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A326551
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.
9
1, -2, 56, -8336, 3985792, -4679517952, 11427218287616, -51793067942397952, 400951893341645930496, -4975999084909976839454720, 94178912073481319162642169856, -2610878440961060713599511173791744, 102545703927828194073741484514193965056, -5548919569628098800740786379865766154469376, 403949193167852851803947801218003477783686152192
OFFSET
0,2
COMMENTS
The hyperbolic analog of the e.g.f. is described by A325291.
The e.g.f. can be derived from the functions described by A326797, A326798, and A326799.
The e.g.f. can be derived from the functions described by A326800, A326801, and A326802.
FORMULA
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.
(1.a) C(x)^2 + S(x)^2 = 1.
(1.b) S'(x)/C(x) = -C'(x)/S(x) = 1/x * Integral C(x) dx.
(2.a) S(x) = Integral C(x)/x * (Integral C(x) dx) dx.
(2.b) C(x) = 1 - Integral S(x)/x * (Integral C(x) dx) dx.
(3.a) C(x) + i*S(x) = exp( i*Integral 1/x * (Integral C(x) dx) dx ).
(3.b) C(x) = cos( Integral 1/x * (Integral C(x) dx) dx ).
(3.c) S(x) = sin( Integral 1/x * (Integral C(x) dx) dx ).
Integration.
(4.a) S(x*y) = Integral C(x*y) * (Integral C(x*y) dy) dx.
(4.b) C(x*y) = 1 - Integral S(x*y) * (Integral C(x*y) dy) dx.
(4.c) S(x*y) = Integral C(x*y) * (Integral C(x*y) dx) dy.
(4.d) C(x*y) = 1 - Integral S(x*y) * (Integral C(x*y) dx) dy.
Exponential.
(5.a) C(x*y) + i*S(x*y) = exp( i*Integral Integral C(x*y) dx dy ).
(5.b) C(x*y) = cos( Integral Integral C(x*y) dx dy ).
(5.c) S(x*y) = sin( Integral Integral C(x*y) dx dy ).
Derivatives.
(6.a) d/dx S(x*y) = C(x*y) * Integral C(x*y) dy.
(6.b) d/dx C(x*y) = -S(x*y) * Integral C(x*y) dy.
(6.c) d/dy S(x*y) = C(x*y) * Integral C(x*y) dx.
(6.d) d/dy C(x*y) = -S(x*y) * Integral C(x*y) dx.
EXAMPLE
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 -+ ...
where C(x) = cos( Integral 1/x * (Integral C(x) dx) dx ),
also, C(x*y) = cos( Integral Integral C(x*y) dx dy ).
RELATED SERIES.
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 -+ ...
where S(x) = sin( Integral 1/x * (Integral C(x) dx) dx ),
also, S(x*y) = sin( Integral Integral C(x*y) dx dy ),
such that C(x)^2 + S(x)^2 = 1.
PROG
(PARI)
{a(n) = my(C=1, S=x); for(i=1, 2*n,
S = intformal( C/x * intformal( C +x*O(x^(2*n)) ) );
C = 1 - intformal( S/x * intformal( C +x*O(x^(2*n)) ) ); ); (2*n)!^2*polcoeff(C, 2*n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Jul 25 2019
STATUS
approved