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Search: id:A136247
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| A136247 |
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Linear Shabat transform on recursive Hermite polynomials: c*H(c*x+d,n)+d->H[x,n]=x*H(x,n-1)-n*H(x.n-2): c=-1;d=1. Triangle of coefficients of Shabat linear transformed recursive orthogonal Hermite polynomials. |
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3
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| 1, 1, -1, 1, -1, 1, 1, 2, 2, -1, 1, 6, -4, -3, 1, 1, -4, -20, 6, 4, -1, 1, -40, 8, 44, -8, -5, 1, 1, -12, 188, -6, -80, 10, 6, -1, 1, 308, 136, -546, -10, 130, -12, -7, 1, 1, 416, -1864, -628, 1256, 50, -196, 14, 8, -1, 1, -2664, -3640, 6696, 1984, -2506, -126, 280, -16, -9, 1
(list; table; graph; listen)
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OFFSET
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1,8
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COMMENT
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Row sums are:
{1, 0, 1, 4, 1, -14, 1, 106, 1, -944, 1}
The Shabat linear tree graph approach is not made for
Hermite, it was developed for Chebyshev polynomials.
The rational for applying it here is that there is a conformal
relationship between Hermites and Chebyshevs and their domains.
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REFERENCES
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page 8 and pages 42 - 43 : Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986
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FORMULA
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h(x, n) = -(-1 - n + (1 - x) - (1 - x)* h(x, n - 1) + n *h(x, n - 2))
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EXAMPLE
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{1},
{1, -1},
{1, -1, 1},
{1, 2, 2, -1},
{1, 6, -4, -3, 1},
{1, -4, -20,6, 4, -1},
{1, -40, 8, 44, -8, -5, 1},
{1, -12, 188, -6, -80,10, 6, -1},
{1, 308, 136, -546, -10, 130, -12, -7, 1},
{1, 416, -1864, -628, 1256, 50, -196, 14, 8, -1},
{1, -2664, -3640, 6696, 1984, -2506, -126,280, -16, -9, 1}
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MATHEMATICA
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Clear[h, a, n, x, y, c, d] (*Solve linear Shabat transform for Hermite type recursion*) Solve[c*x0 + d - x*(c*x1 + d) + n*(c*x2 + d) == 0, x0] c = -1; d = 1; Solve[y = c*x + d == 0, x] h[x, 0] = 1; h[x, 1] = 1 - x; h[x_, n_] := h[x, n] = -(-1 - n + (1 - x) - (1 - x)* h[ x, n - 1] + n *h[x, n - 2]); Table[ExpandAll[h[x, n]], {n, 0, 10}]; a = Table[CoefficientList[h[x, n], x], {n, 0, 10}]; Flatten[a] Table[Apply[Plus, CoefficientList[h[x, n], x]], {n, 0, 10}];
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CROSSREFS
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Cf. A137286.
Adjacent sequences: A136244 A136245 A136246 this_sequence A136248 A136249 A136250
Sequence in context: A092450 A014291 A136587 this_sequence A086610 A141760 A114626
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KEYWORD
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uned,tabl,sign
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 17 2008
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