%I #15 Aug 17 2019 12:17:05

%S 1,-2,56,-8336,3985792,-4679517952,11427218287616,-51793067942397952,

%T 400951893341645930496,-4975999084909976839454720,

%U 94178912073481319162642169856,-2610878440961060713599511173791744,102545703927828194073741484514193965056,-5548919569628098800740786379865766154469376,403949193167852851803947801218003477783686152192

%N E.g.f. C(x), where C(x*y) + i*S(x*y) = exp( i*Integral Integral C(x*y) dx dy ) such that C(x)^2 + S(x)^2 = 1.

%C The hyperbolic analog of the e.g.f. is described by A325291.

%C The e.g.f. can be derived from the functions described by A326797, A326798, and A326799.

%C The e.g.f. can be derived from the functions described by A326800, A326801, and A326802.

%F E.g.f. C(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)!^2, where series C(x) and related series S(x) satisfy the following relations.

%F (1.a) C(x)^2 + S(x)^2 = 1.

%F (1.b) S'(x)/C(x) = -C'(x)/S(x) = 1/x * Integral C(x) dx.

%F (2.a) S(x) = Integral C(x)/x * (Integral C(x) dx) dx.

%F (2.b) C(x) = 1 - Integral S(x)/x * (Integral C(x) dx) dx.

%F (3.a) C(x) + i*S(x) = exp( i*Integral 1/x * (Integral C(x) dx) dx ).

%F (3.b) C(x) = cos( Integral 1/x * (Integral C(x) dx) dx ).

%F (3.c) S(x) = sin( Integral 1/x * (Integral C(x) dx) dx ).

%F Integration.

%F (4.a) S(x*y) = Integral C(x*y) * (Integral C(x*y) dy) dx.

%F (4.b) C(x*y) = 1 - Integral S(x*y) * (Integral C(x*y) dy) dx.

%F (4.c) S(x*y) = Integral C(x*y) * (Integral C(x*y) dx) dy.

%F (4.d) C(x*y) = 1 - Integral S(x*y) * (Integral C(x*y) dx) dy.

%F Exponential.

%F (5.a) C(x*y) + i*S(x*y) = exp( i*Integral Integral C(x*y) dx dy ).

%F (5.b) C(x*y) = cos( Integral Integral C(x*y) dx dy ).

%F (5.c) S(x*y) = sin( Integral Integral C(x*y) dx dy ).

%F Derivatives.

%F (6.a) d/dx S(x*y) = C(x*y) * Integral C(x*y) dy.

%F (6.b) d/dx C(x*y) = -S(x*y) * Integral C(x*y) dy.

%F (6.c) d/dy S(x*y) = C(x*y) * Integral C(x*y) dx.

%F (6.d) d/dy C(x*y) = -S(x*y) * Integral C(x*y) dx.

%e E.g.f. C(x) = 1 - 2*x^2/2!^2 + 56*x^4/4!^2 - 8336*x^6/6!^2 + 3985792*x^8/8!^2 - 4679517952*x^10/10!^2 + 11427218287616*x^12/12!^2 - 51793067942397952*x^14/14!^2 + 400951893341645930496*x^16/16!^2 - 4975999084909976839454720*x^18/18!^2 + 94178912073481319162642169856*x^20/20!^2 -+ ...

%e where C(x) = cos( Integral 1/x * (Integral C(x) dx) dx ),

%e also, C(x*y) = cos( Integral Integral C(x*y) dx dy ).

%e RELATED SERIES.

%e S(x) = x - 8*x^3/3!^2 + 576*x^5/5!^2 - 160768*x^7/7!^2 + 123535360*x^9/9!^2 - 212713734144*x^11/11!^2 + 716196297048064*x^13/13!^2 - 4280584942657732608*x^15/15!^2 + 42250703121584165486592*x^17/17!^2 - 651154631135458759089848320*x^19/19!^2 + 14983590319172065236171175755776*x^21/21!^2 -+ ...

%e where S(x) = sin( Integral 1/x * (Integral C(x) dx) dx ),

%e also, S(x*y) = sin( Integral Integral C(x*y) dx dy ),

%e such that C(x)^2 + S(x)^2 = 1.

%o (PARI)

%o {a(n) = my(C=1, S=x); for(i=1, 2*n,

%o S = intformal( C/x * intformal( C +x*O(x^(2*n)) ) );

%o C = 1 - intformal( S/x * intformal( C +x*O(x^(2*n)) ) ); ); (2*n)!^2*polcoeff(C, 2*n)}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A326552, A325291, A326556 (C^2).

%Y Cf. A326797, A326798, A326799.

%Y Cf. A326800, A326801, A326802.

%K sign

%O 0,2

%A _Paul D. Hanna_, Jul 25 2019