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FORMULA
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Let e.g.f. C(x) = Sum_{n>=0} a(n)*x^(4*n)/(4*n)! and e.g.f. S(x) = Sum_{n>=0} a(n)*x^(4*n+1)/(4*n+1)!, then C(x) and S(x) satisfy:
(1) C(x)^4 + S(x)^4 = 1,
(2) S'(x) = C(x),
(3) C'(x) = -S(x)^3/C(x)^2,
(4) C(x)^3 * C'(x) + S(x)^3 * S'(x) = 0,
(5) S(x)/C(x) = Integral 1/C(x)^4 dx,
(6) S(x)/C(x) = Series_Reversion( Integral 1/(1+x^4) dx ) = Series_Reversion( Sum_{n>=0} (-1)^n * x^(4*n+1)/(4*n+1) ).
(7) S(x)^2/C(x)^2 = tan( 2 * Integral S(x)/C(x) dx ).
(8) C(x)^2 + I*S(x)^2 = exp( 2*I * Integral S(x)/C(x) dx ).
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EXAMPLE
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E.g.f. with offset 0 is C(x) and e.g.f. with offset 1 is S(x) where:
C(x) = 1 - 6*x^4/4! - 1764*x^8/8! - 7700616*x^12/12! - 147910405104*x^16/16! -...
S(x) = x - 6*x^5/5! - 1764*x^9/9! - 7700616*x^13/13! - 147910405104*x^17/17! -...
such that C(x)^4 + S(x)^4 = 1:
C(x)^4 = 1 - 24*x^4/4! + 8064*x^8/8! + 2128896*x^12/12! + 52932870144*x^16/16! +...
S(x)^4 = 24*x^4/4! - 8064*x^8/8! - 2128896*x^12/12! - 52932870144*x^16/16! -...
Related Expansions.
(1) The series reversion of S(x) is Integral 1/(1-x^4)^(1/4) dx:
Series_Reversion(S(x)) = x + 6*x^5/5! + 6300*x^9/9! + 56133000*x^13/13! +...
1/(1-x^4)^(1/4) = 1 + 6*x^4/4! + 6300*x^8/8! + 56133000*x^12/12! + 1992160170000*x^16/16! +...+ A258899(n)*x^(4*n)/(4*n)! +...
(2) d/dx S(x)/C(x) = 1/C(x)^4:
1/C(x)^4 = 1 + 24*x^4/4! + 32256*x^8/8! + 285272064*x^12/12! +...
S(x)/C(x) = x + 24*x^5/5! + 32256*x^9/9! + 285272064*x^13/13! + 8967114326016*x^17/17! +...+ A258901(n)*x^(4*n+1)/(4*n+1)! +...
where
Series_Reversion(S(x)/C(x)) = x - x^5/5 + x^9/9 - x^13/13 + x^17/17 - x^21/21 +...
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PROG
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(PARI) /* E.g.f. Series_Reversion(Integral 1/(1-x^4)^(1/4) dx): */
{a(n)=local(S=x); S = serreverse( intformal( 1/(1-x^4 +x*O(x^(4*n)))^(1/4) )); (4*n+1)!*polcoeff(S, 4*n+1)}
for(n=0, 15, print1(a(n), ", "))
(PARI) /* E.g.f. C(x) with offset 0: */
{a(n)=local(S=x, C=1+x); for(i=1, n, S=intformal(C +x*O(x^(4*n))); C=1-intformal(S^3/C^2 +x*O(x^(4*n))); ); (4*n)!*polcoeff(C, 4*n)}
for(n=0, 15, print1(a(n), ", "))
(PARI) /* E.g.f. S(x) with offset 1: */
{a(n)=local(S=x, C=1+x); for(i=1, n+1, S=intformal(C +x*O(x^(4*n))); C=1-intformal(S^3/C^2 +x*O(x^(4*n+1))); ); (4*n+1)!*polcoeff(S, 4*n+1)}
for(n=0, 15, print1(a(n), ", "))
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