OFFSET
0,3
FORMULA
E.g.f. C(x) + S(x), where series C(x) and S(x) are given by
(0.a) C(x) + S(x) = Sum_{n>=0} a(n)*x^n/(n!)^3,
(0.b) C(x) = Sum_{n>=0} a(2*n)*x^(2*n)/(2*n)!^3,
(0.c) S(x) = Sum_{n>=0} a(2*n+1)*x^(2*n+1)/(2*n+1)!^3,
and satisfy the following relations.
(1.a) C(x)^2 - S(x)^2 = 1.
(1.b) C'(x)/S(x) = S'(x)/C(x) = 1/x * Integral 1/x * Integral C(x) dx dx.
(2.a) S(x) = Integral C(x)/x * Integral 1/x * Integral C(x) dx dx dx.
(2.b) C(x) = 1 + Integral S(x)/x * Integral 1/x * Integral C(x) dx dx dx.
(3.a) C(x) + S(x) = exp( Integral 1/x * Integral 1/x * Integral C(x) dx dx dx ).
(3.b) C(x) = cosh( Integral 1/x * Integral 1/x * Integral C(x) dx dx dx ).
(3.c) S(x) = sinh( Integral 1/x * Integral 1/x * Integral C(x) dx dx dx ).
Integration.
(4.a) S(x*y*z) = Integral C(x*y*z) * Integral Integral C(x*y*z) dx dy dz.
(4.b) C(x*y*z) = 1 + Integral S(x*y*z) * Integral Integral C(x*y*z) dx dy dz.
(4.c) S(x*y*z) = Integral C(x*y*z) * Integral Integral C(x*y*z) dz dy dx.
(4.d) C(x*y*z) = 1 + Integral S(x*y*z) * Integral Integral C(x*y*z) dz dy dx.
Exponential.
(5.a) C(x*y*z) + S(x*y*z) = exp( Integral Integral Integral C(x*y*z) dx dy dz ).
(5.b) C(x*y*z) = cosh( Integral Integral Integral C(x*y*z) dx dy dz ).
(5.c) S(x*y*z) = sinh( Integral Integral Integral C(x*y*z) dx dy dz ).
Derivatives.
(6.a) d/dx S(x*y*z) = C(x*y*z) * Integral Integral C(x*y*z) dy dz.
(6.b) d/dx C(x*y*z) = S(x*y*z) * Integral Integral C(x*y*z) dy dz.
(6.c) d/dy S(x*y*z) = C(x*y*z) * Integral Integral C(x*y*z) dx dz.
(6.d) d/dy C(x*y*z) = S(x*y*z) * Integral Integral C(x*y*z) dx dz.
EXAMPLE
E.g.f.: C(x) + S(x) = 1 + x + 4*x^2/2!^3 + 40*x^3/3!^3 + 832*x^4/4!^3 + 31232*x^5/5!^3 + 1914112*x^6/6!^3 + 178872320*x^7/7!^3 + 24185421824*x^8/8!^3 + 4542993268736*x^9/9!^3 + 1147507517751296*x^10/10!^3 + 379488219034550272*x^11/11!^3 + 160693667742004281344*x^12/12!^3 + 85499599518969496600576*x^13/13!^3 + 56242680517408749713883136*x^14/14!^3 + 45103267674508555161314525184*x^15/15!^3 + 43556364453823048960903288455168*x^16/16!^3 + 50105222938479119498840420930027520*x^17/17!^3 + 68000060622146518553982060676576706560*x^18/18!^3 + 107938578855000557533262550908184207294464*x^19/19!^3 + 198840485174399292764682317537473563673493504*x^20/20!^3 + ...
where C(x*y*z) + S(x*y*z) = exp( Integral Integral Integral C(x*y*z) dx dy dz )
such that C(x)^2 - S(x)^2 = 1.
The e.g.f. as a series of reduced fractional coefficients begins
C(x) + S(x) = 1 + x + 1/2*x^2 + 5/27*x^3 + 13/216*x^4 + 61/3375*x^5 + 7477/1458000*x^6 + 8734/6251175*x^7 + 1476161/4000752000*x^8 + 2166268/22785532875*x^9 + 17509575161/729137052000000*x^10 + 22619260492/3790943032078125*x^11 + 153249423734669/104811992950896000000*x^12 + ...
RELATED SERIES.
C(x) = 1 + 4*x^2/2!^3 + 832*x^4/4!^3 + 1914112*x^6/6!^3 + 24185421824*x^8/8!^3 + 1147507517751296*x^10/10!^3 + 160693667742004281344*x^12/12!^3 + 56242680517408749713883136*x^14/14!^3 + 43556364453823048960903288455168*x^16/16!^3 + 68000060622146518553982060676576706560*x^18/18!^3 + 198840485174399292764682317537473563673493504*x^20/20!^3 + ...
where C(x) = cosh( Integral 1/x * Integral 1/x * Integral C(x) dx dx dx ),
also, C(x*y*z) = cosh( Integral Integral Integral C(x*y*z) dx dy dz ).
S(x) = x + 40*x^3/3!^3 + 31232*x^5/5!^3 + 178872320*x^7/7!^3 + 4542993268736*x^9/9!^3 + 379488219034550272*x^11/11!^3 + 85499599518969496600576*x^13/13!^3 + 45103267674508555161314525184*x^15/15!^3 + 50105222938479119498840420930027520*x^17/17!^3 + 107938578855000557533262550908184207294464*x^19/19!^3 + ...
where S(x) = sinh( Integral 1/x * Integral 1/x * Integral C(x) dx dx dx ),
also, S(x*y*z) = sinh( Integral Integral Integral C(x*y*z) dx dy dz ).
PROG
(PARI) {a(n) = my(C=1, S=x); for(i=1, n,
S = intformal( C/x * intformal( 1/x * intformal( C + x*O(x^n))));
C = 1 + intformal( S/x * intformal( 1/x * intformal( C + x*O(x^n)))); );
n!^3 * polcoeff(E = C + S, n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 21 2019
STATUS
approved