^2). For the expansion coefficients see A038534/A056982 and also A274657/A123854.

			The sequence of rationals {r(n)} begins:
0, 1/2, 21/32, 185/128, 18655/4096, 307503/16384, 12548151/131072, 305496477/524288, 138343008375/33554432, 4464248592375/134217728, 323592065474535/1073741824, ....
The expansion of F_1(1/2,1/2;z) begins:
(1/2)*z + (21/32)*z^2/2! + (185/128)*z^3/3! + (18655/4096)*z^4/4! + (307503/16384)*z^5/5! + ..., or
(1/2)*z + (21/64)*z^2 + (185/768)*z^3 + (18655/98304)*z^4 + (102501/655360)*z^5 + ...

R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Erster Teil, Springer-Verlag, 2012, p. 465, eq. (11) with p.114, eq. (15).
R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Dritter Teil, Springer-Verlag, 2012, p. 2, eq. (3).

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

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

A274655 Numerators of coefficients of z^n for the expansion of Fricke's hypergeometric function F_1(1/2,1/2;z).

Original entry on oeis.org

0, 1, 21, 185, 18655, 102501, 1394239, 33944053, 3074289075, 99205524275, 7190934788323, 4590859955591, 2435122285235549, 23468182735812325, 38870446014205425, 145536272272236993, 280137373064011153371, 1633533514217325226737, 74200692627870055029475

Offset: 0

Wolfdieter Lang, Jul 07 2016

For the denominators see A274656.

The main entry is A274653/A274654. In A274653 Fricke's hypergeometric function F_1(a,b;z) is defined by the recurrence. More details and the Fricke references are also found there.

			The rationals R(n) begin:
0, 1/2, 21/64, 185/768, 18655/98304, 102501/655360, 1394239/10485760, 33944053/293601280, ...

Cf. A274656, A274653/A274654.

a(n) = numerator(R(n)), where the rationals (in lowest terms) are R(n) = [z^n]F_1(1/2,1/2;z), and the recurrence for R(n) = r(n)/n! is obtained from the one given for r(n) in A274653.

R(n) = ((2*n-1)/(2*n))^2*R(n-1) + 2*C(n)/(n*(2*n-1)), n >= 1, R(0) = 0, with C(n) = ((2*n)!)^2 / (n!^4*2^(4*n)).

A274656 Denominators of coefficients of z^n for the expansion of Fricke's hypergeometric function F_1(1/2,1/2;z).

Original entry on oeis.org

1, 2, 64, 768, 98304, 655360, 10485760, 293601280, 30064771072, 1082331758592, 86586540687360, 60473139527680, 34832528367943680, 362258295026614272, 644014746713980928, 2576058986855923712, 5275768805080931762176, 32613843522318487257088

Offset: 0

Wolfdieter Lang, Jul 07 2016

For the numerators see A274655.
For the denominators of the coefficients of z^n/n! for the expansion of F_1(1/2,1/2;z) see A274654.
See the main entry A274653 (with A274654) for the definition of Fricke's hypergeometric function F_1(a,b;z) with the recurrence and details on F_1(1/2,1/2;z).

			See A274653, A274654, A274655.

See A274653.

Cf. A274653.

a(n) = denominator(R(n)), where the rationals (in lowest terms) are R(n) = [z^n]F_1(1/2,1/2;z), and the recurrence for R(n) = r(n)/n! is obtained from the one given for r(n) in A274653.

R(n) = ((2*n-1)/(2*n))^2*R(n-1) + 2*C(n)/(n*(2*n-1)), n >= 1, R(0) = 0, with C(n) = ((2*n)!)^2 / (n!^4*2^(4*n)).

cp's OEIS Frontend

A274653 Numerators of coefficients of z^n/n! for the expansion of Fricke's hypergeometric function F_1(1/2,1/2;z).

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A274655 Numerators of coefficients of z^n for the expansion of Fricke's hypergeometric function F_1(1/2,1/2;z).

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A274656 Denominators of coefficients of z^n for the expansion of Fricke's hypergeometric function F_1(1/2,1/2;z).

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