A158503 Triangle read by rows: numerators of coefficients of the polynomials phi_s(t) used for asymptotic elementary function expansions of parabolic cylinder functions U(a, x), V(a, x).

1, -9, -30, -20, 945, 8028, 19404, 18480, 6160, -1403325, -20545650, -94064328, -200166120, -220540320, -122522400, -27227200, 820945125, 17610977880, 124110533448, 431932849920, 857710030320, 1023307084800, 728175127680, 285558873600, 47593145600

Offset: 0

Author

Chris Kormanyos (ckormanyos(AT)yahoo.com), Mar 20 2009

Keywords

• sign
• tabf

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.

Examples

			The polynomials phi_0, phi_1, phi_2 and phi_3 are:
  1
  -(t/12)*(9 + 30*t + 20*t^2)
  (t^2/288)*(945 + 8028*t + 19404*t^2 + 18480*t^3 + 6160*t^4)
  -(t^3/51840)*(1403325 + 20545650*t + 94064328*t^2 + 200166120*t^3 + 220540320*t^4 + 122522400*t^5 + 27227200*t^6)
		

References

• Amparo Gil, Javier Segura and Nico M. Temme, Computing the real parabolic cylinder functions U(a, x), V(a, x), 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

Crossrefs

For denominators see A001164.

Programs

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}];

Formula

phi_s+1(t) = ( -4*t^2*(t + 1)^2 * d/dt[phi_s(t)] ) - (1/4)*Integral_{T=0..t} (20*T^2 + 20*T + 3)*phi_s(T) dT

phi_0 = 1, phi_-1 = 0