Generating function C(x) = Sum_{n>=0} a(n)*x^(4*n)/(4*n)! and related function S(x) satisfies the following formulas.
For brevity, some formulas here will use C = C(x) and S = S(x), where S(x) = (C(x)^4 - 1)^(1/4) is the e.g.f. of A357804.
(1) C(x)^4 - S(x)^4 = 1.
Integral formulas.
(2.a) S(x) = Integral C(x)^6 dx.
(2.b) C(x) = 1 + Integral S(x)^3 * C(x)^3 dx.
(2.c) S(x)^4 = Integral 4 * S(x)^3 * C(x)^6 dx.
(2.d) C(x)^4 = 1 + Integral 4 * S(x)^3 * C(x)^6 dx.
Derivatives.
(3.a) d/dx S(x) = C(x)^6.
(3.b) d/dx C(x) = S(x)^3 * C(x)^3.
Exponential formulas.
(4.a) C + S = exp( Integral (C^2 - C*S + S^2) * C^3 dx ).
(4.b) C - S = exp( -Integral (C^2 + C*S + S^2) * C^3 dx ).
(5.a) C^2 + S^2 = exp( 2 * Integral S*C^4 dx ).
(5.b) C^2 - S^2 = exp( -2 * Integral S*C^4 dx ).
Hyperbolic functions.
(6.a) C = sqrt(C^2 - S^2) * cosh( Integral (C^2 + S^2) * C^3 dx ).
(6.b) S = sqrt(C^2 - S^2) * sinh( Integral (C^2 + S^2) * C^3 dx ).
(7.a) C^2 = cosh( 2 * Integral S*C^4 dx ).
(7.b) S^2 = sinh( 2 * Integral S*C^4 dx ).
Explicit formulas.
(8.a) S(x) = Series_Reversion( Integral 1/(1 + x^4)^(3/2) dx ).
(8.b) C( Integral 1/(1 + x^4)^(3/2) dx ) = (1 + x^4)^(1/4).
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