1,5

I found this relationship of the recursions to the differential recursions for the Hermite polynomials and Legendre polynomials in Hochstadt.

It makes sense that the Integral form follows the same rule up to a constant of integration chosen here as zero.

Row sums are:

{0, -1, -1, -2, 0, 5, 7, 1, -9, -12, -2}

The doubled functions Integral that indicated orthogonality show these aren't even alternating orthogonal.

Karem Boubaker, On modified Boubaker polynomials..., Trends in Appl. Sci. Research, 2 (2007), 540-544.

Defined differential recursion: page 49 .: Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986

B(x, n) = x*B(x, n - 1) - B(x, n - 2); B'(x,n)=dB(x,n+1)/dx; IB(x, n) = x*B(x, n) - B'(x, n)

{1},

{-1, 1},

{0, -2, 1},

{-1, 2, -3, 1},

{0, 0, 1, -4, 1},

{3, -2, 3, 0, -5, 1},

{0, 6, -3, 8, -1, -6, 1},

{-5, 2, 6, -3, 15, -2, -7, 1},

{0, -16, 5, 0, -2, 24, -3, -8, 1},

{7, -2, -30, 8, -15, 0, 35, -4, -9, 1},

{0, 30, -7, -40, 10, -42, 3, 48, -5, -10, 1},

{-9, 2, 75, -15, -35, 10, -84, 7, 63, -6, -11, 1}

Clear[B, x, n] B[x, 0] = 1; B[x, 1] = x; B[x, 2] = 2 + x^2; B[x, 3] = x + x^3; B[x, 4] = -2 + x^4; B[x_, n_] := B[x, n] = x*B[x, n - 1] - B[x, n - 2] P[x_, n_] := D[B[x, n + 1], x] (* Derived From differential recursion : B''[x, n] = x*B'[x, n] - B[x, n] which Implies integral the Boubaker Polynomial IB[x, n] = x*B[x, n] - B'[x, n] to an Integration conatant*) Table[ExpandAll[x*B[x, n] - P[x, n]], {n, 0, 10}]; a = Join[{{1}}, Table[CoefficientList[x*B[x, n] - P[x, n], x], {n, 0, 10}]]; Flatten[a] Table[Apply[Plus, CoefficientList[x*B[x, n] - P[x, n], x]], {n, 0, 10}]; Table[Table[Integrate[Sqrt[1/(1 - x^2)]*(x*B[x, n] - P[x, n])*(x*B[x, m] - P[x, m]), {x, -1, 1}], {n, 0, 10}], {m, 0, 10}];

Cf. A138034.

Adjacent sequences: A136253 A136254 A136255 this_sequence A136257 A136258 A136259

Sequence in context: A128976 A046772 A114551 this_sequence A144790 A090996 A089309

uned,tabl,sign

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 18 2008