OFFSET
0,2
COMMENTS
Each polynomial phi_s(t) has 2s+1 terms. The signs of the polynomials alternate with s with positive coefficients for s even and negative coefficients for s odd.
REFERENCES
Amparo Gil, Javier Segura and Nico M. Temme, ACM TOMS, Volume 32, Issue 1 (March 2006), pages 70-101.
Amparo Gil, Javier Segura and Nico M. Temme, Numerical Methods for Special Functions, SIAM, 2007, pages 378-385. See Equations 12.121 through 12.125
LINKS
Chris Kormanyos, Rows s = 0..122, flattened
FORMULA
phi_s+1(t) = ( -4t^2(t + 1)^2 * d/dt[phi_s(t)] ) - ( (1/4) Integrate[(20T^2 + 20T + 3) phi_s(T)], {T,0,t}] )
phi_0 = 1, phi_-1 = 0
EXAMPLE
The polynomials phi_0, phi_1, phi_2 and phi_3 are:
1
-(t/12) (9 + 30t + 20t^2)
(t^2/288) (945 + 8028t + 19404t^2 + 18480t^3 + 6160t^4)
-(t^3/51840) (1403325 + 20545650t + 94064328t^2 + 200166120t^3 + 220540320t^4 + 122522400t^5 + 27227200t^6
MATHEMATICA
pktop = {1, -9, -30, -20};
pkbot = {1, 12};
p = (-t/12) (9 + (30 t) + (20 (t^2)));
Do[pk = -(4 (t^2) ((t + 1)^2)) D[p, t] - ((1/4) Integrate[((20 (t^2)) + (20 t) + 3) p, {t, 0, t}]);
p = Together[Simplify[pk]];
Do[pktop = Append[pktop, Coefficient[Expand[Numerator[p]], t^n]], {n, k, (2 k) + k, 1}];
pkbot = Append[pkbot, Denominator[p]];
Print[k], {k, 2, 10, 1}];
CROSSREFS
KEYWORD
sign,tabf
AUTHOR
Chris Kormanyos (ckormanyos(AT)yahoo.com), Mar 20 2009
STATUS
approved