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A268533
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Pascal's difference pyramid read first by blocks and then by rows: T(n,k,m) = 1/(m!) * (d/dx)^m((1-x)^k*(1+x)^(n-k))|_{x=0}.
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2
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1, 1, 1, 1, -1, 1, 2, 1, 1, 0, -1, 1, -2, 1, 1, 3, 3, 1, 1, 1, -1, -1, 1, -1, -1, 1, 1, -3, 3, -1, 1, 4, 6, 4, 1, 1, 2, 0, -2, -1, 1, 0, -2, 0, 1, 1, -2, 0, 2, -1, 1, -4, 6, -4, 1, 1, 5, 10, 10, 5, 1, 1, 3, 2, -2, -3, -1, 1, 1, -2, -2, 1, 1, 1, -1, -2, 2, 1, -1, 1, -3, 2, 2, -3, 1, 1, -5, 10, -10, 5, -1
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OFFSET
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0,7
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COMMENTS
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T(n,k,m) is a pyramidal stack of (n+1) X (n+1)-dimensional matrices, or an infinite-dimensional matrix in block-diagonal form (see examples).
Define triangular slices T_x(i,j) = T(2x+i,x,x+j) with i in {0,1,...} and j in {0,1,... i}. T_0 is Pascal's triangle, and it appears that T_{x} is a triangle of first differences T_{x}(i,j) = T_{x-1}(i+1,j+1)-T_{x-1}(i+1,j) (cf. A007318, A214292).
The so-called "quantum Pascal's pyramid", denoted QT(n,k,m), is obtained from Pascal's pyramid by a complexification of matrix elements: QT(n,k,m) = (-1)^(3m/2) T(n,k,m). QT(n,k,m) effects a Hermite-Cartesian (cf. A066325) to Laguerre-polar change of coordinates (see examples).
Row reversal is complex conjugation: QT(n,n-k,m) = QT(n,k,m)*.
To construct the "normalized quantum Pascal's pyramid", NQT(n,k,m), we need normalization numerators, NumT(n,k,m) as in A269301, and denominators, DenT(n,k,m) as in A269302; then, NQT(n,k,m) = sqrt(NumT(n,k,m) / DenT(n,k,m)) QT(n,k,m). In the context of physics NQT(n,k,m) acting as matrix conjugation effects a cyclic permutation of the infinite-dimensional generators of rotation, so NQT(n,k,m) is essentially equivalent to an infinite-dimensional rotation with (z,y,z) Euler angles (0,Pi/2,Pi/2) (Harter, Klee, see examples).
Normalization or no, Pascal's pyramid also arises in laser optics (Allen et al.) as the paraxial wave equation often admits a useful analogy to the Schrödinger equation for the two-dimensional isotropic quantum harmonic oscillator.
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REFERENCES
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L. Allen, S. M. Barnett, and M. J. Padgett, Optical angular momentum, Institute of Physics Publishing, Bristol, 2003.
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LINKS
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FORMULA
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T(n,k,m) = (1/(m!)) * (d/dx)^m((1-x)^k*(1+x)^(n-k))|_{x=0}.
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EXAMPLE
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First few blocks:
1
. 1, 1
. 1, -1
. . . . . 1, 2, 1
. . . . . 1, 0, -1
. . . . . 1, -2, 1
. . . . . . . . . . . 1, 3, 3, 1
Second triangle . . . 1, 1, -1, -1
slice, T_1: . . . . . 1, -1, -1, 1
0 . . . . . . . . . . 1, -3, 3, -1
1 -1 . . . . . . . . . . . . . . . . 1, 4, 6, 4, 1
2 0 -2 . . . . . . . . . . . . . . 1, 2, 0, -2, -1
3, 2, -2, -3 . . . . . . . . . . . . 1, 0, -2, 0, 1
4, 5, 0, -5, -4 . . . . . . . . . . 1, -2, 0, 2, -1
5, 9, 5, -5, -9, -5 . . . . . . . . 1, -4, 6, -4, 1
n=2 Cartesian/Polar coordinate change using quantum Pascal's pyramid:
| 1 -2 i -1 | | y^2 - 1 | | - (r exp[ I \phi])^2 |
| 1 0 1 | * | x*y | = | r^2 - 2 |
| 1 2 i -1 | | x^2 - 1 | | - (r exp[-I \phi])^2 |
When: x = r cos[\phi], y= r sin[\phi].
Permutation of Pauli Matrices, \sigma_i, using normalized quantum Pascal's pyramid:
| 1 -i |
R = (1/sqrt[2]) * | 1 i |
Then, R * \sigma_j * R^{\dagger} = \sigma_{pi(j)},
where pi(j) is a cyclic permutation: { 1 -> 2, 2 -> 3, 3 -> 1 }.
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MATHEMATICA
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PascalsPyramid[Block_] := Outer[Simplify[Function[{n, k, m}, 1/(m!)(D[(1 - x)^k*(1 + x)^(n - k), {x, m}] /. x -> 0)][Block, #1, #2]] &, Range[0, Block], Range[0, Block]]; PascalsPyramid /@ Range[0, 10]
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CROSSREFS
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KEYWORD
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sign,changed
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AUTHOR
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STATUS
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approved
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