1. Ultraspherical Harmonics

The start and motivation for this mathematical work was when I was unlucky in finding, in the very limited share of mathematical litterature I had access to, any notion of orthogonal functions of higher than three dimensions. The well known Spherical Harmonics give a set of orthogonal functions in the three-dimensional case. But I could not find any description of the corresponding functions in four or higher dimensions. A few years later I in deed did found them, these functions are known as the Gegenbauer or Ultraspherical polynomials. But by then I already had worked through my own approach to the problem, which I give here in hope that it might catch some interest.

I was trying to solve the problem in four dimensions, and it seemed natural to start from the known case of three dimensions: the Spherical Harmonics. But I soon realized that the solution could be made more general, the problem could be solved recoursively in any number of dimensions. These 'papers' describe the recoursive method. In effect it leads to a generalization of the Spherical Harmonics, and Legendre and Gegenbauer polynomials.

2. Hypercomplex Numbers

"Hypercomplex numbers", "hypernumbers", and "quaternions" (discovered by William Hamilton in 1830's) are historical prequisities to vector analysis. Also there is a deep mathematical connection between these, the linear algebra of matrices, and the group theory. A three-dimensional application of such an objects is the rotation operator, which often is expressed in the form of a vector (a pseudovector) of complex-element matrices (called pseudo-scalars or pseudo-numbers). Mathematical objects that obey certain symmetrical multiplication laws are called a group, and rotation operators form a group. This rotation group of pseudo-number matrices has the same "multiplication table" than the hypercomplex numbers, and so represents similar objects in matrix form, a matrix representation, for them.

In stead of the customary complex-element matrix representation, I'd like to introduce here a real-element square matrix representation for the rotation operator group. Why real-element? Because these real-element matrix hypercomplex numbers actually can act as solutions to equations that traditionally call for complex numbers. And as a consequence, the complex numbers themselves too can be given a real-element square matrix representation. The imaginary unit i thus can be represented by a certain real-element square matrix, which in turn represents a member of the rotation group. (Which gives a natural explaination to the peculiar fact that complex multiplications behave like rotations.)

3. Eliminating Physical Units

What is the mathematical meaning of physical units? How are they connected to each other? If length can be expressed in time units, as it is in the modern (1996) definition of length units, using velocity of light as conversion factor, can there be other universal constants that make it possible to express mass in time units, etc.? Can all physical units at end be eliminated and physical expressions be given on an absolute mathematical basis? The answer is found right here. (This is a short article and simple math.)