A136163 Integration of A053120: triangle of coefficients of integration of Chebyshev's T(n,x) polynomials (powers of x in increasing order).

1, -1, -1, -1, -3, 0, 2, 4, 0, -12, 0, 8, -1, 15, 0, -40, 0, 24, -4, 0, 60, 0, -120, 0, 64, -1, -35, 0, 210, 0, -336, 0, 160, 8, 0, -168, 0, 672, 0, -896, 0, 384, -1, 63, 0, -672, 0, 2016, 0, -2304, 0, 896, -8, 0, 360, 0, -2400, 0, 5760, 0, -5760, 0, 2048, -1, -99, 0, 1650, 0, -7920, 0, 15840, 0, -14080, 0, 4608

Offset: 1

Author

Roger L. Bagula, Mar 16 2008, corrected Apr 06 2008

Keywords

• uned
• tabf
• sign

Comments

The row sums are:

{-2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2}

These polynomials are orthogonal:

Table[Table[Integrate[Sqrt[1/(1 - x^2)]*a0[[ n]]*a0[[m]], {x, -1, 1}], {n, 1, 11}], {m, 1, 11}]

Solving for the recurrence:

Table[{c, d} /. Solve[{a0[[n]] -c*x*a0[[n - 1]] + d*a0[[n - 2]] == 0, a0[[n + 1]] - c*x*a0[[n]] + d*a0[[n - 1]] == 0}, {c, d}], {n, 3, 8}];

gives:

Q(x,n)=2*x*Q(x,n-1)-Q(x,n-2)

Examples

			{1},
{-1, -1},
{},
{-1, -3, 0, 2},
{4,0, -12, 0, 8},
{-1, 15, 0, -40, 0, 24},
{-4, 0, 60, 0, -120, 0, 64},
{-1, -35, 0, 210, 0, -336, 0, 160},
{8, 0, -168, 0,672, 0, -896, 0, 384},
{-1, 63, 0, -672, 0, 2016, 0, -2304, 0, 896}.
{-8, 0, 360, 0, -2400, 0, 5760, 0, -5760, 0, 2048},
{-1, -99, 0, 1650, 0, -7920, 0, 15840, 0, -14080, 0, 4608}
		

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795.
• Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986, page 8 and pages 42 - 43;

Crossrefs

Cf. A053120.

Programs

Mathematica

P[x, 0] = 1; P[x, 1] = x; P[x_, n_] := P[x, n] = 2*x*P[x, n - 1] - P[x, n - 2]; a0 = Table[ExpandAll[P[x, n]] /. x -> y, {n, 0, 10}]; b0 = Table[n*(n - 2)*Integrate[a0[[n]], {y, -1, x}], {n, 1, 11}] a = Join[{{1}}, Table[CoefficientList[b0[[n]], x], {n, 1, 11}]] Table[Apply[Plus, CoefficientList[b0[[n]], x]], {n, 1, 11}] Flatten[a]

Formula

T(x,n)=2*x*T(x,n-1)-T(x,n-2); Q(x,n)=Integrate[T(y,n-1),{y,-1,x}]