A136265 Integral form of A053120 :Triangle of coefficients of Integral form Chebyshev's T(n, x) polynomials (powers of x in increasing order); Much improved version by use of the integro-differential recursive form over a previous attempt.

1, -1, 2, 0, -4, 2, 3, -2, -12, 4, 0, 16, -6, -32, 8, -5, 2, 60, -16, -80, 16, 0, -36, 10, 192, -40, -192, 32, 7, -2, -168, 36, 560, -96, -448, 64, 0, 64, -14, -640, 112, 1536, -224, -1024, 128, -9, 2, 360, -64, -2160, 320, 4032, -512, -2304, 256, 0, -100, 18, 1600, -240, -6720, 864, 10240, -1152, -5120, 512, 11, -2

Offset: 1

Author

Roger L. Bagula, Mar 18 2008

Keywords

• uned
• tabl
• sign

Comments

Row sums are:

Join[{1}, Table[Apply[Plus, CoefficientList[2*x*P[x, n] - Q[x, n], x]], {n,

0, 10}]];

{1, 1, -2, -7, -14, -23, -34, -47, -62, -79, -98, -119}

Integration of the doubled functions is not orthogonal:

Table[Table[Integrate[Sqrt[1/(1 - x^2)]*(2*x*P[x, n] - Q[x, n])*(2*x*P[x, m] -

Q[x, m]), {x, -1, 1}], {n, 0, 10}], {m, 0, 10}]

Examples

			{1},
{-1, 2},
{0, -4, 2},
{3, -2, -12, 4},
{0, 16, -6, -32, 8},
{-5,2, 60, -16, -80, 16},
{0, -36, 10, 192, -40, -192, 32},
{7, -2, -168, 36, 560, -96, -448, 64},
{0, 64, -14, -640, 112, 1536, -224, -1024, 128},
{-9, 2, 360, -64, -2160, 320, 4032, -512, -2304, 256},
{0, -100, 18, 1600, -240, -6720, 864, 10240, -1152, -5120, 512},
{11, -2, -660, 100,6160, -800, -19712, 2240, 25344, -2560, -11264, 1024}
		

References

• Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986,Pages 42-50

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]; Q[x_, n_] := D[P[x, n + 1], x]; Table[ExpandAll[2*x*P[x, n] - Q[x, n]], {n, 0, 10}]; a = Join[{{1}}, Table[CoefficientList[2*x*P[x, n] - Q[x, n], x], {n, 0, 10}]]; Join[{1}, Table[Apply[Plus, CoefficientList[2*x*P[x, n] - Q[x, n], x]], {n, 0, 10}]]; Flatten[a]

Formula

P(x, n) = 2*x*P(x, n - 1) - P(x, n - 2); Q(x, n) := D[P[x, n + 1], x]=dp(x,n)/dx Output Integral form: IP(x,n)=2*x*p(x,n)-Q(x,n)