A268533 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}.

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

Offset: 0

Author

Bradley Klee, Feb 22 2016

Keywords

• sign

Comments

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.

Examples

			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 }.
		

References

• L. Allen, S. M. Barnett, and M. J. Padgett, Optical angular momentum, Institute of Physics Publishing, Bristol, 2003.

Links

• L. Allen et al., Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes, Physical Review A, 45 (1992), 8185-8190.
• William G. Harter, Principles of Symmetry, Dynamics, Spectroscopy, Wiley, 1993, Ch. 5, page 345-348.
• Brad Klee, Quantum Angular Momentum Matrices, Wolfram Demonstrations Project, 2016.
• Mohamed Sabba, A quantum Pascal pyramid and an extended de Moivre-Laplace theorem, arXiv:2404.03560 [quant-ph], 2024. See pp. 1-2.

Crossrefs

Cf. A007318, A214292, A269301, A269302.

Programs

Mathematica

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]

Formula

T(n,k,m) = (1/(m!)) * (d/dx)^m((1-x)^k*(1+x)^(n-k))|_{x=0}.