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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}. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

T(n,k,m) is a pyramidal stack of (n+1)-by-(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) affects 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 affects 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.

REFERENCES

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

LINKS

Table of n, a(n) for n=0..90.

L. Allen et al., Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes, Physical Review A, 45 (1992), 8185-8190.

W. Harter, Principles of Symmetry, Dynamics, Spectroscopy, Wiley, 1993, Ch. 5, page 345-348.

B. Klee, Quantum Angular Momentum Matrices, Wolfram Demonstrations Project, 2016.

FORMULA

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

EXAMPLE

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 }.

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]

CROSSREFS

Cf. A007318, A214292, A269301, A269302.

Sequence in context: A173432 A101675 A051764 * A275849 A025906 A213369

Adjacent sequences:  A268530 A268531 A268532 * A268534 A268535 A268536

KEYWORD

sign

AUTHOR

Bradley Klee, Feb 22 2016

STATUS

approved

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Last modified January 27 02:46 EST 2020. Contains 331291 sequences. (Running on oeis4.)