Duality between Bessel Functions and Chebyshev Polynomials in Expansions of Functions

In expansions of arbitrary functions in Bessel functions or Spherical Bessel functions, a dual partner set of polynomials play a role. For the Bessel functions, these are the Chebyshev polynomials of first kind and for the Spherical Bessel functions the Legendre polynomials. These two sets of functions appear in many formulas of the expansion and in the completeness and (bi)-orthogonality relations. The analogy to expansions of functions in Taylor series and in moment series and to expansions in Hermite functions is elaborated. Besides other special expansion, we find the expansion of Bessel functions in Spherical Bessel functions and their inversion and of Chebyshev polynomials of first kind in Legendre polynomials and their inversion. For the operators which generate the Spherical Bessel functions from a basic Spherical Bessel function, the normally ordered (or disentangled) form is found.

Eigenfunctions for a Quantum Wire on a Single Electron at Its Surface and in the Quantum Well with Beaded Fractional Quantized States for the Fractional Charges

We developed energy profiles for the fractional quantized states both on the surface of electron due to overwhelming centrifugal potentials and inside the electron at different locations of the quantum well due to overwhelming attractive electrodynamic potentials. The charge as a physical constant and single entity is taken as density and segments on their respective sub-quanta (floats on sub quanta) and hence the fractional charge quantiz at in. There is an integrated oscillatory effect which ties all fractional quantized states both on the surface and in the interior of the volume of an electron. The eigenfunctions, i.e., the energy profiles for the electron show the shape of a string or a quantum wire in which fractional quantized states are beaded. We followed an entirely different approach and indeed thesis to reproducing the eigenfunctions for the fractional quantized states for a single electron. We produced very fascinating mathematical formulas for all such cases by using Hermite and Laguerre polynomials, spherical based and Neumann functions and indeed asymptotic behavior of Bessel and Neumann functions. Our quantization theory is dealt in the momentum space.