Properties and possibilities of quantum shapelets

Quantum shapelets arise as the solution of a d-dimensional harmonic oscillator or D-dimensional Coulomb problem and may be obtained by requiring scale-space invariance. These functions have application to image processing in conventional or quantum contexts. We recall the scale-space-based derivation of shapelets and present novel properties of these functions, including integral relations, infinite series and finite convolution sums. Many of these relations also have application to the combinatorics of zero-dimensional quantum field theory