A136262 Integral form of A137286: Triangle of coefficients of Integral form of recursive orthogonal Hermite polynomials given in Hochstadt's book: n*IP(x, n) = x*P(x, n ) - n*P'(x, n - 2); derived to a constant from the differential recursion: P''(x,n)=x*P'(x,n)-n*P(x,n).
1, -1, 1, 0, -2, 1, 5, -2, -3, 1, 0, 18, -5, -4, 1, -33, 8, 42, -9, -5, 1, 0, -174, 33, 80, -14, -6, 1, 279, -48, -555, 87, 135, -20, -7, 1, 0, 1950, -279, -1380, 185, 210, -27, -8, 1, -2895, 384, 7920, -975, -2940, 345, 308, -35, -9, 1, 0, -25290, 2895, 24360, -2640, -5628, 588, 432, -44, -10, 1, 35685, -3840, -125055
Offset: 1
Examples
{1},
{-1, 1},
{0, -2, 1},
{5, -2, -3, 1},
{0, 18, -5, -4, 1},
{-33, 8, 42, -9, -5, 1},
{0, -174, 33,80, -14, -6, 1},
{279, -48, -555, 87, 135, -20, -7, 1},
{0, 1950, -279, -1380, 185, 210, -27, -8, 1},
{-2895,384, 7920, -975, -2940, 345, 308, -35, -9, 1},
{0, -25290, 2895, 24360, -2640, -5628, 588, 432, -44, -10, 1},
{35685, -3840, -125055,12645, 62790, -6090, -9954, 938, 585, -54, -11, 1}
References
- Defined :page 8 and pages 42 - 43 and page 49: Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986
Crossrefs
Cf. A137286.
Programs
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Mathematica
P[x, 0] = 1; P[x, 1] = x; P[x_, n_] := P[x, n] = x*P[x, n - 1] - n*P[x, n - 2]; DP[x_, n_] := D[P[x, n + 1], x]; Table[ExpandAll[x*P[x, n] - DP[x, n]], {n, 0, 10}]; a = Join[{{1}}, Table[CoefficientList[x*P[x, n] - DP[x, n], x], {n, 0, 10}]]; Flatten[a]
Formula
The Hermite Integral form is: IH[x,n]=(x*H[x,n]-H'[x,n])/n Which can be done as an integer form: n*IH[x,n]
Comments