OFFSET
0,2
COMMENTS
The functions C = C(x) and S = S(x) such that C(x)^(1/2) - S(x)^(1/2) = 1 may be generated by the following method.
(Start) Set C = 1, S = x^2, then iterate
C = 1 + Integral S'*sqrt(C/S) dx and
S = Integral C'*sqrt(S/C) dx.
FORMULA
The functions C = C(x) and S = S(x) satisfy:
(1a) sqrt(C) - sqrt(S) = 1.
(1b) C'*sqrt(S) = S'*sqrt(C) = 72*x.
(1c) C' = 72*x/sqrt(S).
(1d) S' = 72*x/sqrt(C).
Integrals.
(2a) C = 1 + Integral 72*x/sqrt(S) dx.
(2b) S = Integral 72*x/sqrt(C) dx.
(2c) C = 1 + Integral S'*sqrt(C/S) dx.
(2d) S = Integral C'*sqrt(S/C) dx.
Exponentials.
(3a) sqrt(C) = exp( Integral 36*x/(C*sqrt(S)) dx ).
(3b) sqrt(S) = 6*x*exp( Integral 36*x/(S*sqrt(C)) - 1/x dx ).
(3c) C - S = exp( Integral 72*x/(C*sqrt(S) + S*sqrt(C)) dx ).
(3d) C - S = exp( Integral C'*S'/(C*S' + S*C') dx).
Functional equations.
(4a) C = 1/3 - 36*x^2 + (2/3)*C^(3/2).
(4b) S = 36*x^2 - (2/3)*S^(3/2).
Explicit solutions.
(5a) C(x) = 1 + Sum_{n>=1} 2*(-4)^n*binomial(3*n/2,n)/((3*n-2)*(3*n-4)) * x^n.
(5b) S(x) = 36*x^2 + Sum_{n>=3} 18*(-4)^n*(3*n-3)*binomial(3*n/2-2,n)/((3*n-4)*(3*n-6)) * x^n.
(5c) sqrt(C(x)) = 1 + Sum_{n>=1} -(-4)^n * binomial(3*n/2,n)/(3*n-2) * x^n.
Formulas for terms.
a(n) = 2*(-4)^n * binomial(3*n/2,n) / ((3*n-2)*(3*n-4)) for n>=1, with a(0) = 1.
EXAMPLE
G.f.: C(x) = 1 + 12*x + 12*x^2 - 24*x^3 + 96*x^4 - 504*x^5 + 3072*x^6 - 20592*x^7 + 147456*x^8 - 1108536*x^9 + 8650752*x^10 + ...
RELATED SERIES.
S(x) = 36*x^2 - 144*x^3 + 864*x^4 - 6048*x^5 + 46080*x^6 - 370656*x^7 + 3096576*x^8 - 26604864*x^9 + 233570304*x^10 + ...
C(x)^(1/2) = 1 + 6*x - 12*x^2 + 60*x^3 - 384*x^4 + 2772*x^5 - 21504*x^6 + 175032*x^7 - 1474560*x^8 + 12748164*x^9 - 112459776*x^10 + ...
sqrt(S(x)) = 6*x - 12*x^2 + 60*x^3 - 384*x^4 + 2772*x^5 - 21504*x^6 + 175032*x^7 - 1474560*x^8 + 12748164*x^9 - 112459776*x^10 + ...
where C(x)^(1/2) - S(x)^(1/2) = 1
and C'*sqrt(S) = S'*sqrt(C) = 72*x.
PROG
(PARI) {a(n) = my(C=1, S=x^2); for(i=0, n, C = 1 + intformal( 72*x/sqrt(S +x^3*O(x^n)) ); S = intformal( 72*x/sqrt(C) ) ); polcoeff(C, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = if(n==0, 1, 2*(-4)^n * binomial(3*n/2, n) / ((3*n-2)*(3*n-4)) )}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Feb 20 2018
STATUS
approved